APPLICATIONS OF DYNAMICS
TO
PHYSICS AND CHEMISTRY
BY
J. J. THOMSON, M.A., F.R.S.
FELLOW OF TRINITY COLLEGE AND CAVENDISH PROFESSOR
OF EXPERIMENTAL PHYSICS, CAMBRIDGE.
(j UNJVEKSITY
MACMILLAN AND CO.
AND NEW YORK.
1888
[All Rights reserved.]
PRINTED BY C. J. CLAY, M.A. & SONS,
AT THE UNIVERSITY PRESS.
3
J
PREFACE.
THE following pages contain the substance of a
course of lectures delivered at the Cavendish Labora-
tory in the Michaelmas Term of 1886.
Some of the results have already been published
in the Philosophical Transactions of the Royal
Society for 1886 and 1887, but as they relate to
phenomena which belong to the borderland between
two departments of Physics, and which are generally
either entirely neglected or but briefly noticed in
treatises upon either, I have thought that it might
perhaps be of service to students of Physics to
publish them in a more complete form. I have
included in the book an account of some investiga-
tions published after the delivery of the lectures
which illustrate the methods described therein.
There are two modes of establishing the connexion
between two physical phenomena ; the most obvious
as well as the most interesting of these is to start
with trustworthy theories of the phenomena in ques-
tion and to trace every step of the connexion between
them. This however is only possible in an exceed-
ingly limited number of cases, and we are in general
compelled to have recourse to the other mode in
VI PREFACE.
which by methods which do not require a detailed
knowledge of the mechanism required to produce the
phenomena, we show that whatever their explanation
may be, they must be related to each other in such
a way that the existence of the one involves that
of the other.
It is the object of this book to develop methods
of applying general dynamical principles for this
purpose.
The methods I have adopted (of which that used
in the first part of the book was suggested by
Maxwell's paper on the Electromagnetic Field) make
everything depend upon the properties of a single
function of quantities fixing the state of the system, a
result analogous to that enunciated by M. Massieu
and Prof. Willard Gibbs for thermodynamic pheno-
mena and applied by the latter in his celebrated paper
on the "Equilibrium of Heterogeneous Substances"
to the solution of a large number of problems in
thermodynamics.
I wish in conclusion to thank my friend Mr L. R.
Wilberforce, M.A., of Trinity College, for his kindness
in correcting the proofs and for the many valuable
suggestions he has made while the book was passing
through the press.
J. J. THOMSON.
TRINITY COLLEGE, CAMBRIDGE,
May 2nd, 1888.
CONTENTS.
CHAPTER I.
PAGE
Preliminary Considerations ........ i
CHAPTER II.
The Dynamical Methods to be employed ..... 8
CHAPTER III.
Application of these Principles to Physics 16
CHAPTER IV.
Discussion of the terms in the Lagrangian Function . . -31
CHAPTER V.
Reciprocal Relations between Physical Forces when the Systems
exerting them are in a Steady State 80
CHAPTER VI.
Effect of Temperature upon the Properties of Bodies ... 89
CHAPTER VII.
Electromotive Forces due to Differences of Temperature . . 106
CHAPTER VIII.
On " Residual" Effects .... .128
Vlll CONTENTS.
CHAPTER IX.
PAGE
Introductory to the Study of Reversible Scalar Phenomena . .140
CHAPTER X.
Calculation of the mean Lagrangian Function . . . -151
CHAPTER XL
Evaporation 158
CHAPTER XII.
Properties of Dilute Solutions 1 79
CHAPTER XIII.
Dissociation 193
CHAPTER XIV.
General Case of Chemical Equilibrium . . . . .215
CHAPTER XV.
Effects produced by Alterations in the Physical Conditions on the
Coefficient of Chemical Combination 233
CHAPTER XVI.
Change of State from Solid to Liquid 243
CHAPTER XVII.
The Connexion between Electromotive P'orce and Chemical
Change 265
CHAPTER XVIII.
Irreversible Effects 281
APPLICATIONS OF DYNAMICS TO
PHYSICS AND CHEMISTRY.
CHAPTER I.
PRELIMINARY CONSIDERATIONS.
i. IF we consider the principal advances made in the
Physical Sciences during the last fifty years, such as the
extension of the principle of the Conservation of Energy
from Mechanics to Physics, the development of the Kinetic
Theory of Gases, the discovery of the Induction of Electric
Currents, we shall find that one of their most conspicuous
effects has been to intensify the belief that all physical
phenomena can be explained by dynamical principles and
to stimulate the search for such explanations.
This belief which is the axiom on which all Modern
Physics is founded has been held ever since men first began
to reason and speculate about natural phenomena, but, with
the remarkable exceptions of its successful application in
the Corpuscular and Undulatory Theories of Light, it
remained unfruitful until the researches of Davy, Rumford,
Joule, Mayer and others showed that the kinetic energy
possessed by bodies in visible motion can be very readily
converted into heat. Joule moreover proved that whenever
this is done the relation between the quantity of kinetic
T. D. I
2 DYNAMICS.
energy which disappears and the quantity of heat which
appears in consequence is invariable.
The ready conversion of kinetic energy into heat con-
vinced these philosophers that heat itself is kinetic
energy; and the invariable relation between the quantities
of heat produced and of kinetic energy lost, showed that
the principle of the Conservation of Energy, or of Vis- Viva
as it was then called, holds in the transformation of heat into
kinetic energy and vice versa.
This discovery soon called attention to the fact that
other kinds of energy besides heat and kinetic energy
can be very readily converted from one form into another,
and this irresistibly suggested the conclusion that the various
kinds of energy with which we have to deal in Physics, such
for example as heat and electric currents, are really forms of
kinetic energy though the moving bodies which are the
seat of this energy must be indefinitely small in comparison
with the moving pieces of any machine with which we are
acquainted.
These conceptions were developed by several mathema-
ticians but especially by v. Helmholtz, who, in his treatise
Ueber die Erhaltung der Kraft, Berlin, 1847, applied the
dynamical method of the Conservation of Energy to the
various branches of physics and showed that by this prin-
ciple many well-known phenomena are connected with
each other in such a way that the existence of the one
involves that of the other.
2. The case which from its practical importance at first
attracted the most attention was that of the transformation
of heat into other forms of energy and vice versa.
In this case it was soon seen that the principle of the
Conservation of Energy the First Law of Thermodynamics
as it was called was not sufficient to obtain all the relations
PRELIMINARY CONSIDERATIONS. 3
existing between the effects of heat on the various pro-
perties of a body and the heat produced or absorbed
when certain changes take place in the body, but that
these relations could be deduced by the help of another
principle, the Second Law of Thermodynamics which states
that if to a system where all the actions are perfectly rever-
sible a quantity of heat dQ be communicated at the absolute
temperature 0, then
r '!-
the integration being extended over any complete cycle of
operations.
This statement is founded on various axioms by different
physicists, thus for example Clausius bases it upon the
" axiom" that heat cannot of itself pass from one body to
another at a higher temperature, and Sir William Thomson
on the " axiom " that it is impossible by means of inanimate
material agency to derive mechanical effect from any por-
tion of matter by cooling it below the temperature of the
coldest of the surrounding objects.
Thus the Second Law of Thermodynamics is derived
from experience and is not a purely dynamical principle.
We might have expected a priori from dynamical con-
siderations that the principle of the Conservation of Energy
would not be sufficient by itself to enable us to deduce all
the relations which exist between the various properties of
bodies. For this principle is rather a dynamical result than
a dynamical method and in general is not sufficient by
itself to solve completely any dynamical problem.
Thus we could not expect that for the dynamical treat-
ment of Physics the principle of the Conservation of Energy
would be sufficient by itself, since it is not so in the much
simpler cases which occur in ordinary Mechanics.
4 DYNAMICS.
The resources of dynamics however are not exhausted
even though the principle of the Conservation of Energy has
been tried. Fortunately we possess other methods, such as
Hamilton's principle of Varying Action and the method of
Lagrange's Equations, which hardly require a more detailed
knowledge of the structure of the system to which they
are applied than the Conservation of Energy itself and yet
are capable of completely determining the motion of the
system.
3. The object of the following pages is to endeavour
to see what results can be deduced by the aid of these
purely dynamical principles without using the Second Law
of Thermodynamics.
The advantages of this method in comparison with that
of the two laws of Thermodynamics are
(1) that it is a dynamical method, and so of a much more
fundamental character than that involving the use of the
Second Law ;
(2) that one principle is sufficient instead of two ;
(3) that the method can be applied to questions in which
there are no transformations of other forms of energy from
or into heat (except the unavoidable ones due to friction),
while for this case the other method degenerates into the
principle of the Conservation of Energy, which is often not
sufficient to solve the problem.
The disadvantages of the method on the other hand are
that, since the method is a dynamical one, the results are
expressed in terms of dynamical quantities, such as energy,
momentum, or velocity, and so require further knowledge
before we can translate them in terms of the physical
quantities we wish to measure, such as intensity of a
current, temperature, and so on : a knowledge which in all
cases we do not possess.
PRELIMINARY CONSIDERATIONS. 5
The Second Law of Thermodynamics, on the other hand,
being based on experience does not involve any quantity
which cannot be measured in the Physical Laboratory.
For this reason there are some cases where the Second
Law of Thermodynamics leads to more definite results than
the dynamical methods of Hamilton or Lagrange. Even
here I venture to think the results of the application of the
dynamical method will be found interesting, as they show
what part of these problems can be solved by dynamics, and
what has to be done by considerations which are the results
of experience.
4. Many attempts have been made to show that the
Second Law of Thermodynamics is a consequence of the
principle of Least Action ; none of these proofs seem quite
satisfactory ; but even if the connexion had been proved in
an unexceptionable way it would still seem desirable to
investigate the results of applying the principle of Least
Action, or the equivalent one of Lagrange's Equations,
directly to various physical problems.
If these results agree with those obtained by the use of
the Second Law of Thermodynamics, it will be a kind of
practical proof of the connexion between this law and the
principle of Least Action.
5. Considering our almost complete ignorance of the
structure of the bodies which form most of the dynamical
systems with which we have to deal in physics, it might
seem a somewhat unpromising undertaking to attempt to
apply dynamics to such systems. But we must remember
that the object of this application is not to discover the
properties of such systems in an altogether a priori fashion,
but rather to predict their behaviour under certain circum-
stances after having observed it under others.
A dynamical example may illustrate what the application
DYNAMICS.
of dynamics to physical problems may be expected to do,
and the way in which it is likely to do it. Let us suppose
that we have a number of pointers on a dial, and that
behind the dial the various pointers are connected by a
quantity of mechanism of the nature of which we are
entirely ignorant. Then if we move one of the pointers, A
say, it may happen that we set another one, B, in motion.
If now we observe how the velocity and position of B
depend on the velocity and position of A, we can by the aid
of dynamics foretell the motion of A when the velocity and
position of B are assigned, and we can do this even though
we are ignorant of the nature of the mechanism connecting
the two pointers. Or again we may find that the motion of
B when A is assigned depends to some extent upon the
velocity and position of a third pointer C : if in this case we
observe the effect of the motion of C upon that of A and B
we may deduce by dynamics the way in which the motion
of C will be affected by the velocities and positions of the
pointers A and B.
This illustrates the way in which dynamical considera-
tions may enable us to connect phenomena in* different
branches of physics. For the observation of the motion of
B when that of A is assigned may be taken to represent the
experimental investigation of some phenomenon in Physics,
while the deduction by dynamics of the motion of A when
that of B is assigned may represent the prediction by the
use of Hamilton's or Lagrange's principle of a new phenome-
non which is a consequence of the one investigated experi-
mentally.
Thus to take an illustration, suppose we investigate
experimentally the effect of a current of electricity both
steady and variable upon the torsion of a longitudinally
magnetized iron wire along which the current flows, then we
PRELIMINARY CONSIDERATIONS. 7
can deduce by dynamics the effects of torsion and variations
of torsion in the wire upon a current flowing along it.
The method is really equivalent to an extension and
generalization of the principle of the equality of action and
reaction, as when we have two bodies A and B acting upon
each other if we observe the motion of B which results when
A moves in a known way we can deduce by the aid of this
principle the motion of A when that of B is known. The
more general case which we have to consider in Physics is
when instead of two bodies attracting each other we have
two phenomena which mutually influence each other.
CHAPTER II.
THE DYNAMICAL METHODS TO BE EMPLOYED.
6. As we do not know the nature of the mechanism of
the physical systems whose action we wish to investigate, all
that we can expect to get by the application of dynamical
principles will be relations between various properties of
bodies. And to get these we can only use dynamical
methods which do not require an intimate knowledge of the
system to which they are applied.
The methods introduced by Hamilton and Lagrange
possess this advantage and, as they each make the behaviour
of the system depend upon the properties of a single func-
tion, they reduce the subject to the determination of this
function. In general the way that we are able to connect
various physical phenomena is by seeing from the behaviour
of the system under certain circumstances that there must be
a term of a definite kind in this function, the existence
of this term will then often by the application of Lagrangian
or Hamiltonian methods point to other phenomena besides
the one that led to its detection.
7. We shall now for convenience of reference collect the
dynamical equations which we shall most frequently have to
use.
The most generally useful method is Hamilton's principle
DYNAMICAL METHODS. 9
of Varying Action according to which (see Routh's Advanced
Rigid Dynamics, p. 245)
where T and V are respectively the kinetic and potential
energies of the system, / the time, and q a coordinate of
any type. In this case / and /j are each supposed to be
constant.
In some cases it is convenient to use the equation in this
form but in others it is more convenient to use Lagrange's
Equations, which may be derived from equation (i) (Routh's
Advanced Rigid Dynamics, p. 249) and which may be
written in the form
d dL dL - , x
dtTq-^q-^ (2) "
where L is written for T- V and is called the Lagrangian
function and Q is the external force acting on the system
tending to increase q.
In the preceding equations the kinetic energy is sup-
posed to be expressed in terms of the velocities of the
coordinates. In many cases however instead of working
with the velocities corresponding to all the coordinates it is
more convenient to work with the velocities corresponding
to some coordinates but with the momenta corresponding to
the others. This is especially convenient when some of
the coordinates only enter the Lagrangian function through
their differential coefficients and do not themselves occur
explicitly in this function. In a paper " On some Applica-
tions of Dynamical Principles to Physical Phenomena"
(Phil. Trans. 1885, Part n.) I have called these "kinos-
thenic " coordinates. In the following pages the term
"speed coordinates" will for the sake of brevity be used
instead wherever it will not lead to ambiguity.
10 DYNAMICS.
The most important property of such a coordinate is
that whenever no external force of its type acts upon the
system, the momentum corresponding to it is constant.
For if x be a speed coordinate, since
dL
* = '
we have by Lagrange's equation since no external force
acts on the system
d dL
as the momentum corresponding to x is dLldy^ this equa-
tion shows that it is constant.
8. Routh {Stability of Motion, p. 61) has given a general
method which enables us to use the velocities of some
coordinates and the momenta corresponding to the remain-
der, and which is applicable whether these latter coordinates
are speed coordinates or not.
The method is as follows : suppose that we wish to use
the velocities of the coordinates q^ ^ 2 ...and the momenta
corresponding to the coordinates </> n </> 2 ...then Routh has
shown that if we use instead of L the new function L given
by the equation.
and eliminate <j> 19 < 2 ...by means of the equations
dT dT
* i '';zr'*-;/r '-**
#<, d<p 2
then as far as the coordinates q^ q 2 , are concerned we may
use Lagrange's equations if we substitute L' for L. Thus we
have a series of equations of the type
-.
dt dq dq
DYNAMICAL METHODS. I I
If we call
the part of the kinetic energy corresponding to the coordi-
nate <j then we see by (4) that L = the kinetic energy of
the system minus its potential energy minus twice the kinetic
energy corresponding to the coordinates whose velocities are
eliminated.
9. If we do not know the structure of this system all
that we can determine by observing its behaviour will be
the Lagrangian function or its modified form, and since
this function completely determines the motion of the
system it is all we require for the investigation of its
properties. We see however that when we calculate the
'energy" corresponding to any physical condition the in-
terpretation may be ambiguous if the energy is not entirely
potential. For what we really calculate is the Lagrangian
function or its modified form and this is the kinetic energy
minus the potential energy minus twice the kinetic energy
corresponding to the coordinates whose velocities are elimi-
nated. So that the term in the energy which we have cal-
culated may be any one of these three things. Thus to
take an example, it is said that the energy of a piece of
soft iron of unit volume, throughout which the intensity
of magnetization is uniform and equal to /, is - / 2 /2/,
where k is the coefficient of magnetic induction of the
iron, but all that this means is that the term I*j2k occurs
in the Lagrangian function (modified or otherwise) of
the system whose motion or configuration produces the
phenomenon of magnetization. And without further con-
siderations we do not know whether this represents an
amount of kinetic energy I* 1 2k or potential energy - / 2 /2/,
or some kinetic energy corresponding to coordinates whose
12 DYNAMICS.
velocities have been eliminated, or some combination of all
three of these.
10. This ambiguity however does not occur if we have
the system completely mapped out by coordinates, because
in this case whenever we find a term in the Lagrangian
function it must be expressed in terms of these coordinates
and their velocities, or it may be the momenta corresponding
to them, and we can tell by inspection whether the term
expresses kinetic or potential energy. Two investigations in
the second volume of Maxwell's Electricity and Magnetism
afford a good illustration of the way in which this ambiguity
is cleared away by an increase in the precision of our ideas
about the configuration of the system. In the early part of
the volume by considering the mechanical forces between two
circuits carrying electric currents, it is shown that two such
circuits conveying currents /, / possess a quantity of potential
energy Mij where J/"is a quantity depending on the shape
and size of the two circuits and their relative position. Later
on however when coordinates capable of fixing the electrical
configuration of the system have been introduced it is shown
that the system instead of possessing Mij units of potential
energy really possesses + Mij units of kinetic.
11. The following considerations may be useful as
helping to show that this ambiguity is largely verbal and is
probably mainly due to our ignorance of what potential
energy really is.
Suppose that we have a system fixed by n coordinates,
q^ q 2 ,...q n of the ordinary kind, that is, coordinates which
occur explicitly in the expressions for the kinetic or potential
energies, and which we shall call positional coordinates,
and m kinosthenic or speed coordinates </> l5 < 2 ,...< w . Let
us further suppose that there are no terms in the expression
for the kinetic energy which involve the product of the
DYNAMICAL METHODS. 13
velocity of a q and a </> coordinate and that the system has
no potential energy.
Then by Routh's method we can use Lagrange's equation
for the q coordinates if instead of the ordinary Lagrangian
function L which reduces in this case to the kinetic energy
we use the modified function L' given by the equation
L' = i-^~.. ....................... (6),
d(f>
or
and where <j> lt < 2 are to be eliminated by the aid of the
equations
Thus since the expression for L does not contain any
terms involving the product of the velocity of a # and a tf>
coordinate, L will be of the form
T( 99 ) ~ ^W)
where T {qq} is the kinetic energy arising from the motion of
the q or positional coordinates, T^ that arising from the
motion of the kinosthenic or speed coordinates.
By Routh's modification of the Lagrangian equations
we have
but
so that equation (8) reduces to
d_ dT (qq} _ dT (99} = _ dTw / v
dt dq, ' dq, ', dq,
14 DYNAMICS.
If the system fixed by the positional coordinates q
had possessed a quantity of potential energy equal to F, the
equations of motion would have been of the type,
*
_ ._<w) !?!___ . , (IQ).
dt dq^ dq x dq l
By comparing equations (9) and (10) we see that the
system fixed by the positional coordinates q will behave
exactly like a system whose kinetic energy is T (qq] and whose
potential energy is T($$) .
Thus we may look on the potential energy of any system
as kinetic energy arising from the motion of systems con-
nected with the original system the configurations of these
systems being capable of being fixed by kinosthenic or speed
coordinates.
Thus from this point of view all energy is kinetic, and all
terms in the Lagrangian function express kinetic energy, the
only thing doubtful being whether the kinetic energy is due
to the motion of ignored or positional coordinates; this
can however be determined at once by inspection.
12. Some of the theorems in dynamics become very
much simpler from this point of view. Let us take for
example the principle of Least Action that for the uncon-
strained motion of a system whose energy remains constant
Tdt
is a minimum from one configuration to another and apply
it to the system we have been considering in which all the
energy is kinetic but some of it is due to the motion of a
system whose configuration can be fixed by kinosthenic
coordinates.
As all the energy is kinetic its magnitude remains
constant by the principle of the Conservation of Energy,
DYNAMICAL METHODS. 15
and so the principle of Least Action takes the very simple
form, that with a given quantity of energy any material
system will by its unguided motion go along the path which
will take it from one configuration to another in the least
possible time. The material system must of course include
the kinosthenic systems whose motion produces the same
effect as the potential energy of the original system ; and
two configurations are not supposed to coincide unless the
configuration of these kinosthenic systems coincide also.
This view which regards all potential energy as really
kinetic has the advantage of keeping before us the idea
that it is one of the objects of Physical Science to explain
natural phenomena by means of the properties of matter in
motion. When we have done this we have got a complete
physical explanation of any phenomenon and any further
explanation must be rather metaphysical than physical. It
is not so however when we explain the phenomenon as due
to changes in the potential energy of the system; for potential
energy cannot be said, in the strict sense of the term, to
explain anything. It does little more than embody the
results of experiments in a form suitable for mathematical
investigations.
The matter whose motion constitutes the kinetic energy
of the kinosthenic systems, the " </>" systems, which we regard
as the potential energy of the "q" systems, may be either
that of parts of the system, or the surrounding ether, or
both ; in many cases we should expect it to be mainly the
ether.
CHAPTER III.
APPLICATION OF THESE PRINCIPLES TO PHYSICS.
13. IN our applications of Dynamics to Physics it will
be well to begin with the cases which are the most nearly
allied to those we consider in ordinary Rigid Dynamics.
Now in this subject when there is no friction all the motions
are reversible and are chiefly relations between vector quan-
tities. We shall therefore begin by considering reversible
vector effects and afterwards go on to reversible effects
involving scalar as well as vector relations ; those for
example in which a scalar quantity such as temperature is
prominently involved : lastly we shall consider irreversible
effects. Thus the order in which we shall consider the
subject will be
1. Reversible vector phenomena.
2. Reversible scalar phenomena.
3. Irreversible phenomena.
14. We shall begin by considering the relations between
the phenomena in elasticity, electricity, and magnetism and
the way in which these depend upon the motion and
configuration of the bodies which exhibit the phenomena.
These phenomena differ from some we shall consider
later on in that we have the quantities concerned in them
entirely under our control and can by applying proper
COORDINATES. I/
external forces make them take any value we please (sub-
ject of course to such limitation as the strength of the
material and the saturation of magnets may impose). The
other phenomena on the other hand depend upon a multi-
tude of coordinates over whose individual motion we have
no control though we have some over their average motion.
As the first kind of phenomena most closely resemble those
we have to do with in ordinary dynamics we shall begin
with them.
1 5. The first thing we have to do when we wish to apply
dynamical methods to investigate the motion of a system is
to choose coordinates which can fix its configuration.
We shall find it necessary to give a more general
meaning to the term "coordinate" than that which obtains
in ordinary Rigid Dynamics. There a coordinate is a
geometrical quantity helping to fix the geometrical con-
figuration of the system.
In the applications of Dynamics to Physics however,
the configurations of the systems we consider have to be fixed,
with respect to such things as distributions of electricity and
magnetism, for example, as well as geometrically, and to do
this we have in the present state of our knowledge to use
quantities which are not geometrical.
Again the coordinates which fix the configurations of the
systems in ordinary dynamics are sufficient to fix them
completely, while we may feel pretty sure that the coordi-
nates which we use to fix the configuration of the system
with respect to many of its physical properties, though they
may fix it as far as we can observe it, are not sufficient to
fix it in every detail ; that is they would not be sufficient
to fix it if we had the power of observing differences
whose fineness was comparable with that of molecular
structure.
T. D. 2
1 8 DYNAMICS.
Hydrodynamics furnishes us with many very good illus-
trations of this latter point. For example, when a sphere
moves through an incompressible fluid, we can express the
kinetic energy of the system comprising both sphere and
fluid in terms of the differential coefficients of the three
coordinates which fix the centre of the sphere, though it
would require a practically infinite number of coordinates
to fix the configuration of the fluid completely.
Now Thomson and Tait (Natural Philosophy, vol. I.
p. 320) have shown how we can "ignore" these coordinates
when the kinetic energy can be expressed without them, and
that we may treat the system as if it were fully determined
by the coordinates in terms of whose differential coefficients
the kinetic energy is expressed.
And again Larmor (Proceedings of London Mathemati-
cal Society, xv. p. 173) has proved that if Z' be Routh's
modification of the Lagrangian function, that is q^ q 2 -,--
being the coordinates retained, < < 2 ,... those ignored, (?
<2 2 .--, 3> 15 $2 the momenta corresponding to these coordinates
respectively, if
[ * L'dt=o .................. (IT).
Jt
then
t
If all the kinetic energy vanishes when the positional
coordinates q^ ,,... are constant, as is the case when a
number of solids move through a perfect fluid in which
there is no circulation, Z' is the difference between the
kinetic and potential energies of the system. If however
the kinetic energy does not vanish when the velocities of
the positional coordinates all vanish, as for example when
a number of solids are moving through a fluid in which there
is circulation, Z' no longer equals the difference between
the kinetic and potential energies of the system.
UJNIVERSITS
COORDINATES.
It follows from (i i), by the Calculus of Variations, that if
L' be expressed in terms of a series of quantities q^ q z ,...
and their first differential coefficients, then whatever these
quantities may be, we must have a series of equations of the
type
d dL' dL'
Thus we see that we may treat any variable quantities
as coordinates if the modified Lagrangian function can be
expressed in terms of them and their first differential coeffi-
cients. We shall take this as our definition of a coordinate.
1 6. When we introduce a symbol to fix a physical
quantity we may not at first sight be sure whether it is a
coordinate or the differential coefficient of one with respect
to the time.
For example, we might feel uncertain whether the symbol
representing the intensity of a current was a coordinate or
the differential coefficient of one. The simplest dynamical
considerations however will enable us to overcome this
difficulty. Thus if when there is no dissipation of energy
by irreversible processes, the quantity represented by the
symbol remains constant under the action of a constant
force tending to alter its value, the energy at the same time
remaining constant, then the symbol is a coordinate.
Again, if it remains constant and not zero when no force
acts upon it, there being no dissipation and the energy
remaining constant, the symbol represents a velocity, that is,
the differential coefficient of a coordinate with respect to
the time.
Let us apply these considerations to the example men-
tioned above ; as the intensity of a current flowing through
a perfect conductor, the only circumstances under which
there is no dissipation, does not satisfy the first of these
2 2
20 DYNAMICS.
conditions, while it does satisfy the second, we conclude
that the intensity of a current ought to be represented by
the rate of change of a coordinate and not by the coordinate
itself.
Specification of Coordinates.
17. To fix the configuration of the system so far as the
phenomena we are considering are concerned we shall use
the following kinds of coordinates.
(1) Coordinates to fix the geometrical configuration of
the system, i.e. to fix the position in space of any bodies of
finite size which may be in the system. For this purpose
we shall use the coordinates ordinarily used in Rigid Dy-
namics and denote them by the letters x lt x a , x 3 ...; and
when we want to denote a geometrical coordinate generally
without reference to any one in particular we shall use the
letter x.
(2) Coordinates to fix the configuration of the strains
in the system. We shall use for this purpose, as is ordinarily
done in treatises on elasticity, the components parallel to
the axes of x, y, z of the displacements of any small portion
of the body, and denote them by the letters a, (3, y respec-
tively. For the strains
*Y,df*\ fJ: + <ty\ f^^^\
dy + dz) ' \dz + dx) ' \dx + dy) '
we shall use the letters e, f, g, a, b, c respectively. It will
be convenient to have a letter typifying these quantities
generally without reference to any one in particular, we
shall use the letter w for this purpose.
(3) Coordinates to fix the electrical configuration of
the system. For this purpose we shall use coordinates
COORDINATES. 2 1
denoted by the letters y } , y a ..., the typical coordinate
being denoted by y, where y in a dielectric is what Maxwell
calls an electric displacement, and in a conductor the time
integral of a current flowing through some definite area.
(4) Coordinates to fix the magnetic configuration. We
might do this by specifying the intensity of magnetization
at each point, but it is clearer 1 think to regard the magnetic
configuration as depending, even in the simplest case, upon
two coordinates, one of which is a kinosthenic or speed
coordinate.
This way of looking at it brings it into harmony with the
two most usual ways of representing the magnetization of a
body, viz. Ampere's theory and the hypothesis of Molecular
Magnets.
According to Ampere's theory the magnetization is due
to electric currents flowing through perfectly conducting
circuits in the molecules of the magnets. In this case the
differential coefficient of the kinosthenic coordinate would
fix the intensity of the current, and the other coordinate the
orientation of the planes of the circuits.
According to the Molecular Magnet theory, any magnet
of finite size is built up of a large number of small magnets
arranged in a polarized way. Here the momentum corre-
sponding to the kinosthenic or speed coordinate may be
regarded as fixing the magnetic moment of a little magnet,
which it is well fitted to do by its constancy ; the other co-
ordinate may be regarded as fixing the arrangement of the
little magnets in space.
We shall denote the kinosthenic coordinate by and
the geometrical one by rj and suppose that they are so
chosen that the intensity of magnetization at any point is r?,
where is the momentum corresponding to the kinosthenic
coordinate . rj is a vector quantity and may be resolved
22 DYNAMICS.
into three components parallel to the axes of x, y, z respec-
tively.
1 8. Having chosen the coordinates there are two ways
in which we may proceed. We may either write down the
most general expression for the Lagrangian function in terms
of these coordinates and their differential coefficients, and
then investigate the physical consequences of each term
in this expression. If these consequences are contradicted
by experience we conclude that the term we are considering
does not exist in the expression for the Lagrangian function.
Or we may know as the result of experiment that there
must be a certain term in the expression for the Lagrangian
function and proceed by the application of Lagrange's Equa-
tions to develop the consequences of its existence. Thus, for
example, we know by considering the amount of work
required to establish the electric field that there must be in
the Lagrangian function of unit volume of the dielectric a
term of the form
where K is the specific inductive capacity of the dielectric
and D the resultant electric displacement. We can then by
applying Lagrange's equation to this term see what are the
consequences of the specific inductive capacity of the
dielectric being altered by strain (see 39).
We shall make use of both methods but commence with
the first as being perhaps the most instructive, and also
because we shall have a great many examples of the second
method later on since the scalar phenomena do not admit
of being treated by the first method.
19. In using the first method the first thing we have to
do is to write down the most general expression for the
Lagrangian function in terms of the coordinates #, y, 17, , w.
LAGRANGIAN FUNCTION. 23
Let us suppose that is eliminated by means of the
equation
and that we work with Routh's modification of the Lagran-
gian function.
The most general expression for the terms correspond-
ing to the kinetic part of this function, which is the only
part we can typify, is of the form
+ 2 (x^X a ) X^ + ......
2 + 2 +
+ 2 (xy) xy + 2 (xw) xw + 2 (xrj) xrj
+ 2 (xg) xg+2 (yw) yw+ 2 (yrj) yrj
+ 2 (yg) y+ 2 (wrj) wi] + 2 (o/|) a>
These terms may be divided into fifteen types.
There are five sets which are quadratic functions of the
velocity or momentum corresponding to one kind of coordi-
nate. Each of these five sets must exist in actual physical
systems if there is anything analogous to inertia in the
phenomena which the corresponding coordinates typify.
Again, there are ten sets of terms of the type
(xy) xy or (x) x,
involving the product of two velocities or a velocity and
a momentum of two coordinates of different kinds.
To determine whether any particular term of this type
exists or not we must determine what the physical conse-
quences of it would be ; if these are found to be contrary
to experience we conclude that this term does not exist.
24 DYNAMICS.
20. We can determine the consequences of the exist-
ence of a term of this kind in the expression for the kinetic
energy in the following way.
Let us suppose that we have a term in the modified
Lagrangian function of the type
where/ and q may be any of the five kinds of coordinates we
are considering.
Then we have by Routh's modification of Lagrange's
equations
idL' _dL' _
dt dp ~ ~
where P is the external force of type/ acting on the system.
Thus the effect of the term
is equivalent to the existence of a force of the type / equal
to
that is,
-{(/?) ? + |(/?)? 2 + 2|.(/ ? )^ ...... (14);
a force of type q equal to
that is,
- {(#)/ + | (/?)/+ S^(/ ? )r/} ...... (15);
and a force of type r equal to
LAGRANGIAN FUNCTION. 25
that is, pq-fc (pq) (16);
where r is a coordinate of any type other than that of
poiq.
Each of the terms in these expressions would correspond
to some physical phenomenon ; and as it is clearer to take a
definite case to illustrate this, let us suppose that p is the
geometrical coordinate symbolized by x, and q the electrical
coordinate y.
Then if the term (xy) xy occurred in the expression for
the Lagrangian function, the mechanical force produced by
a steady current would not be the same as that produced by
a variable one momentarily of the same intensity. This is
so because by the expression (14) there is the term
(xy)y
in the expression for the force of type x, that is the
mechanical force, and as y is zero if the current is steady,
there would be a mechanical force depending on the rate of
variation of the current if this term existed.
Again, we see from the term -=- (xy) y 2 in the expression
(14), remembering that p stands for x and q for y, that if
(xy) were a function of y the current would produce a
mechanical force proportional to its square, so that the force
would not be reversed if the direction of the current was
reversed.
Or again, if we consider the expression for the force of
type y or q t that is the electromotive force, we see that the
existence of this term implies the production of an electro-
motive force by a body whose velocity is changing,
depending upon the acceleration of the body ; this is shown
by the existence of the term (xy) x in (15), the expression
for the electromotive force.
26 DYNAMICS.
If (xy) were a function of x, the term (xy) x 2 in (15)
shows that a moving body would produce an electromotive
force proportional to the square of its velocity, and therefore
one that would not be reversed when the direction of motion
of the body was reversed.
As none of these effects have been observed we conclude
that this term does not exist in the expression for the
Lagrangian function of a physical system (see Maxwell,
Electricity and Magnetism, 574).
21. We shall now go through the various types of terms
which involve the product of the velocities of two coordi-
nates of different kinds, or a velocity of one kind and the
momentum of another, in order to see whether they exist or
not in the expression for the Lagrangian function.
The reasoning to be used is of the same nature as that
just given, and we may leave it to the reader to show by the
consideration of the expressions (14) and (15) that the
existence of the several terms carries with it the con-
sequences we describe.
Taking the terms in order we have
1. Terms of the form
(xy) xy.
We have just seen that terms of this kind cannot exist in
the expression for the Lagrangian function. See also Max-
well, Electricity and Magnetism, n. part iv. chap. 7.
2. Terms of the form
(xw) xw.
Terms of this form may exist in the case of a vibrating solid
body which is also moving as a whole. For the velocity of
any point in the solid equals the velocity of the centre of
gravity plus the velocity of the point relatively to its centre of
gravity. This latter velocity will involve w, so that the
LAGRANGIAN FUNCTION. 2/
square of the velocity and therefore the kinetic energy may
involve xw.
3. Terms of the form
(xrj) xri
cannot exist, for we can prove that they would involve the
existence of a magnetizing force in a moving body depend-
ing upon the acceleration of the body. It would also require
that the mechanical force exerted by a magnet should
depend upon the rate of change of the magnetization.
None of these effects have been observed.
4. Terms of the form
(*) ^
apparently do not exist, for they would require that the
mechanical force exerted by a magnet should depend upon
the rate of variation of the magnetic intensity, and this
effect has not been observed.
5. Terms of the form
(yw) yu>.
If these terms existed it would be possible to develop
electromotive forces by vibration, and these forces would
depend upon the acceleration of the vibration and not
merely upon the velocity ; as these have not been observed
we conclude that this term does not exist in the Lagrangian
function of physical systems.
6. Terms of the form
(yn) yn-
If these terms existed there would be electromotive forces
depending upon the rate of acceleration of the changes in
the magnetic field.
They also indicate magnetic forces depending upon the
rate of change of the current. As neither of these effects
28 DYNAMICS.
have been observed we conclude that terms of this form do
not exist in the expression for the Lagrangian function.
7. Terms of the form
00 #
Terms of this kind only involve the production of an
electromotive force in a varying magnetic field, the electro-
motive force varying as the rate of change of the magnetic
field. This is the well-known phenomenon of the produc-
tion of an electromotive force round a circuit whenever the
number of lines of magnetic force passing through it is
changed.
As the term we are considering is the only one in the
Lagrangian function which could give rise to an effect of
this kind without also giving rise to other effects which have
not been verified by experience, we conclude that this term
does exist.
8. Terms of the form
(wrj) wfi.
If we take any molecular theory of magnetism, such as
Ampere's, where the magnetic field depends upon the
arrangement of the molecules of the body, we should rather
expect this term to exist. The consequences of its existence
have however not been detected by experiments.
If this term existed, then considering in the first place
its effect upon the magnetic configuration we see that a
vibrating body should produce magnetic effects depending
upon the vibrations. Secondly, considering the effects of
this term on the strain configuration we see that there
should be a distorting force depending upon the rate of
acceleration of the magnetic field. As neither of these
effects have been observed there is no evidence of the
existence of this term.
LAGRANGIAN FUNCTION. 29
9. Terms of the form
(wg) w.
These would involve the existence of distorting forces
depending upon the rate of change of the magnetic field,
and we have no evidence of any such effect.
10. Terms of the form
(of) tf .
If we assume Ampere's hypothesis of molecule currents this
term is of the same nature as the term (xg) xg which we
discussed before, so that unless the properties of these
molecular circuits differ essentially from those of finite size
with which we are acquainted this term cannot exist.
21. Summing up the results of the foregoing considera-
tions, we arrive at the conclusion that the terms in the
Lagrangian function which represent the kinetic energy
depending upon the five classes of coordinates we are
considering must be of one or other of the following types :
(xx) x 2
(17).
(ww) w*
(m)^
() f
(xw)xw
22. We might make a model with five degrees of
freedom which would illustrate the connection between
these phenomena which are fixed by coordinates of five
types.
And if we arrange the model so that its configuration
being defined by the five coordinates x, y, w^ 77, , only
those terms which are in the expression (17) shall exist in
3O DYNAMICS.
the expression for its kinetic energy, and make the potential
energy of the model corresponding to each coordinate
analogous to that possessed by the physical system, then
the working of this model will illustrate the interaction of
phenomena in electricity, magnetism, elasticity &c., and any
phenomenon exhibited by the model will have its counter-
part in the phenomena exhibited in these subjects.
When however we know the expression for the energy
of such a model, there is no necessity to construct it in
order to see how it will work, as we can deduce all the rules
of working by the application of Lagrange's Equations.
And from one .point of view we may look upon the method
we are using in this book as that of forming, not a model,
but the expression for the Lagrangian function of a model
every property of which must correspond to some actual
physical phenomenon.
CHAPTER IV.
DISCUSSION OF THE TERMS IN THE LAGRANGIAN
FUNCTION.
23. WE must now proceed to examine the terms in the
expression (17) more in detail, and find what coordinates
enter into the various coefficients (xx), (yy) When we
have proved that these coefficients involve some particular
coordinates we must go on to see what the physical
consequences will be. In this way we shall be able to
obtain many relations between the phenomena in electri-
city, magnetism and elasticity.
24. The first term we have to consider is {xx} x 2 , which
corresponds to the expression for the ordinary kinetic energy
of a system of bodies. We know that {xx} may be a function
of the geometrical coordinate typified by x, but we need not
stop to consider the consequences of this as they are fully
developed in treatises on the Dynamics of a System of Rigid
Bodies.
Next {xx} may involve the electrical coordinate y, for
in a paper "On the Effects produced by the Motion of
Electrified Bodies," Phil. Mag. Apr. 1881, I have shown
that the kinetic energy of a small sphere of mass m charged
32 DYNAMICS.
with a quantity of electricity e and moving with a velocity v
15 a
where a is the radius of the sphere and /u, the magnetic
permeability of the dielectric surrounding it.
The existence in the kinetic energy of this term, which is
due to the "displacement currents" started in the surrounding
dielectric by the motion of the electrification on the sphere,
shows that electricity behaves in some respects very much
as if it had mass. For we see by the expression (18) that
the kinetic energy of an electrified sphere is the same as
if the mass of the body had been increased by 4fjLe 2 /i$a.
Thus whenever a moving body receives a charge of
electricity its velocity will be impulsively changed, for the
momentum will remain constant, and as the apparent mass
is suddenly increased the velocity must be impulsively
diminished.
The apparent increase in mass cannot exceed a very
small quantity because air or any other dielectric breaks
down when the electric force gets very intense. If we take
75 as the intensity in electrostatic measure in C.G.S. units of
the greatest electric force which a fairly thick layer of air
can stand, which is the value given by Dr Macfarlane (Phil.
Mag., Dec. 1880), we have, since the electric force at the
surface of the sphere must be less than 75,
K being the specific inductive capacity of the medium.
So that the ratio of the increase in mass to the original
mass, which by (18) is equal to
15 a
ELECTRIFIED SPHERES. 33
cannot exceed K* 1500
and since in air i / ^K= 9 x io 2
we see that the ratio cannot exceed
i . 6 x icT 18 a 3 /m,
or about 4 x io~ 19 /p,
where p is the mean density of the substance enclosed by the
electrified surface.
Thus the alteration in mass, even if the mean density
inside the surface is as small as that of air at the atmospheric
pressure and oC., is only about 5 x io~ 16 of the original
mass, and is much too small to be observed.
Let us now consider the electrical effects of this term.
Let Q be the electromotive force acting on the sphere.
The energy of the system, using the same notation as before,
is
1 2 u<A i e 2
- m + v~ H ^ .
2 15 # / 2 Ka
If v be increased by v and e by 8e, the increment in the
energy is
4 fA 2 \ 4 y^ 2 o, $
1 5 a ) 1 5 # Ka '
and by the Conservation of Energy this must equal
so that
/ 4 fu? s
[m+. -
\ 15 a
15 a ) 15 a Ka
Since no mechanical force acts upon the system the
momentum will be constant, so that
w *MS ^ / \
') H vbe=o (20).
15 a J 15 a
T. D. 3
34 DYNAMICS.
Eliminating v and Be between equations (19) and (20) we
find
So that the capacity of the sphere is increased in the
ratio of i to i / i -- '- ^Kv 2 , or since according to the
Electromagnetic Theory of Light ^K- i/F 2 where V is the
velocity of light through the dielectric, the capacity of the
sphere is increased in the ratio of i to i i ------- ^ , Thus
15 V*
the capacity of a condenser in motion will not be the same
as that of the same condenser at rest, but as the difference
depends on the square of the ratio of the velocity of the
condenser to the velocity of light it will be exceedingly
small.
If the earth does not carry the ether with it, a point on
the earth's surface will be moving relatively to the ether, and
the alterations in the velocity of such a point which occur
during the day will produce a small diurnal variation in the
capacities of condensers.
25. When we have two spheres of radius a and a
moving with velocities v and v respectively the kinetic
energy (see the paper on the " Effects produced by the
Motion of Electrified Bodies," Phil. Mag., April, 1881),
assuming Maxwell's theory, is
e cos e
Ki 2 /A A 2 / 1 , 2 /A/ S \ /2
-m+ r }v +(-m+- r -^}v 2 +
2 15 a ) \2 15 a J
where R is the distance between the centres of the two
spheres and e the angle between their directions of motion ;
m, m, e, e are respectively the masses and charges of the
spheres.
ELECTRIFIED SPHERES. 35
By expressing v 9 v and cos c in terms of the coordinates
of the centres of the spheres and their differential coefficients
with respect to the time, we can, by using Lagrange's
equations in the way explained in 20, show that these terms
require the existence of the following forces on the two
spheres, v and v being the accelerations of the spheres
respectively.
On the first sphere.
a. An attraction
u^e'
= vv cos e
3^~
along the line joining the centres of the spheres.
/?. A force
in the direction opposite to the acceleration of the second
sphere.
y. A force
i , , d
3 ^ et c ' dt
in the direction opposite to the direction of motion of the
second sphere.
There are corresponding forces on the second sphere, and
we see that unless the two spheres move with equal and
uniform velocities in the same direction the forces on the
two spheres are not equal and opposite. The sum of the
momenta of the two spheres will not increase indefinitely
however, since the sum of the actions and reactions is not
constant but is a function of the accelerations.
We may easily prove that if x, y, z are the coordinates of
the centre of one sphere, x, y', z' those of the other, then
| 4 ne* /jiee' \ dx ( , 4 />u?' 2 /A ee' \ dx'
36 DYNAMICS.
is constant, along with symmetrical expressions for the y and
z coordinates.
26. Other electrical theories besides Maxwell's lead to
the conclusion that the coefficient [xx] is a function of the
electrification of the system.
Thus according to Clausius' theory (Crelle, 82, p. 85)
the forces between two small electrified bodies in motion are
the same as if, using the same notation as before, there was
the term
, cos e ee'
in the expression for the Lagrangian function. The first of
these is the same as the term we have just been considering.
The forces which according to Weber's theory (Abhand-
limgen der Koniglich Sdchsischen Gesellschaft der Wissen-
schaften, 1846, p. 211. Maxwell's Electricity and Magnetism,
2nd Edit. vol. n. 853) exist between two electrified bodies
in motion may easily be shown to be the same as those
which would exist if in the Lagrangian function there was
the term
ee'(x-x' .. y-y , z z' .. Y ee
(U - U') (v - ,<) + __._ (W _ ,) _ _ ,
-g __._
where x, y, z, x, /, z' are the coordinates of the centres of
the electrified bodies and w, v, w, u', v', w' the components
of their velocities parallel to the axes of coordinates.
This term leads however to inadmissible results, as we
can see by taking the simple case when the bodies are moving
in the same straight line which we may take as the axis of
x. In this case the term in the kinetic energy reduces to
ee'
or -- (u* -2?tu'+ u' 2 )
K
ELECTRIFIED SPHERES. 37
so that the electrified bodies will behave as if their masses
were in consequence of the electrification increased by
zeelR since the coefficients of u* and it' 2 are each increased
by half this amount. Hence if we take e and / of opposite
signs and suppose the electrifications are great enough to
make zee'lR greater than the masses of one or both of the
bodies, then one of the bodies at least will behave as if its
mass were negative. This is so contrary to experience that
we conclude the theory cannot be right. This consequence
of Weber's theory was first pointed out by v. Helmholtz
( Wissenschaftliche Abhandlnngen, i. p. 647).
The forces which according to Riemann's theory, given
in his posthumous work Schwere, Elektricitdt und Magnetis-
mus, p. 326, exist between two moving electrified bodies may
easily be shown to be the same as those which would exist if
there were the term
in the expression for the Lagrangian function. We can
easily see that this theory is open to the same objection
as Weber's, that is, it would make an electrified body
behave in some cases as if its mass were negative.
27. If we regard the expression for the kinetic energy
from the point of view of its bearing on electrical phenomena
we shall see that it shows that if we connect the terminals
of a battery to two spheres made of conducting material, the
quantity of electricity on the spheres will depend upon their
velocities.
We see from the expression (22) for the kinetic energy of
a moving conductor that if we have a number of conductors
moving about in the electric field there will be a positive term
in the Lagrangian function depending upon the square of
the electrification. And the same is true to a smaller
38 DYNAMICS.
extent if the moving bodies are not conductors but
substances whose specific inductive capacity differs from
that of the surrounding medium. This is equivalent to a
decrease in the potential energy produced by a given
electrification, since an increase in the potential energy
corresponds to a decrease in the Lagrangian function.
Thus the presence of the moving conductors is equivalent
to a diminution in the stiffness of the dielectric with respect
to alterations in its state of electrification. And therefore
the speed with which electrical oscillations are propagated
across any medium will be diminished by the presence of
molecules moving about in it ; the diminution being pro-
portional to the square of the ratio of the velocity of the
molecules to the velocity with which light is propagated
across the medium. Thus if the electromagnetic theory of
light is true the result we have been discussing has an
important bearing on the effect of the molecules of matter
on the rate of propagation of light.
28. We can see that {xx} may be a function of the
strain coordinates, for let us take the case when {xx} is the
moment of inertia of a bar about an axis through its centre :
then it is evident if the bar be compressed in the middle
and pulled out at the ends that the moment of inertia will
be less than if the bar were unstrained, for the effect of
the strain has practically been to bring the matter forming
the bar nearer to the axis. Thus the moment of inertia
and therefore {xx} may depend upon the strain coordinates.
These coordinates will in general only enter {xx} through
the expression for the alteration in the density of the strained
body, i.e. through
da dB dy
+-+-/- (23),
dx dy dz
and this will only enter {xx} linearly.
ELECTRIC CURRENTS. 39
If we form the equations of elasticity by using Hamilton's
principle
3 (T- V) dt = o
Jta
\ve shall easily find that the presence of (23) in {xx}
leads to the introduction of the so-called "centrifugal forces"
into the equation of elasticity for a rotating elastic solid.
This however we shall leave as an exercise for the reader.
29. Let us now consider that part of the Lagrangian
function which depends upon the velocities of the electrical
coordinates, i.e. the part denoted by
Let us take the case of two conducting circuits whose
electrical configuration is fixed by the coordinates y } , y 2 ,
where j,, j> 2 are the currents flowing through the circuits
respectively.
This part of the Lagrangian function may in this case be
conveniently written
Now we can fix the geometrical configuration of the two
circuits if we have coordinates which can fix the position
of the centre of gravity and the shape and situation of the
first circuit, the shape of the second circuit and its position
relatively to the first.
Let us denote by x 2 - x l any coordinate which helps to
fix the position of one circuit relatively to the other, and by
,, 2 coordinates helping to fix the shape of the first and
second circuits respectively.
It is evident that the kinetic energy must be expressible
in terms of these coordinates, for the only coordinates neces-
sary to fix the system which we have omitted are those fixing
the centre of gravity and situation of the first circuit, and
40 DYNAMICS.
since a motion of the whole system as a rigid body through
space cannot alter this part of the kinetic energy of the
system, the expression for the kinetic energy cannot involve
these coordinates.
If we write for a moment x instead of x z - x l (a coordi-
nate helping to fix the position of one circuit relatively to
the other) then by Lagrange's Equations we see that these
terms in the kinetic energy correspond to the existence of a
force tending to increase x equal to
i dL . dM . i dN .
--X (24) -
We see from this expression that dLldx, and dNIdx
must vanish, otherwise there would be a force between the
two circuits even though the current in one of them
vanished. The quantities L and N are by definition the
coefficients of self-induction of the two circuits, and hence
we see that the coefficient of the self-induction of a circuit is
independent of the position of other circuits in its neighbour-
hood and is therefore the same as if these circuits were re-
moved 1 .
By (16) the force tending to increase x is
dM . .
that is there is a force between the two circuits proportional
to the product of the currents flowing through them, and also
to the differential coefficient with respect to the coordinate
along which the force is reckoned of a function which does
not involve the electrical coordinates. This corresponds
exactly to the mechanical forces which are actually observed
1 This is quite consistent with the apparent diminution in the self-
induction caused by a neighbouring circuit when an alternating current
is used.
ELECTRIC CURRENTS. 4!
between the circuits, and a little consideration will show that
these forces could not arise from any other terms in the
Lagrangian function. Thus the consideration of the mechan-
ical forces which two circuits carrying currents are known to
exert upon each other proves that the term My\y 2 exists in
the expression for the Lagrangian function.
Let us now go on to consider the effect of these terms
on the electrical configuration of the two circuits.
By Lagrange's Equation for the coordinate^, we have
d dL 1 dL'
where Y } is the external electromotive force tending to
increase y^. Now as we shall prove directly dL'\dy\ = o, so
that the effects on the electrical configuration of the first
circuit, arising from the term
are the same as would be produced by an external electro-
motive force tending to increase y } equal to
-~(l^ + My,) ..................... (26).
Thus if any of the four quantities Z, M, j\, y. 2 vary in
value there is an electromotive force acting round the
circuit through which the current jy, flows. And the
expression (26) gives the E. M. F. produced either by the
motion of neighbouring circuits conveying currents or by
alterations in the magnitudes of the currents flowing through
the circuits.
This example is given in Maxwell's Electricity and
Magnetism, vol. n. part iv. chapter vi., and it is one -which
illustrates the power of the dynamical method very well.
The existence of the mechanical force shows that there is
42 DYNAMICS,
the term
My* y
in the expression for the Lagrangian function and then the
law of the induction of currents follows at once by the
application of Lagrange's Equations.
The problem we have just been considering is dynamic-
ally equivalent to finding the equations of motion of a
particle with two degrees of freedom when under the action
of any forces. We know that these cannot be deduced by
the aid of the principle of the Conservation of Energy alone,
for to take the simplest case of all, that in which no forces
act upon the particle, the principle of the Conservation of
Energy is satisfied if the velocity is constant whether the
particle moves in a straight line or not. From this analogy
we see that when we have two circuits the principle of the
Conservation of Energy is not sufficient to deduce the
equations of motion, and that some other principle must be
assumed implicitly in those proofs which profess to deduce
these equations by means of the Conservation of Energy
alone.
30. There is no experimental evidence to show that
{yy} is a function of the electrical coodinates y, and it
certainly is not when the electrical systems consist of a
series of conducting circuits, for if it were the coefficients of
self and mutual induction would depend upon the length of
time the currents had been flowing through the circuits.
And in any case it would require the existence of electro-
motive forces which would not be reversed if the direction
of all the electric displacements in the field were re-
versed.
31. Similar reasoning will show that {yy} cannot be a
function of the magnetic coordinates, for if it were there
would be magnetic forces produced by electric currents
EFFECT OF STRAIN. 43
which would not be reversed if the directions of all the
currents in the field were reversed.
32. We- must now consider whether {yy} is a function
of the strain coordinates or not. If it is then the coefficients
of self and mutual induction of a number of circuits must
depend upon the state of strain of the wires forming the
circuits. This result though not impossible has never been
detected, and it is contrary to Ampere's hypothesis that the
force exerted by a current depends only upon its strength
and position and not upon the nature or state of the
material through which it flows.
Then again, if we consider what the effect on the elastic
properties of the substance would be if {yy} were a function
of the strain coordinates, we see at once that it would
indicate that the elastic properties of a wire would be
altered while an electric current was passing through it.
The evidence of various experimenters on this point is
somewhat conflicting. Both Wertheim (Ann. de Chim. et
de Phys. [3] 12, p. 610, Wiedemann's Elektricitcit, n. p. 403)
and Tomlinson have observed that the elasticity of a wire is
diminished when a current passes through it and that this
diminution is not due to the heat generated by the current.
Streintz ( Wien. Ber. [2] 67, p. 323, Wiedemann's Elektrititat,
n. p. 404) on the other hand was unable to detect any such
effect.
But even if this effect were indisputably established it
would not prove rigorously that {yy} is a function of the
strain coordinates, for as we shall endeavour to show when
we consider electrical resistance this effect might have been
due to another cause.
To sum up we see that {yy} is a function of the
geometrical coordinates but not of the electric or magnetic
ones and probably not of the strain ones.
44 DYNAMICS.
33. We shall now consider the part of the Lagrangian
function which depends upon the magnetic coordinates and
which does not involve the velocities of the geometrical,
electrical or strain coordinates. Thus the terms we are
about to consider in the Lagrangian function of unit
volume of a substance are those we have denoted by
we may have in addition to these terms arising from the
potential energy.
In order to begin with as simple a case as possible let
us suppose that all the magnetic changes take place indefi-
nitely slowly ; in this case we may neglect the term
and confine our attention to the terms
ijgtf'+jtfte
or as it is more convenient to write them
(27).
Let us take first the case when the magnetization is
parallel to one of the axes, x for example, and let us denote
the magnetic force parallel to this direction by H and the
intensity of magnetization by /, where by definition
/=irf ........................ (28).
The investigation in 389 of Maxwell's Electricity and
Magnetism shows that if we suppose that all the energy in
the magnetic field is resident in the magnets, there is in the
Lagrangian function for unit volume of a magnet the term
HI.
The result of this investigation is stated in the Electricity
MAGNETIZATION. 45
and Magnetism to be that the potential energy of unit
volume of the magnet is
-HI,
but we have seen in 9 that the question whether energy
determined in this manner is kinetic or potential is really
left unsettled : what is actually proved is that a certain term
exists in the Lagrangian function.
If we suppose that the energy is distributed throughout
the whole of the magnetic field, including unmagnetized
substances as well as magnets, then the investigation in 635
of the Electricity and Magnetism shows that the Lagrangian
function of unit volume anywhere in the magnetic field con-
tains the term
_
where B is the magnetic induction.
These two ways of regarding the energy in the magnetic
field lead to identical results ; and as we shall for the
present confine our attention to the magnetized substances
we shall find it more convenient to adopt the first method
of looking at the question.
We have seen that the Lagrangian function for unit
volume of a magnet contains the term
HI,
or in our notation
and this is the term we previously denoted by
Mflfc
Since the magnetic changes are supposed to take place
indefinitely slowly, Lagrange's equation for the >? coordinate
reduces to
46 DYNAMICS.
Applying this to the expression (27) and substituting
for i {} we get
^=o ............... (30),
and since is supposed to remain constant and therefore
&fy=*r/,
this may be written
+ H = o ............... (31).
So that if k be the coefficient of magnetic induction and
defined by the equation
I
we have by (31)
-/;
and therefore
(33).
If we know the way in which 7 varies with H we could
by this equation express A as a function of /. The relation
between / and H is however in general so complicated that
there seems but little advantage to be gained by taking some
empirical formula which connects the two and determining
A by its help.
For small values of H, Lord Rayleigh (Phil. Mag. 23,
p. 225, 1887) has shown that II H is constant, so by
equation (33) A in this case is also constant.
34. The mechanical force parallel to the axis of x
acting on unit volume of the magnet is
dIJ
MAGNETIZATION. 47
The only quantity in the terms we are considering which
involves x explicitly is J7, so that dL'Idx reduces to
,dH
*-fc
T dH
or I -^ (34),
and this is the mechanical force parallel to the axis of x
acting on unit volume of the magnet. This expression may
also be written
1 , dH*
- k j
2 ax
with similar expressions for the components parallel to the
axes ofy and z.
These are the same expressions for this force as those
given in Maxwell's Electricity and Magnetism, vol. IT. p. 70,
the consequences of which are as is well known in harmony
with Faraday's investigations on the way in which para-
magnetic and diamagnetic bodies move when placed in a
variable magnetic field.
35. We have just investigated the mechanical forces
produced by a magnetic field ; we shall now proceed to
investigate some of the stresses produced by it.
Let us take the case of a cylindrical bar of soft iron
whose axis coincides with the axis of x, and suppose that it
is magnetized along its axis. Let e, /, g be the dilatations
of the bar parallel to the axes of x, y, z respectively. We
shall at present assume that there is no torsion in the bar.
We shall suppose that the changes in the strains take place
so slowly that we may neglect the kinetic energy arising
from them.
The- potential energy due to these strains is
}* + \n {e 2 +/ 2 +/ - 2ef- 2eg - 2/g\,
48 DYNAMICS.
where n is the coefficient of rigidity and m - ;//3 the
modulus of compression.
Thus the terms in the Lagrangian function involving the
magnetic and strain coordinates are
- \m (e +/+ g} 2 - \n (e 2 +f 2 + g 2 - 2ef- 2eg
neglecting those depending on the rate of variation of these
quantities which rate we shall assume to be indefinitely small.
The experiments of Villari and Sir William Thomson
(Wiedemann's Elektricitdt, in. p. 701) have shown that k
depends upon the strain in the magnet, hence by equation
(32) A will be a function of the strains. We shall pro-
ceed to investigate the stresses which arise in consequence
of this. Using the Hamiltonian principle
and substituting da-ldx, dfi/dy, dyfdz for e, /, g respectively,
we get the following equations by equating to zero the
variation caused by changing a into a + Sa
dL d dL
-dx-dx-de =Q > ................... (35)
inside the bar,
dL
at the boundary.
By equating to zero the variation caused by changing ft
into ft + 8/3 we get
dL d dL
inside the bar,
dL
at the boundary.
STRAIN AND MAGNETIZATION. 49
And by equating to zero the variation caused by changing
y into y + By we get
dL d dL
- = (39)
inside the bar ;
at the boundary.
The first and second terms in the equations (35), (37)
and (39) may conveniently be considered separately. Since
H is the only quantity in the expression for L which can
involve the coordinates x, y, or z explicitly the terms
dL dL dL
dx ' dy ' dz
reduce to
^dH .dH .dH
^ ^
or kH ,-, kH -y- ,
dx dy dz
respectively.
These are the expressions for the components of the
mechanical force acting on the body, and it is shown in
Maxwell's Electricity and Magnetism, 642, that this dis-
tribution of force would strain the body in the same way as
"a hydrostatic pressure H 2 j%ir combined with a tension
BHI^T? along the lines of force," B being the magnetic
induction. Thus we may suppose that the strains arising
from these terms are known. If e, f, g are the strains due
to the second term in equations (35) (37) and (39), we have
T. D. 4
DYNAMICS.
if? -y-m(' +f+g) - n (f-e-g) = o
dA
if? -- - m (e+f+g] -n(g-e-f) = o
-.(41).
Solving these equations and putting r; = / we get
2** = -^-4 (AF) -^T n {^( A ^ 2 ) + ^ (4J 2 )}
_m^n_(d a r J
$mn {de x ' dg ^ ' )
2m
2Jlg =
2m d
m-n (d
dg
d
\
/ x
(42).
and
If the magnet is symmetrical about its axis we have
*/ / , ^
So that equations (42) reduce to
m i T
ne =
3;^ n de
-n df
\ .-.(43).
- n df
The dilatation e + zf is equal to
3;^ - 11 (de ' df
Differentiating equations (43) with regard to / 2 , we get
approximately since ^ (AI 2 ) and -^.(AI 2 ) must be
small compared with m or n, or the changes in the
elasticity caused by magnetization would not be so small
as to have escaped detection,
STRAIN AND MAGNETIZATION. 51
de m d d m-n d d
^ ' ' (
'
_ m-n
^ ' ^ ' ' \ (
'
df m-n d d tjr^.n + nd d (AP\\
~dl^~Z^-n de dl* ( ' + -^n JfTP^ ' } J
Now by equation (32)
So that these equations become
de m i dk m-n i dk
dl* ~ yn-nk* de yn-n k 2 df , . .
df _ m-n i dk m + n i dk \
Now if the coefficient of magnetization depends upon
the strains, the intensity of magnetization of the bar
when under the action of a constant magnetizing force
will be altered by strain, and in order to compare the
formulae with the results of experiments we shall find it
more convenient to express defd! 2 , dfldl* in terms of the
changes which take place in the intensity of magnetization
when the bar is stretched rather than in terms of dk'de
and dkjdf.
We have /=//,
so that when H is supposed to be constant
dl r^dk r - r dk dl
and equations (46) may be written
de _/ dk\f m i dl m-n i dl\ 1
* dl* \ di)\yn-nklde yn-nkl d/)\ , ox
, , . ,, , T i-r r (4w
df (IT "k\ ( tn n \ dl m + n i dl\
dl 2 \ dl )\ yn n kl de yn n kl df) j
These expressions give the strains which result from the
dependence of the intensity of magnetization on the state
42
52 DYNAMICS.
of strain of the magnetized body. In addition there are
the strains arising from Maxwell's distribution of stress.
Kirchhoff (Wied. Ann. xxiv. p. 52, xxv. p. 601) has
investigated the effect of this on a small soft iron sphere
placed in a uniform magnetic field and has shown that it
would produce an elongation of the sphere along the lines
of force and a contraction at right angles to them. We may
therefore assume that in general this distribution of stress
causes an expansion of the magnet in the direction of the
lines of force and a contraction in all directions perpen-
dicular to this.
The expressions for the strains in a magnetizable
substance placed in the magnetic field have also been in-
vestigated by v. Helmholtz (Wied. Ann. xin. p. 385). The
object of the investigations of v. Helmholtz and Kirchhoff
was rather different from that of Maxwell. Maxwell's object
was to show that his distribution of stress would produce
the same forces between magnetized bodies as those which
are observed in the magnetic field, while v. Helmholtz and
Kirchhoff's object was to show that it follows from the prin-
ciple of the Conservation of Energy that, whatever theory
of electricity and magnetism we assume, the bodies in the
electric or magnetic field must be strained as if they were
acted upon by a certain distribution of stress which in the
simplest case is the same as that given by Maxwell.
We have in addition to the strain produced by these
stresses, the strains depending upon the alteration of the
intensity of magnetization with stress along and perpen-
dicular to the lines of force.
The effect of stress along the lines of force on the
magnetization of iron has been investigated by Villari (Pogg.
Ann. 126, p. 87, 1868) and Sir William Thomson (Proc.
Roy. Soc. 27, p. 439, 1878) ; both these physicists found that
STRAIN AND MAGNETIZATION. 53
the intensity of magnetization was increased by stretching
when the magnetizing force was small, but that when the
magnetization exceeds about 10 when measured in C.G.S. units
the intensity of magnetization is diminished by stretching.
Sir William Thomson also investigated the effect of
stress at right angles to the lines of magnetic force on
the intensity of magnetization and found that this was in
general opposite to that of tension along the lines of force,
so that for small values of the magnetizing force extension
at right angles to the lines of force diminishes the mag-
netization, while for larger values of this force it increases
it. The critical value of the force in this case however is
higher than that for tension along the lines of force.
Thus, except when the magnetizing force is between the
critical values, dljde and dlldf have opposite signs, hence
we see by equation (48) that except in this case, since
Prof. Ewing's measurements show that Hdkldl is always
less than unity,
de , dl
dP and de
have the same sign, and
df
have opposite signs.
Now dllde is positive or negative according as the
magnetizing force is less or greater than the critical value,
so that when the magnetizing force is less than the critical
value the extension we are investigating will increase with
the magnetic force, but when the magnetizing force is
greater than this value the extension will diminish as the
force increases.
As we mentioned before the strain produced by Maxwell's
distribution of stress, which is the other cause tending to
54 DYNAMICS.
strain the body, has been shown by Kirchhoff to produce
an expansion along the lines of force and a contraction at
right angles to them. Thus when the magnetizing force is
less than the critical value this strain and the strain we have
just investigated act in the same way, but when the force is
greater they act in opposite directions.
Joule's investigations (Phil. Mag. 30, pp. 76, 225, 1847)
prove that the length of an iron bar increases when it
is magnetized and as far as the experiments went the
increase in the length was proportional to the square of the
magnetizing force. Mr Shelford Bidwell (Proc. Roy. Soc.
XL. p. 109) however has lately shown that when the mag-
netizing force is very large the magnet shortens as the
magnetizing force increases.
Comparing these experimental results with our theoretical
conclusions we see that they are in accordance when the
magnetizing force is small, and that when the magnetizing
force is large they indicate that the strains due to the same
cause as that which causes the intensity of magnetization to
alter with strains are more powerful than those arising from
Maxwell's distribution of stress. Prof. Ewing's experiments
on the effect of strain on magnetization (" Experimental
Researches in Magnetism," Phil. Trans. 1885, part n. p.
585) would seem to show that this must be the case. For
Kirchhoff (Wiedemann's Annalen, xxv. p. 60 1) has shown
that the greatest increase in length which Maxwell's stresses
can produce in a soft iron sphere whose radius is R, placed
in a uniform magnetic field where the force at an infinite
distance from the sphere is H, is
'53 H* R
I 7 67T E
where E is one of the constants of elasticity for soft iron
and is equal in the c. G. s. system of units to r8x io 12 .
STRAIN AND MAGNETIZATION. 55
Thus in this case supposing k to be constant we have
de i53 i
1-
Now according to Prof. E wing's experiments the intensity
of magnetization of a soft iron wire which was represented by
181 when there was no load was increased to 237 when the
wire was loaded with a kilogramme, so that in this case
87 i
7 = -.--nearly ............ (50).
The diameter of the wire was such that the load of a
kilogramme corresponded to a stress of about 2 x io 8 per
square centimetre in C.G.S. units, so that if q be Young's
modulus for the wire and Se the extension produced by the
load
q^e = 2 x i o 8 :
for wrought iron q\n is about 2-5, so that
nBe 8 x io 7
and therefore by (49)
\_dl__ i
nl de 2 '4 x io 8 '
so that by equation (48) if e be the elongation due to the
magnetization
de i i , ,
<// 8> 7xio 8 l "
Comparing this with (49) we see that the part of de\dl* due
to the cause we are now considering is very much greater
than that due to Maxwell's distribution of stress. The value
of dl\de is probably exceptionally large in this case, and
near the critical value it is doubtless very much less, so that
in this case it is conceivable that the effect of the Maxwell
5 6 DYNAMICS.
stress may be comparable with that due to the alteration of
intensity of magnetization with strain.
Since the Maxwell effect is in general so small compared
with the other we should expect the critical value of the
magnetizing force to be approximately the same as the
value of the force when the extension is a minimum ; it is
however much less. There seems however to be reason
to think that the critical value when the magnet is free
from strain has been very much underestimated. Indeed
Prof. Ewing (loc. tit.) expresses his opinion that "if we
deal only with very small stresses it is doubtful whether
any reversal of the positive effect of stress would be
reached even at the highest obtainable value of the magneti-
zation." By the positive effect of stress Prof. Ewing means
an increase of magnetization with an increase of stress, the
magnetizing force remaining constant.
Bid well's discovery that dejdl* is negative when the
magnetization exceeds a certain value, in conjunction with
the theoretical results we have been investigating in this
paragraph, shows that when the magnetization reaches this
value the positive effects of stress must be reversed. The
magnet in this case however is not free from stresses as it is
acted on by those called into play by the magnetization.
36. If the dilatation in volume e + 2/ be denoted by 8,
then the part of 8 due to the same cause as that which
makes the intensity of magnetization depend upon strain is
by (48) given by the equation
</8 _ i j_ fdT_ a
dl*~ yn-n kl \de * 2 ~i
Joule's experiments show that the dilatation in the
volume if it exists at all must be very small compared with
the elongation, as he was not able to detect it though his
STRAIN AND MAGNETIZATION. 57
apparatus would have enabled him to do so if it had
amounted to one part in 4500000. Hence as the greater
part of the strain is that given by equation (48)
dl dl_
de df
must be small, so that dl\de and dl\df must have opposite
signs except when they are very small. This agrees with
the results of Sir William Thomson's experiments on the
effects of traction along and perpendicular to the lines of
force on the intensity of magnetization ; as, except in the
neighbourhood of the critical magnetic forces when dllde
and dlldf are both small, traction along and perpendicular
to the lines of force produced opposite results.
If we assume that Joule's experiments prove that there
is no change in volume then by equation (52)
dl_ dl_
de + 2 df ~
and equation (48) reduces to
de i dl
37. The critical value of the intensity of magnetization,
i.e. the intensity when the magnetization is neither increased
nor decreased by a small strain will, since by (32) and (47)
d d , ,, , .
(54),
be given by the equation
* In my paper on " Some Applications of Dynamics to Physical
Phenomena," Part I. Phil. Trans. Part II. 1885 this equation has the
wrong sign, which was carried down from equation (51) in the same
paper.
58 DYNAMICS.
The experiments of Sir William Thomson and Prof. Ewing
have shown that the critical value of / depends upon the
state of strain. Hence we see by means of equation (55) that
dAlde must be a function of e, so that if A be expanded in
powers of e it must contain powers about the first. We may
therefore write
A = a + fie + ye* + $e s + ...
Now if the term ye 2 exists, the coefficient of e 2 in the
Lagrangian function will contain the term \yP and so will
involve the state of magnetization of the body. The
coefficients of elasticity however are linear functions of the
coefficient of e 2 in the Lagrangian function so that if this
latter quantity depends upon the state of magnetization, the
coefficients of elasticity will do the same. We conclude
therefore that the elasticity of an iron bar must be altered by
magnetization. This effect does not seem to have been
observed.
If in the expression for A we neglect powers above the
second we have
= + 2 e
de
and therefore
The right hand side of this equation changes sign when
e passes through the value
Now the effects of strain on the intensity of magnetization
and of magnetization upon strain depend by (32) and (47)
upon the value of,
TORSION AND MAGNETIZATION. 59
so that we should expect the effect of magnetization on the
strain of an iron rod to depend upon the strain previously
existing in the rod, in such a way that when the strain was
less than a critical value magnetization would increase the
length of the rod, and when it was greater than this value
magnetization would have the contrary effect and tend to
shorten the rod. This agrees with the result of Joule's
experiments as he found that when the soft iron wires were
stretched beyond a certain limit they became shorter instead
of longer when they were magnetized.
38. So far we have only considered the effect of expan-
sion and contraction upon the intensity of magnetization
and vice-versa. We can however in a similar way discuss
the effects of torsion upon the magnetic properties of iron
wire.
Let us now suppose that twist is the only strain in an iron
wire which is longitudinally magnetized and has a twist c
about its axis, then, using the same notation as before, the
terms in the Lagrangian function depending upon strain
and magnetization are
By a similar method to that employed in the case of
dilatations we can prove that the twist c due to the same
cause as that which makes the intensity of magnetization
alter with the torsion is given by the equation
............... (53).
Now when a twisted bar is magnetized it untwists to
a certain extent if the magnetization is intense, but the twist
increases if the magnetization is weak. If however the bar
60 DYNAMICS.
initially has no twist in it then it neither twists nor untwists
when it is magnetized (Wiedemann's Elektricitat, in. p. 692).
This shows that if A be expanded in powers of c the
first power must be absent, otherwise by equation (58) an
untwisted bar would twist when it was magnetized. Hence
A must contain a term in c and therefore the coefficient of
c 2 in the Lagrangian function must contain a term proportional
to I 2 . Now the coefficient of <? in the Lagrangian function
is proportional to the coefficient of rigidity and hence we see
that the rigidity of iron wire will be altered by magnetization.
Since the twist diminishes with strong magnetization we
see by equation (58) that the coefficient of <: in
- d (AP\
n "dc dl* (AJ >
must be negative when / is large and hence that the co-
efficient of in A must be negative. Let us call this
coefficient /, the coefficient of c 2 in the Lagrangian
function is
.
but the apparent coefficient of rigidity is twice the coefficient
of - f in the Lagrangian function so that in this case the
apparent coefficient of rigidity is
Thus in this case the effect of strong magnetization is to
increase the rigidity, so that the same couple will not twist
the wire as much when it is strongly magnetized as when it
is unmagnetized.
When the intensity of magnetization is small the opposite
will be the case, as in this case the twist in a wire increases
when it is longitudinally magnetized.
Since =-
TORSION AND MAGNETIZATION. 6l
and by (47)
and also
we have
dc
-JTT - - -j* < = r- r -J- \--T
dl \ 11 dc v ' } nkl dc \ dc
or approximately since d*(AI z )lndc 2 is very small
dc i dl / __ ^\
............ (6o) "
We see by this equation that when the magnetization is
so strong that magnetizing the wire diminishes the twist in
it, then twisting the wire will diminish the intensity of
magnetization. On the other hand when the intensity of
magnetization is so small that magnetizing the wire increases
the twist in it then twisting the wire will increase the
intensity of magnetization.
The reciprocal relations between torsion and magnetiza-
tion have been experimentally investigated by Wiedemann
(Lehre von der Elektriritat, in. p. 692) and he arranges
the corresponding results in parallel columns. These are
also quoted in Prof. Chrystal's article on Magnetism in the
Encyclopedia Britannica. The following is one set of the
corresponding statements.
" 5. If a wire under the influence of a twisting strain is
magnetized, the twist increases with weak but diminishes
with strong magnetization."
"V. If a bar under the influence of a longitudinal
magnetizing force is twisted the magnetization increases
with small twists but decreases with large ones."
Comparing these statements with the results we have
previously obtained we see that whether the first part of V is
62 DYNAMICS.
true or not depends upon the intensity of magnetization. If
the twist be of such a magnitude that 5 is true, then the first
part of V is true if the magnetization is weak, but the opposite
is true if the magnetization is strong. Further since by V
the influence of twist on magnetization depends upon the
size of the twist, it follows by equation (60) that the influence
of magnetization upon twist must depend upon the size of
the twist so that 5 is only true when the twist is on one
side of a critical value, when it is on the other side the
contrary is true.
The existence of a critical twist as well as a critical
magnetization makes the verbal enunciation of the relations
between torsion and magnetization cumbrous; they are all
however expressed by equation (60).
39. Strains in a dielectric produced by the
electric field. The strains produced in a dielectric by
the electric field can be found by a method so similar to
that used in the last two paragraphs that we shall consider
them here though they have no connexion with the terms in
the Lagrangian function which we have been considering.
Let/, , r be the electric displacements parallel to the
axes of x t y, z respectively, then if the body is isotropic, the
terms in the Lagrangian function of unit volume of the
dielectric which depend upon the coordinates fixing the
strains and electric configuration if the dielectric is free
from torsion are,
where e, /, g are the dilatations parallel to the axes of x, y, z
respectively, ^Tthe specific inductive capacity of the dielectric
and X, Y, Z the electromotive forces parallel to the axes
STRAIN AND MAGNETIZATION. 63
of x t y, z. Then we see as in 35 that e,f, g the strains due
to the dependence of K upon the strains in the dielectric
are given by the equations
= - Z = o ~ = o
where L stands for the expression (61).
Substituting for L its value these equations become re-
spectively
27r{/ 2 + q* + r 2 } ^ ~ 4- ;// (e +/+ g) + n (e -f-g) - o
n (f-. e -X\ = Q v. (62).
\*/ 3 / / \ /
Now / = KX, q=~ KY, r = -~ KZ.
47r 4?r 4?r
So that if R 2 =X 2 + Y 2 + Z 2 ,
we get from equations (62)
dK , x (dK . dK\\
de
, . ,, .
-(m-n) (-^. + -- )\ -..(63)
\df dgj)
with symmetrical expressions for g and //.
The expansion in volume
is given by the equation
dK dK
Just as in the analogous case of magnetism these are
not the only strains produced in the dielectric by the
electric field. The term (Xp + Yq + Zr) which occurs in
the Lagrangian function can be shown to involve the same
64 DYNAMICS.
distribution of strain in the dielectric as would be produced
by the distribution of stress which Maxwell supposes to
exist in the electric field, viz. a tension KR 2 !^ along the
lines of force and a pressure of the same intensity at right
angles to them. The effect of this distribution of stress
will be of the same character for all dielectrics, and its
nature depends more upon the distribution of force
throughout the electric field than upon the nature of the
dielectric. The experiments of Quincke (Phil. Mag. x. p.
30, 1880) and others show that the behaviour of different
dielectrics when placed in the same electric field is very
different. Thus, for example, though most dielectrics
expand when placed in an electric field, the fatty oils on
the contrary contract. This difference of behaviour shows
that in many cases at any rate, the strains due to the same
cause as that which makes the specific inductive capacity
depend upon the strain are greater than those produced by
Maxwell's distribution of stress.
Quincke has shown that the coefficients of elasticity of a
dielectric are altered when an electric displacement is pro-
duced in it, this shows that \IK when expanded in powers
of e must contain a term in e z and is another proof that
the specific inductive capacity depends upon the strain in
the dielectric. Since part of the strain of a dielectric in an
electric field is due to the same cause as that which makes
the specific inductive capacity depend upon strain, the
expression for i/^when expanded in powers of e must con-
tain the first power of the strains as well as the second, as if
it only contained the second powers placing the dielectric
in an electric field would merely be equivalent to changing
the coefficients of elasticity of the body and so could not
strain the body if it were previously free from strain.
No experiments seem to have been made to determine
MAGNETIC INERTIA. 65
directly the values of dKlde, dK/df&c., and the experimen-
tal difficulties which would have to be overcome in order to
do this are much greater than those in the corresponding
case in magnetism. The dependence of K upon strain is
probably much less than that of /, the coefficient of
magnetic induction. For the specific inductive capacity
seems to be much more independent of the molecular state
of the dielectric than the coefficient of magnetic induction
is of the molecular state of soft iron. Thus there is a com-
paratively small difference between the specific inductive
capacities of various substances, while the coefficient of
magnetic induction of iron is enormously greater than that
of any other substance. Again, the coefficient of magnetic
induction is known to be much affected by changes in
temperature; while some recent experiments made by Mr
Cassie in the Cavendish Laboratory have shown that the
effect of changes of temperature on the specific inductive
capacities of ebonite, mica and glass is small, amount-
ing in the case of glass, for which it is largest, to i part in
400 for each degree centigrade of temperature. No experi-
ments seem to have been made on the effect of torsion on
electrification or of electrification upon torsion.
40. Influence of inertia on magnetic pheno-
mena. In the preceding investigations we have supposed
the magnetic changes to take place so slowly that the
effects of inertia may be neglected. If however a change in
the magnetization involves, as it does according to all
molecular theories of magnetism, motion of the molecules
of the magnet, then magnetism must behave as if it possessed
inertia.
In soft iron and steel the conditions are made so com-
plex by the effects of magnetic friction, magnetic retentive-
ness and permanent magnetism, that it would be difficult to
T.D. 5
66 DYNAMICS.
disentangle the effects of inertia proper from other compli-
cations. The effect, if it exists, would probably be detected
most easily in the case of crystals, as only one of these,
quartz, has ever been suspected of showing residual mag-
netism (see Tumlirz, Wied. Ann. xxvn. p. 133, 1886). The
effect of inertia would be to introduce into the equations
of magnetization a term
as** 1
M w*
where / is the intensity of magnetization. The equations
of magnetization would therefore be of the form
where 77 is the external magnetic force.
If ffis periodic and varies as e ipt then by (65)
(60,
so that if/ be so large that kMp 2 >i, the crystal if para-
magnetic for a steady magnetic force will be diamagnetic for
a variable one and vice versa.
Changes of this kind could be detected very readily if
the crystal were freely suspended in the magnetic field, for
when / 2 passed through the value \lkM the crystal would
swing through a right angle.
41. The term (y) y in the Lagrangian func-
tion. We have considered the terms depending upon the
squares of the velocities of the electrical coordinates, and
those depending solely on the magnetic coordinates, let us
now consider those terms in the expression for the kinetic
energy which involve the product of the velocities of a mag-
netic and an electrical coordinate.
It is proved in Maxwell's Electricity and Magnetism
ELECTROMOTIVE FORCE. 6/
{ 634) that when a current whose components are #, v, w
flows through the element of volume dxdydz and the volume
dxdy'dz' is magnetized to the intensities A, B, C parallel to
the axes of x, y, z respectively, then the kinetic energy L
possessed by the system is
where / is the reciprocal of the distance between the
elements dxdydz and dx'dy'dz.
Now we represent the intensity of magnetization by
TJ where is the momentum corresponding to a kinosthenic
or speed coordinate and rj is a vector quantity.
Since rj is a vector quantity it may be resolved into com-
ponents parallel to the axes of x, y, z. Let us denote these
components by X, //,, v respectively, then we may put
Making this substitution we have
. v f\ +v ( v ^. x ^
dz' dy } \ dx' dz
So that these terms are of the form
Considering the Lagrangian Equation for the electrical
coordinate, we see that there is an electromotive force
parallel to the axis of x on the element dxdydz equal to
dtdu 9
52
68 DYNAMICS.
so that, per unit volume, this force equals
-'*{*%- c
with corresponding expressions for the electromotive forces
parallel to the axes of y and z.
These are the usual expressions for the electromotive
forces due to the variations of the magnetic field.
The magnetic force parallel to x acting on the element
dx'dy'dz' is by 33 equal to
idL
d\
so that the magnetic force parallel to x per unit volume
is equal to
* -*** ............... (7o)>
with similar expressions for the magnetic forces parallel to
the axes of y and z. These expressions agree with those
given by Ampere for the magnetic force produced by a
system of currents.
Again there is a mechanical force acting on the element
dxdydz whose component parallel to the axis of x is
d_L
Jx'
If we call
.
G, H respectively, then F, G, H are the same as the
MECHANICAL FORCE DUE TO A CURRENT. 69
quantities denoted by the same symbols in Maxwell's
Electricity and Magnetism.
Since the force on the element dxdydz is
dL
dx'
we see that the force on unit volume may be written
dF dG dH
U -- + v -j + W r- ,
dx dx dx
or
(dG dF\ (dF dH\ dF dF dF
V\ . -j- \-W \-j } + U -j- + V-j- + W- r ...( f J2\
{dx dy } {dz dx } dx dy dz w
This differs from Maxwell's expression for the same force
by the term
dF dF dF
U +V -j- +W -T-.
dx dy dz
du dv dw
Since -j- + -j- + --j- = o
ax ay dz
it follows that
MdF dF dF\ ,
u +v -j- +w ^ ) dxdydz = o
dx dy dz)
if all the circuits are closed. So that as long as the circuits
are closed the effect of the translator}' forces is the same as
if they were given by Maxwell's expressions.
In the above investigation we have assumed that we
could move the element without altering the current ; if we
suppose the current to move with the elements we shall get
Maxwell's expression exactly.
The components parallel to y and z of the force on the ele-
ment dxdydz are given by expressions corresponding to (72).
The force parallel to x on the magnetized volume
Jx'dy'dz', is
*L
dx'
7O DYNAMICS.
so that the force parallel to oc per unit volume is
with corresponding expressions for the forces parallel to y
and z.
Thus the force on the magnet is equal and opposite to
that on the current.
We see by this example how from the existence of a
single term in the expression for L we can deduce the laws
of the induction of currents, the production of a magnetic
field by a current, the mechanical force on a current in a
magnetic field and the mechanical force on a magnet placed
near a current.
42. Twist in a magnetized iron wire produced
by a current. Prof. G. Wiedemann (Elektricitat, in.
p. 689) has shown that when a current flows along a longi-
tudinally magnetized wire, it produces a couple tending to
twist the wire. This shows that there must be a term in the
Lagrangian function for the wire of the form
/*rf ........................ (74),
where y is the current flowing along the wire, >; the intensity
of magnetization /, and c the twist about the axis of the
wire,/(<:) being some function of c. Applying Hamilton's
principle to this term we see that it indicates the existence
of a couple tending to twist the wire equal to
>l ........................ (75).
Applying Lagrange's equation for the ^coordinate to this
term we see that since the electromotive force tending to
TORSION AND MAGNETIZATION. 7 1
increase y is
<L dL
dtdy*
the existence of this term shows that there is an electro-
motive force along the wire equal to
that is -J t {f(f}1} ..................... (?6) '
Thus twisting a longitudinally magnetized iron wire must
produce an electromotive force which lasts as long as the
twist is changing, and any alteration in the longitudinal
magnetization of a twisted iron wire must produce one lasting
as long as the magnetization changes. Hence Faraday's rule
that the electromotive force round the circuit due to induc-
tion equals the rate of diminution in the number of lines of
force passing through it, will not apply to the case of a twisted
iron wire, for we might get an electromotive force round a
circuit made of such a wire by moving it in the plane of
the magnetic force, and in this case there is no alteration in
the number of lines of force passing through the circuit.
The production of an electromotive force by twisting
a longitudinally magnetized iron wire has been experi-
mentally verified.
Again, if we consider Lagrange's equations for the
coordinates fixing the magnetic configuration, since any term
in the Lagrangian function indicates an effect similar to
that which would be produced by an external magnetic force
equal to
we see that the term we are considering indicates the
72 DYNAMICS.
existence of a magnetizing force on the wire equal to
/y ........................ (77)
tending to magnetize it longitudinally. So that if a current
of electricity passes along a twisted wire or if a wire
conveying a current of electricity be twisted the wire will be
longitudinally magnetized. These effects have been ob-
served by Prof. G. Wiedemann (Elektriritat, in. p. 692).
43. Hall's phenomenon. The terms we are con-
sidering, involving both the electric and magnetic coordinates,
are also interesting from their connexion with Hall's phe-
nomenon, for as we shall see directly this phenomenon
indicates the existence in the Lagrangian function of terms
of this kind. Hall discovered (Phil. Mag. x. 301,) that
when currents are flowing through a conductor placed in a
magnetic field, there is an electromotive force due to the
field even though it remains constant, and that this electro-
motive force at any point is parallel and proportional to the
mechanical force acting on the conductor conveying the
current at that point. Thus the electromotive force is at
right angles both to the direction of the current and the
magnetic induction, and its components parallel to the axes
of x, y, z are respectively given by the expressions
where C' is a constant depending upon the nature of the
medium through which the current is flowing, a, /?, y are
the components of the magnetic force and /, g, h are the
components of the electric displacement if the medium is a
dielectric, if the medium is a conductor /, g, h are the
components of the electric current
HALL EFFECT. 73
Prof. Fitzgerald ("On the Electromagnetic Theory of
the Reflection and Refraction of Light," Phil. Trans.
1880, Part ii.) and Mr Glazebrook (Phil. Mag. XL p. 397,
1881) have shown that the existence of this force proves
that there is a term equal to
KM/te-/M)+^-y/M(/3/-?)} .-.(78)
in the expression for the Lagrangian function of unit
volume of the medium.
Let us consider the Lagrangian equation for the electric
displacement/. It indicates the existence of an electromo-
tive force parallel to the axis of x equal to
_d_dL L dL
-~dt~tf + ~df
or in this case
C'(P/i-yg) + kC(yg-ph) (79).
The first of these terms corresponds to the Hall effect,
the second to an electromotive force tending to displace the
lines of electrostatic force.
This latter force is at right angles both to the direction
of electric displacement and to that in which the change in
the magnetic force is greatest ; the magnitude of the force
is
\C'HD sin (9,
where D is the resultant electrostatic displacement, H the
rate of change of the magnetic force and the angle
between the electric displacement and the direction in
which the change in the magnetic force is greatest.
If P be the original electromotive force then since the
Hall electromotive force is very small we have approximately
where K is the specific inductive capacity of the medium.
74 DYNAMICS.
Thus the ratio of the disturbing force we are considering to
the original electromotive force is
~ C'KHsm 6.
07T
Now Hall's experiments show that C in electromagnetic
measure is at most of order io~ 5 ; and K is of the order io~ 31
so that the ratio of the disturbing force to the original force
is of the order
io- 27 H sin 0,
and is thus much too small for there to be any chance of its
detection by experiment.
We see too from the expression for this force that it
absolutely vanishes when both the electric displacement and
the magnetic force are stationary, and these were the con-
ditions when Hall tried unsuccessfully to detect the existence
of his effect in an insulator (Phil. Mag. x. p. 304, 1880.)
Let us now consider Hall's effect from the point of view
of magnetic instead of electromotive force. Perhaps the
easiest way to do this will be to suppose that the magnetic
forces are produced by an element of volume dx'dy'dz'
magnetized to intensities A, B, C parallel to the axes of
x, y, z respectively. If O is the magnetic potential of this
element at a distance r, then, for a point outside the magnet
and
-- + JS~+C- dx'dy'dz' ...
dxr dy r dz r
Substituting these values for a, /?, y in the expression (78)
we see that there is a term in the Lagrangian function equal
to
HALL EFFECT. 75-
dxdy'dz'.
If as on page (67) we put
^ = ^X; ^ = 6*; CWv
the magnetic force on the element dx'dy'dz' parallel to the
axis of x will be
idL
so that in this case the magnetic force parallel to x per unit
volume is
if. d a i d* i - d z i~|
or if
then the magnetic force at the points #', y, / parallel to the
axis of x due to the electric displacements /, g, h through
unit volume at the point (xyz) is
similarly the magnetic forces parallel to y and z are
respectively.
76 DYNAMICS.
If we have electric displacements distributed throughout
a volume of any size, then the components of the magnetic
force parallel to the axes of x t y, z due to the same cause
as that which produces Hall's effect are
d* d* d*
dx' ' dy ' dz'
respectively, where
* = SS5tdxdydz .................. (85),
the integration being extended over the volume throughout
which there are electric displacements.
If the point at which we wish to find the magnetic force
is inside the volume occupied by the electric displacements
we must modify the preceding results. Let us suppose that
we have a small sphere whose centre is at the point where
we require the magnetic force, magnetized to the intensities
A, B, C parallel to the axes of x, y, z respectively. Then
inside the sphere
' So that the Lagrangian function for an element of volume
dxdydz inside the sphere is
1 C ^ {A (gh-gh) + B (hf- hf) + C (fg -fg)} dxdydz,
hence the components of the magnetic force due to the
electric displacement at the point where the force is measured
are
%*C f (gh-gh),
So that the general expression for the components of the
magnetic force due to the cause producing the Hall effect
are
HALL EFFECT.
77
-, -
+ f *C' (hf- hf)
(86),
where /, g, h are the components of the electric displace-
ment at the point where the magnetic force is measured and
^ is given by equation (81). Since C r is a very small
quantity, as are also / g, h, these forces will be very small,
and it is only when f, g, h vary very rapidly that we could
expect to have any chance of detecting them. We shall
therefore calculate the magnitude of these forces when the
electric displacement changes with the greatest rapidity we
can produce in an experiment. This if the Electromagnetic
Theory of Light is true will be when the electric displace-
ments are those which accompany the propagation of light.
Let us suppose that we have a circularly polarized ray
travelling along the axis of z and that the electric displace-
ments are given by the equations
(87),
/ 27r / , \
/= w cos (vt - z)
A
27r / J. \
g = w sm (vt z)
A
where w is the amplitude of the oscillation, X the wave length
and v the velocity of propagation of light.
Substituting these values we see
hf-hf=o
fg-fg=-*^-\
,(88).
78 DYNAMICS.
If we consider a long cylindrical beam of light
# = o
and thus by equation (86) the circularly polarized ray
produces a magnetic force in the direction along which it is
propagated equal to
we can deduce the value of w for strong sunlight from
the data given in Maxwell's Electricity and Magnetism, Vol.
ii. 793. The maximum electromotive force in this case
is given as
6 x io 7
in electromagnetic measure, the maximum value w, of the
displacement corresponding to this is
47T
or
2TTV*
Assuming the wave length to be 6 x io~ 5 , which is a little
greater than that of the D line and C to be io~ 5 , we see
that the magnetic force produced by circularly polarized
light as intense as strong sunlight cannot be greater than
2 x io~ 18 ,
which is much too small to be detected by experiments.
The direction of the magnetic force is related to the
direction of rotation of the electric displacement in a
circularly polarized ray like translation and rotation in a
left-handed screw.
Prof. Rowland has shown (Phil. Mag. Apr. 1881) that the
Hall effect if it existed in transparent bodies (which with the
HALL EFFECT. 79
exception of electrolytes are all insulators) would account for
the rotation of the plane of polarization of light passing
through such bodies placed in a magnetic field in which the
lines of magnetic force are more or less parallel to the
direction of propagation of the light. In this case by the
aid of an external magnetic force we rotate the plane of
polarization ; in the case we have just investigated, which
may be looked upon as the converse of this, a beam of
circularly polarized light produces a magnetic force parallel
to the direction in which it is travelling.
CHAPTER V.
RECIPROCAL RELATIONS BETWEEN PHYSICAL FORCES
WHEN THE SYSTEMS EXERTING THEM ARE IN A
STEADY STATE.
44. THE preceding methods are applicable to systems
in all states, whether steady or variable. When however the
system is in a steady state the reciprocal relations between
the various physical forces become so simple that they seem
deserving of special treatment, and we shall accordingly
consider them separately.
Let us consider the mutual effect of two quantities fixed
by the coordinates / and q upon each other. Let us
suppose that we have a force P of type / acting upon the
system, then P will alter the coordinate p in a definite way
and the amount of the alteration may depend upon the
value of the other coordinate q. Let us suppose that q
suffers a small alteration 8^ and that 8P is the amount by
which P must be increased in order to keep p the same as
before. Then since the system is in a steady state if L be
the Lagrangian function we have
and
RECIPROCAL RELATIONS. 8 1
SO that &P=-^-8? ( 8 9)-
Now if we have a force Q of type q producing a definite
change in the coordinate q then if we alter p by S/ we must
in order to keep q constant alter Q by some quantity 8(2,
and since
e-S
dq
and Q + *Q = -~-^ty,
dq dpdq *
we have %Q = -^ (9)
dpdq
so that by (89) and (90) we have
fdP\ -C*Q\ ( \
\ dq J p constant \ dp ) q constant
Or the alteration in P when q is increased by unity, p being
constant, is the same as the alteration in Q when / is
increased by unity, q being constant. Thus if P depends
upon q then Q will depend upon / and vice versa. And
we notice that if by increasing q we increase the "spring"
of / then by increasing p we shall increase the " spring "
Off.
Equation (91) is analogous to the " thermodynamical
relations" given in Maxwell's Theory of Heat, p. 169 and
forms one of those reciprocal relations which exist in
physics and which so often enable us to duplicate discoveries
in Physical Science. The consequences of reciprocal rela-
tions of a different kind are considered by Lord Rayleigh
in the Theory of Sound, Vol. i. Chapter 5.
As an example of the application of this equation we
T. D. 6
82 DYNAMICS.
may take the case of a wire bent into any shape by the
action of any number of forces two of which are P and Q,
then the increase in Q required to keep its point of
application at rest when / is increased by unity, will also
be the amount by which P must be increased to prevent its
point of application moving when q is increased by unity.
Or again, we see by this equation that if the force
required to produce a given extension in an iron wire is
altered by magnetizing the wire then the magnetic force
required to magnetize the wire to a given intensity will be
altered by straining the wire : and that these alterations will
be connected by the following relation, P being the tension,
<rthe extension of the wire, /f the magnetic force and /the
intensity of magnetization,
fdP\ = tdff\
\fl-f Je constant \ u //constant
Again when a current passes through an electrolyte in
solution it decomposes it and the strength of the solution
changes, this change in the strength of the solution may,
and in general will, change the coefficient of compressibility,
the volume and the surface tension of the solution, and in
this case equation (91) shows that the electromotive force
required to send a given current through a cell containing
the solution will be altered by pressure and by any change
in the free surface of the solution. Let E be the electro-
motive force, y the current, v the volume of the solution,
S its surface, and T its surface tension, then in this case for
the effect of pressure / we have
fdE\ = -( d -\ ( }
\ (IV / y constant V*y/ v constant
The negative sign is taken because/ tends to diminish v.
EFFECT OF PRESSURE ON ELECTROMOTIVE FORCE. 83
If k be the modulus of compression, v the volume of
the solution when free from pressure, then
>=*K)
'dp\ f v\ dk
when y is constant, k is also constant so that
koV
o
and therefore from (93)
>= v Ji-l\ d d A
vj dy k
I dk
so that if the pressure is increased from P l to P 2 the
increase E in the electromotive force required to keep the
current constant is given by
T fib
*E=\v a \P*-P?\ ............ (94).
To get an idea of the magnitude of this effect let us take
the case of a solution of chloride of lithium, the volume of
the solution being i cubic centimetre.
The data for calculating dkjdy in this case are the fol-
lowing :
The passage of unit quantity of electricity corresponds
to the decomposition of about 4'3xio~ s grammes of
lithium chloride, we shall suppose that none of this is redis-
solved, then the passage of a unit quantity of electricity will
withdraw this quantity of salt from the solution.
Rontgen's and Schneider's experiments (Wiedemann's
Annalen, xxix. p. 186, 1886) show that the addition of 6
grammes of lithium chloride to 100 cubic centimetres of
62
84 DYNAMICS.
water increases the modulus of compression by about 15
parts in 100, so that if the increase in the modulus is
proportional to the quantity of salt, then the subtraction of
4-3 x 10 ~ 3 grammes from i cubic centimetre will diminish
the modulus by about i part in 100, hence
i dk
T -r =- 10 , approximately.
Now k for water is about 2-2 x io 10 , so that if B is the
change produced by a pressure of 1000 atmospheres, which
in absolute measure is about io 9 , we have
TO 18 T
& = -J- ^ TO 2 = -Jxi0 6 ,
2 2'2 X IO 10 IO 2
that is the pressure of 1000 atmospheres would diminish the
counter electromotive force by about 1/400 of a volt.
The numbers given by Rontgen and Schneider for the
effect of carbonate of soda on the coefficient of compres-
sibility, show that the effect of pressure on a solution of this
salt would be much greater than that on the lithium chloride
solution.
Let us now suppose that the volume of the solution is
altered by the passage of an electric current, but that the
coefficient of compressibility is unaltered.
Then since
if the passage of the unit of electricity increases the volume
by dvjdy we must apply an additional pressure kdvjv Q dy to-
keep the volume constant, so that
/dp\
\<ty).
and the equation (91) becomes
EFFECT OF PRESSURE ON ELECTROMOTIVE FORCE. 85
fdE\ _^fa
\dv) y constant" # d 1 '
so that since kdv\v Q = djt>, we see from this equation that
the change BE in the counter electromotive force is given
by the equation
J B = S/g (95)-
When the electric current goes through a salt solution
the changes which take place and which alter the volume
are so numerous that it is not possible to calculate from
existing data the change which takes place in the volume
when unit quantity of electricity passes through the solution.
In order to see of what order this effect is likely to be, let us
suppose that the change in the volume is comparable with
the volume of the salt electrolysed. When unit quantity of
electricity goes through a solution of sulphate of potassium
it electrolyses about 9 x io~ 3 grammes of salt, and since the
specific gravity of the salt is 2*6, the volume of this is about
3 -5 x zo" 3 , hence in this case we may suppose that dvldy is
comparable with 3*5 x io~ 3 and that the change in the
counter electromotive force produced by 1000 atmospheres
is of the order
3'5 x i 6 ,
or about 1/28 of a volt.
We will now consider the case when gas is given off.
Let us suppose we are electrolysing water, above which
we have air, enclosed by a cylinder with a moveable piston.
If unit quantity of electricity goes through the water,
9 x io~ 4 grammes of water are electrolysed, the volume of
the water therefore diminishes by 9x10** cubic centi-
metres. At one terminal io~ 4 grammes of hydrogen will be
liberated, and 8 x io~ 4 grammes of oxygen at the other.
86 DYNAMICS.
Let us proceed to find the change in the pressure, the
volume remaining constant when unit of electricity passes.
The diminution in pressure due to the disappearance of
the water is, if v be the volume of the gas above the water
the increase in pressure due to the io~ 4 grammes of hydro-
gen is if the temperature is o C.
and the increase due to the oxygen is one half of this, hence
so that by (95)
dE
But
so that
E i,., K _ 41
=- = -- {1-65 x io - 9 x/ x 10 }.
dv v l
dE i'6q x io 6
- 9 x io .
dp p
If the pressure is increased from P l to P a the change
& in E is given by the equation
&E= 1-65 x io 6 x log 5* - 9 x io- 4 x (P 2 -P,).
*1
For a thousand atmospheres the counter electromotive
force is increased by
i '65 x i o 6 x 6*9 - 9 x io~ 4 x io 9 approximately,
= i'2 x io 7 9 x io 5 ,
EFFECT OF SURFACE TENSION ON E. M. F. 8/
so that the counter electromotive force is increased by about
one-eighth of a volt.
The effect of surface tension is given by
I 7 Ql / I 7 I
\o / y constant \ #y /^S 1 constant
This effect will in general be very small, for example in the
case of chloride of lithium, the experiments of Rontgen
and Schneider (Wiedemann's Annalen, xxix. p. 209, 1886),
show that the addition of 6 parts by weight of lithium chloride
to 100 of water increases the surface tension by about 3 parts
in 100. The passage of i unit of electricity decomposes
about 4 '3 x io~ 3 grammes of lithium chloride, so that if v
be the volume of the solution
i dT 13 io 2
r- = - X -^-_ X X 4'7 X IO
T dy v io 2 6
= - 2 x io~ 3 - approximately,
and for water T= 81,
so that
dT 16-2 x io~ 2
and therefore by (96)
dE 16-2 x io
or if the volume remains constant the effect of increasing
the surface by S is to diminish the counter electromotive
force by
l6'2 X I0~ 2 6*
v
Suppose that the liquid is squeezed out into a thin film
88 DYNAMICS.
whose thickness is / then
v = St
and
~ l6'2 X I0~ 2
S= - -
If t were of the order of molecular distances say io~ 7 then
&" = - 1 6*2 x io 5 ,
or the counter electromotive force is diminished by about
'016 volts.
The preceding investigation is on the supposition that
the electrolyte is in contact with the air; if it were in
contact with a solid such as glass the withdrawal of the
electrolyte from the solution on the passage of the current
would increase the surface tension between the liquid and
the solid, so that the electromotive force required to
decompose an electrolyte in a porous plate would be larger
than that required to decompose it when it is in bulk.
Again, the surface tension of liquids is altered when they
absorb gases, so that the electromotive force required to
decompose an electrolyte which absorbs a gas produced by
the passage of the current will be different when the
electrolyte fills the interstices of a porous plate from that
required when it is in an ordinary electrolytic cell.
CHAPTER VI.
EFFECT OF TEMPERATURE UPON THE PROPERTIES
OF BODIES.
45. WE have only considered so far the relations
between the phenomena in electricity, magnetism and elas-
ticity and have not discussed any phenomenon in which
temperature effects occur. We shall now go on however to
endeavour to extend the methods we have hitherto used to
those cases in which we have to consider the effects of
temperature upon the properties of bodies.
Before doing this however we must endeavour to arrive
at some dynamical interpretation of temperature. The only
case in which a dynamical conception of temperature has
been attained is in the Kinetic Theory of Gases, and there
the temperature is the mean energy due to the translatory
motion of the molecules of the gas. So that if N be the
number of molecules of the gas in unit volume NB is the
energy of translatory motion of the molecules at the tempe-
rature 0.
There seems good reason for believing that NO is a part
of the kinetic energy of the molecules when these are
aggregated so as to form a solid or liquid as well as when
.they form a gas.
9O DYNAMICS.
The experiments and ideas which led to the establish-
ment of the principle of the Conservation of Energy at the
same time led to the conclusion that the energy of sensible
heat is energy due to the motion of the molecules and is
therefore part of the kinetic energy of the system. The
reader should refer on this point to Maxwell's Theory of
Heat, p. 301. Another reason for supposing that the
temperature in the liquid as well as in the gaseous
condition is measured by the mean energy of translation
of the molecules is, that Van der Waals (Die Continidtdt des
gasformigen und fliissigen Zustandes] has given a theory of
the molecular constitution of bodies in those states which
are intermediate between the liquid and gaseous, in which
this supposition is made, and that this theory agrees well
with the facts in many important respects. And again
since most solids and liquids are capable of getting into a
state where their specific heat is constant, that is, where the
rise in temperature is proportional to the energy communi-
cated to the system, we are led to suppose that the kinetic
energy of some particular kind is a linear function of the
temperature.
This following illustration will show that it is probable
that when we have two bodies in contact the collisions-
between the molecules will tend to equalize the mean
energy of this translatory motion when these bodies are
solids and liquids as well as when they are gases. The
mean translatory energies of two substances in contact thus
tend to become equal, so that in this important respect the
mean translatory energy has the same property as tempe-
rature.
Let us suppose that we have two different substances,
composed of molecules A and B respectively, and that the
molecules of the two substances are separated by a material
TEMPERATURE. 91
plane surface. Let us also suppose that the mass of this,
plane is large compared with that of a molecule of either
substance and that it is prevented by perfectly elastic stops
from moving through more than a distance comparable
with molecular distances. Since the mass of the plane is
very much greater than that of a molecule and since it can
only move through a small distance in one direction the
velocity of the plane will be very small compared with that
of the molecules we shall suppose that it is so small that
the number of molecules which are moving more slowly
than the plane may be neglected, or what amounts to the
same thing that all the molecules on the surface of the
substances which are moving towards the plane strike it,
and that none of those which are moving away from the
plane do so. Let us suppose that the action between the
molecule and the plane is the same as that between a
perfectly elastic sphere and plane.
Let m be the mass of an A molecule, v the velocity of
the molecule, and a the angle its direction of motion makes
with the normal to the plane before impact, V the velocity
after impact, M the mass of the plane, w and W its velocity
before and after it is struck by the molecule. Then we
may easily show that
- m{ V 2 -v*}= -r-rj -AMw 2 - mv 2 cos 2 a-(M- m) vw cos a}.
2 (M+m)* {
Let us take the sum of the equations representing the
effects of all the collisions which take place in unit time, we
have
y$m { V 2 - v*\
= (M+ m} 2 ^ ^ Mw * ~ v 2cosSa -( M - m ) zwcos a}... (9 7).
If N be the number of A molecules which come in con-
92 DYNAMICS.
tact with the plane in unit time and O l the mean translatory
kinetic energy of such molecules, then if B0 l denotes the
change in 1 in unit time
If N' be the number of collisions and the mean
kinetic energy of the plane, then
Since the directions of motion of the A molecules are
equally distributed
Since the plane is supposed to move so slowly that all
the molecules moving towards it strike it, and since its
average velocity is zero, we have
5 (M m) vw cos a = o,
so that equation (97) becomes
{2N'0 - J W0\ ..... (98).
m) 2 *
If 2 be the average translatory kinetic energy of the
molecules which strike the plane in unit time, ^ the num-
ber of such molecules and TV/ the number of collisions, m'
the mass of a molecule, we have similarly
' e -
and we have also
2 Mm
TEMPERATURE. 93
Now we can make the average kinetic energy of the
plane what we please by giving it the proper initial velocity.
For our purpose we wish the plane to act as a transmitter
and not as a storer of energy, and it will do so if we give it
such an initial velocity that the mean kinetic energy of the
plane does not alter in unit time. If this is the case 8#
vanishes and we have by (100)
so that
m'N
~
mN'
b
(M+m) 2 '
Substituting these values for 20-^/3, and 20-0 2 /3 in
equations (98) and (99), we have
Thus if 2 is greater than W O l will increase and 2 will
diminish, and vice versa, and if O l is equal to 2 they will
remain equal ; thus the mean translatory energy behaves in
these respects exactly like temperature. There seems
nothing in the above illustration to restrict it to the case of
gases, and we should expect it would hold equally well for
solids or liquids.
94 DYNAMICS.
46. We are thus led to assume that part of the kinetic
energy of a system, whether that system be a portion of a
solid, liquid or gas, is proportional to the temperature.
Let us denote this part of the kinetic energy by
where u is a coordinate helping to fix the position or con-
figuration of a molecule. We see at once that there is an
essential difference between these coordinates and those we
have hitherto been considering and which fix the geometrical,
strain, electric and magnetic configuration of the system.
We have these latter coordinates entirely under our control
and subject to certain limitations imposed by the finite
strength of materials, the strength of dielectrics, and
magnetic saturation ; we may make them take any value we
please. We may therefore fitly call these coordinates con-
trollable coordinates. It is quite different, on the other
hand, with the coordinates fixing the separate moving parts
of the systems whose kinetic energy constitutes the tempera-
ture of the body. We can it is true affect the average
value of certain functions of a large number of these coor-
dinates, but we have no control over the coordinates indivi-
dually. We may therefore call these coordinates "uncon-
strainable" coordinates. Their fundamental property is
that we can not oblige any individual coordinate to take
any value which may be assigned. Since we have no power
of dealing with individual molecules, the "controllable"
coordinates must merely fix the position of a large number
of molecules as a whole.
If the term
involves any "controllable" coordinate <, then it is evident
TEMPERATURE. 95
that this coordinate <f> must enter as a factor into all the
terms in the form expressed by the equation
J {(uu)u* + ...} = J/(0) \(uu)'u* + ...} (101),
where the coefficients (uu)' do not involve < : otherwise the
phenomenon would be influenced more by the motion of
some particular molecule than by that of others. We
shall assume that 0, the temperature, is proportional to
that is that
= JC" {()** + } , (102),
where C does not involve any of the "controllable" coordi-
nates which fix the configuration of the system.
47. We may conveniently divide the kinetic energy of
a system into two parts, one depending on the motion of
"unconstrainable" coordinates, which we shall denote by
T u , and we shall assume that this is proportional to the
absolute temperature 6, the other depending on the motion
of the "controllable" coordinates, we shall denote by T c ,
T c corresponds to what v. Helmholtz in his paper on
11 Die Thermo dynamik chemischer Vorgange" ( Wissenschaftliche
Abhandlungen, n. p. 958) calls "die freie Energie." There
will not be any terms in the kinetic energy involving the
product of the velocities of an " unconstrainable" and a
"controllable" coordinate, otherwise the energy of the system
would be altered by reversing the motion of all the "uncon-
strainable" coordinates.
Let us suppose that <j> is a controllable coordinate which
enters into the expression for that part of the kinetic energy
which expresses the temperature, then if & be the. external
force of this type acting on the system we have by Lagrange's
equations, V being the potential energy,
96 DYNAMICS.
a_d_dT_dT dV
~ dt~d$ d<$>* d<$>'
Now T=T C +T H
dT u
and p- = o,
#(/>
so that
to_d_dT c _dT c __dT d_V
~ dt d$ d$ d<j> + d<t>"
Now by equation (103) T u is of the form
where (ww)' does not involve <, so that we have
dT u
and therefore
dV .
'"
differentiating this equation on the supposition that all the
controllable coordinates are constant and that the only
variable is the energy depending on the motion of " uncon-
trollable " coordinates, we have
and therefore by (104)
48. Now let us suppose that energy is communicated to
the system, partly by the action of the external forces on the
"controllable" coordinates, and partly through the "uncon-
strainable" coordinates: let the quantity of work commu-
nicated in the latter way be S<2- If the motion of the
DYNAMICAL EQUATIONS. 97
"unconstrainable" coordinates is that which gives rise to the
energy corresponding to temperature, SQ may be regarded
as a quantity of heat communicated to the system.
We have by the Conservation of Energy, if </> denotes a
" controllable " coordinate,
S7; + S7;+SF ......... (107).
Now *T e =$8<l> + :S<j> .......... (108),
I 9 a<f> )
and since T c is a quadratic function of the velocities of the
" controllable " coordinates, we have
and therefore
........... (109);
a<f> )
so that by subtracting (108) from (109) we get
Since the change in the configuration is that which
actually takes place in the time S/> we have
<8/=S0,
so that
ddT e dT\
-
and therefore equation (107) becomes
Now, if V be completely fixed by the controllable
coordinates, we have
T. D.
98 DYNAMICS.
So that
Substituting for <l> the value given by (103) we have
S<2 = 2S<A -~ + 8T U ( JI 3);
but by (106)
d _Tn = _r L
so that
uJ 4> constant
87; ......... (114).
Let us suppose that the quantity of work communicated
to the system is just sufficient to prevent T u from changing,
then
or
V d<f> } T U constant \dT u ) $ constant
Remembering that T u is proportional to the absolute
temperature 0, we see that equation (115) becomes
/dQ\ _
\d<j>/6 constant
dO
$. constant
where in finding d^fdO we must take care that is the only
quantity which varies.
In this form equation (116) is identical with the third
thermodynamical relation given in Maxwell's Theory of
Heat, p. 169, and v. Helmholtz in his paper "Die Thermo-
DYNAMICAL EQUATIONS. 99
dynamik Chemischer Vorgange" ( Wissenschaftliche Abhand-
hmgen, 2, p. 962) deduces this equation from the Second
Law of Thermodynamics and applies it to the case of
the variation of the electromotive force of galvanic cells
with temperature. The conclusions at which he arrives
have been verified by the experiments of Czapski (Wied.
Ann. 21, p. 209) and Jahn (Wied. Ann. 28, pp. 21, 491).
If 8(2 = o, that is if all the work done on the system is done
by means of forces of the types of the various controllable
coordinates, then we have by equation (114)
dT H ) $ constant \ d<$> /> constant
49. Since
we see by (113) that
or
^=28 log/ W + Slog T; ......... (118),
* u
so that
BQ
T u
is a perfect differential. This is analogous to the Second
Law of Thermodynamics, and we see by the analogy that
it shews that energy arising from the motion of quantities
fixed by "unconstrainable" coordinates can only be partly
converted into work spent in moving the quantities fixed by
the "controllable" coordinates. The amount which can be
converted follows laws analogous to those which regulate
the conversion of heat into mechanical work.
72
100 DYNAMICS.
In the preceding work we have assumed that the
potential energy of the system was not changed if the
"controllable" coordinates remained unchanged. When
however the system is a portion of a solid or liquid the
potential energy may by some alteration in the state of
aggregation be changed without there being any corre-
sponding change in the controllable coordinates. To
include this case we must suppose that V is a function of
the temperature as well as of the <'s, and that its value in
the neighbourhood of the temperature corresponding to a
change of state in the substance varies very rapidly.
In this case we have 6 being the temperature,
v dV dV
8F =^ + ^ 8 '
and instead of (114)
Since 80 and &T H vanish together we see that equation
(116) still holds. Equations (117) and (118) however require
modification. We have now (&Q-S V($> constant))/ 7L a perfect
differential instead of $Q/T, e .
50. Relations between heat and strain. We
shall now apply equation (116) to determine the effects due
to the variation of various physical quantities with tempera-
ture, and shall begin by considering the effects produced by
the variation of the coefficients of elasticity m and n with
temperature.
In equation (116) let us suppose that <J> is a stress of
type e, then using the same notation as in 35, we have
$ = m (e +f + g) + n(e -f-g\
dn . f .
THERMAL EFFECTS DUE TO STRAIN. IOI
So that by equation (116), 8Q, the heat which must be
supplied to unit volume of the bar to keep its temperature
from changing when e is increased by Be is given by the
equation
and thus if the coefficients of elasticity diminish as the
temperature increases, heat must be supplied to keep the
temperature of a bar constant when it is lengthened, and
hence if the bar is left to itself and not supplied with heat it
will cool when it is extended.
If $ is a couple tending to twist the bar about the axis
of x, we have, if a is the twist about that axis,
dn
and therefore by (116) 8(), the heat required by unit
volume of the bar to keep the temperature from changing
when a is increased by &a is given by the equation
so that if a rod which is already twisted is twisted still further
it will cool if left to itself, provided, as is usually the case,
the coefficient of rigidity diminishes as the temperature
increases.
The preceding results were first obtained by means of
the Second Law of Thermodynamics by Sir William Thomson
in his paper on the Dynamical Theory of Heat (Collected
Papers, Vol. i. p. 309).
51. Thermal Effects produced by Electrifica-
tion. Let us now consider the case when < is an electric
IO2 DYNAMICS.
force parallel to the axis of x, producing an electric displace-
ment / in that direction. In this case if K be the specific
inductive capacity of the dielectric, we have
_
)f constant ~ ~ K* dQ J '
so that 8(2, the heat which must be supplied to unit volume
of the dielectric in order to prevent its temperature changing
when the electric displacement is increased by S/j is by
(116) given by the equation
Some recent experiments made by Mr Cassie in the Caven-
dish Laboratory on the effect of temperature on the specific
inductive capacities of glass, mica and ebonite, have shewn
that the specific inductive capacity of these dielectrics
increases as the temperature increases, and that at about
30 C.
i dK , .
for glass,
i dK
T r --, = '0004 for mica,
li. uv
i dK
-^j:- = '0007 for ebonite.
A av
Thus the heat which must be supplied to unit volume of a
piece of glass to enable its temperature to remain constant
when it is electrified is by (123)
002 .
and this at 30 C.
THERMAL EFFECTS DUE TO MAGNETIZATION. IO3
/'
is the work supplied from electrical sources, hence in charg-
ing a Leyden jar, we see that the mechanical equivalent of
the heat absorbed by it during charging, if its temperature
remains constant, is about two-thirds of the work supplied
to it from electrical sources.
We see also by equation (123) that a piece of glass will
be cooled when it moves from a place where the electric
force is weak to one where it is strong.
52. Thermal effects of Magnetization. Let us
now suppose that $ is a magnetic force magnetizing a piece
of soft iron or other magnetizable substance to the intensity
/. Then if k be the coefficient of magnetic induction
so that
\d(t.
/constant </
And therefore &Q the heat which must be supplied to
unit volume of the magnet to keep its temperature con-
stant when the intensity of magnetization is increased by
87 is by equation (116) given by the equation
so that if the coefficient of magnetization decreases as
the temperature increases then a magnet will get heated
when its intensity of magnetization is increased, and there-
fore when it moves from weak to strong parts of the
magnetic field. This was pointed out by Sir William
Thomson in the paper just quoted.
IO4
DYNAMICS.
The experimental investigation of the heating effects
produced by the motion of magnetizable bodies in variable
magnetic fields is rendered difficult from the heating effect
produced by the electric currents induced in the magnet by
the alteration in the number of lines of magnetic force pass-
ing through it. '
Another thing which would increase the difficulty is
the phenomenon called by Ewing hysteresis (Experimental
Investigation on Magnetism, Phil. Trans. 1885, p. n).
This causes the intensity of magnetization to depend not
only on the strength of the magnetic force, but also on the
previous magnetic history of the substance : so that the
curve representing the relation between intensity of magneti-
zation (ordinate) and magnetic force (abscissa), as the mag-
netic force goes through a complete cycle, will be of the
kind shewn in the accompanying figure, and will enclose
a finite area, indicating the dissipation of a finite quantity
Of energy proportional to the area of the curve, and this
dissipated energy will appear as heat.
Experiments made on the effects of temperature upon
THERMAL EFFECTS DUE TO MAGNETIZATION. IO$
the coefficient of magnetization of iron have shewn that
these are rather complex. Baur (Wiedemann's Elektridtdt.
iii. p. 750) from his experiments on this subject has arrived
at the following results.
The influence of temperature upon the magnitude of the
coefficient of magnetization depends upon the magnitude of
the magnetizing force.
The coefficient of magnetization increases with the
temperature if the magnetizing force does not exceed a
certain value, but when the magnetizing force exceeds this
value the coefficient of magnetization diminishes as the
temperature increases.
The smaller the magnetizing force the greater the influ-
ence of temperature upon the coefficient of magnetization.
Taking these results in conjunction with equation (125)
we see,
1. That when a magnetizable body moves in a magnetic
field where the force is everywhere less than the critical
value, its temperature will tend to fall when it moves from
places of weak to places of strong magnetic force and vice
versa.
2. That when the body is placed in a magnetic field
where the magnetic force is everywhere greater than the
critical value, its temperature will rise when it moves from
places of weak to places of strong magnetic force and vice
versa.
The coefficient of magnetization of nickel diminishes as
the temperature increases, so that a piece of nickel will get
warmer when it moves from a weak to a strong part of the
magnetic field. The coefficient of magnetization of cobalt
on the other hand increases as the temperature increases, so
that a piece of cobalt will cool as it moves from a weak to a
strong part of the magnetic field.
CHAPTER VII.
ELECTROMOTIVE FORCES DUE TO DIFFERENCES
OF TEMPERATURE.
53. WE shall now go on to consider various cases in
which inequalities of temperature in a substance give rise
to electromotive forces.
Sir William Thomson has shewn that when a current
of electricity flows along an unequally heated bar it carries
with it as it flows from a hot to a cold place either heat
or cold : heat if the bar is made of brass or copper, cold
if it is made of iron. Sir William Thomson expressed this
result by saying that the specific heat of electricity in copper
and brass is positive, since the electricity in this case carries
heat with it just as if it were a real fluid possessing specific
heat ; the " specific heat " of electricity in iron on the other
hand is said to be negative, since electricity in iron behaves
with regard to heat in the opposite way to a fluid possessing
specific heat.
It follows from this result, by the consideration of the
reciprocal relations, that electromotive forces must be de-
veloped in any conductor the temperature of which is not
uniform throughout. We shall now endeavour to find what
terms in the Lagrangian function these phenomena corre-
THERMOELECTRICITY. I O/
spond to, or rather we shall shew that if there was a certain
term in the Lagrangian function an unequally heated body
would exhibit similar phenomena.
Let us suppose that in the term
i{(uu)u 2 +...},
which expresses the part of the kinetic energy of unit
volume of the substance due to sensible heat, the coeffi-
cients (uu) are functions of
d , ... d , , d . 7 .
i^-n + Ty^*-^^
where <r x , <r y , <r z are quantities not explicitly involving
f, g, h, the quantities of electricity which have passed
through planes of unit area at right angles to the axes.
x, y, z respectively
Let us write for the sake of brevity
d . .. d , N d , ,.
<./) + <ow) + a M)Bc
Then, since / g, h are controllable coordinates, and
by hypothesis (uu) involves e, we may write
${()+...}=/()${()+...},
where f(e) denotes some function of e. The coefficients-
(uu)' are supposed not to involve /, g, h explicitly.
By Hamilton's principle any term in the Lagrangian
function indicates the existence of effects which are the
same as those which would be produced by electromotive
forces parallel to the axes of x, y, z, and equal to the
coefficients of Sf, 8g, h which this term contributes when
the variation of the Lagrangian function is taken.
The term we are considering is, taking the whole
volume,
"f/WJ
IIP
108 DYNAMICS.
When c is increased by Se the alteration in this term is
() {()' if + ...} dxdydz ...... (126).
Since 8. = M/) + (o>%) + (<r.M),
we see that if we integrate (126) by parts the terms
in S/"are
Since
is proportional to the temperature, we may put
and then (127) may be written
So that by Hamilton's principle if X be the force per
unit length which would produce the same effects as this
term indicates
To take the simplest case let us suppose that /(e) is a
linear function of , so that
As ^e/ is the alteration in the energy made by the
electrification, it can only be a small quantity, so that we
THERMOELECTRICITY. 109.
have approximately
and therefore
or if b and a remain constant throughout the substance,
So that this term indicates the existence of an electromotive
force parallel to x and proportional to the rate of alteration
of the temperature in that direction. If Y and Z are the
electromotive forces parallel to y and z respectively we
have
'**- dy
bf$ dO
Z = a z - -
a dz
The occurrence of Bf in the surface integral shews that
there is a discontinuity in the potential at the surface of
separation of two media and that the potential in the first
medium is higher than that in the second by
where the suffix attached to the bracket indicates the medium
for which the value of the quantity inside the bracket is to
be taken.
54. Thermal effects of this term. Let us suppose
that 8(2 is the quantity of heat that must be supplied to-
unit volume of the conductor to keep its temperature from
changing when a quantity of electricity 8f flows through it,,
that is when /is increased by 8f.
IIO DYNAMICS.
We see by equation (113) that
8(2 = the increase in T u when / is increased by 8/
bft dO s _ ,
= - <r x -=-_ Sf by equations 127 and 129.
If u be the current parallel to x and 8/ the time it has
l3een flowing
8/ = z/8/,
-so that BQ = -a- x -j-ut ( I 3)-
If the current flows in the direction in which heat is
flowing, that is from hot to cold, 8(? will have the same sign
as vjb, since J3 and u are necessarily positive. Hence if
vjb is positive heat must be supplied to unit volume to keep
the temperature constant when a current flows into it from
a hotter place, that is a current from a hot to a cold place
carries cold with it, so that in this case the electricity
behaves as if it had a negative specific heat. Hence vjb is
of the opposite sign to the specific heat of electricity in the
substance.
We see from equation (130) that the electromotive
force at any part of the circuit always tends to produce a
current in the same direction as one which would cause a
fall in temperature at this part of the circuit.
If we produced a distribution of electricity throughout
the volume of a body, some very peculiar results would
follow if this term existed.
Let us take the case of an isotropic body whose
temperature is uniform, then we may suppose that a-*, o^,,
<r x are each equal to <r and independent of x, y, z, then
d i~ f\ , d /- ~\ , d t- i\ - ~ S4f , * , dh
1 : dy dz
THERMOELECTRICITY. I II
but if p be the volume density of the distribution of
electricity,
df dg dh_
So that the energy in unit volume corresponding to the
heat energy equals
*)' <*+} ............. (131),
and thus when we alter the volume density of the electricity
we alter the energy due to the heat and therefore the
temperature.
To calculate the amount and even the sign of this
alteration in temperature we must observe that u... will be
altered if we suddenly alter p. The case is quite analogous
to that of a moving body the effective mass of which is
suddenly increased, we may suppose, by the tightening of a
string attached to another mass. In this case it is the
momentum of the system and not its velocity which remains
constant.
If we express the term (131) in terms of the momenta
z>j, v z ... corresponding to the various coordinates u^ u z , we
see, since
dT
that it will be of the form
where f(v lt v 2 --) denotes a quadratic function of z/ x , v 2
&c., which does not involve p. As this expression is pro-
portional to the temperature 0, we see that if p be suddenly
increased by Sp, the increase BO in the temperature is given
112 DYNAMICS.
by the equation
80
so that if bo- is positive the temperature of the body is
increased by communicating a charge of electricity to it,
that is the electricity behaves like a body whose specific
heat was negative. But we saw that bo- was of the opposite
sign to what Sir William Thomson has defined as the
specific heat of electricity in the substance. Hence we
see that the analogy between the behaviour of electricity
and that of a fluid possessing either positive or negative
specific heat can be extended to cover the case when
a bodily charge of electricity is communicated to the
body.
We can shew however that if the charge of electricity be
of the same order of magnitude as those which occur in
electrostatic phenomena this heating effect must be ex-
tremely small. For multiplying both sides of equation (132)
by /?, we have
($W fibo-
%- = - op, approximately.
u a
Now fSba-ja is by equation (130) the "specific heat" of
electricity. The value of this for antimony at the tempera-
ture 27C. is (see Tait's Heat, p. 180) about io~ 2 x 300
when the unit is io~ 5 of the E. M. F. of a Grove's cell. As
the E. M. F. of a Grove's cell is about 2 x io 8 in absolute
measure the "specific heat" of electricity in antimony in
absolute measure will be about 6000.
We must now find a limiting value for 8p. Let us sup-
pose that electricity is uniformly distributed through a sphere
of radius r, then if p be the density of the electrical distribu-
THERMOELECTRICITY. 1 1 3
tion, K the specific inductive capacity, the force just outside
the sphere is
Now the greatest value this can have in air is (see
Everett's Units and Physical Constants, p. 142) about 4 x io 12 ,
so that a limiting value of p will be given by
Now
so that P = - o- approximately.
9 x zoV J
Hence substituting this value of p for Sp, we get at the
temperature 27C.
9 x io x r
2
Now (3S6 is the mechanical equivalent of the heat
available for changing the temperature, so that the change
in temperature will be of the order
since 4-2 x io 7 is the mechanical equivalent of heat.
Thus the change of temperature which can be produced in
this way by any statical charge of electricity is infinitesimal.
55. Thermoelectric effects of strain. If the
quantity b in the expression for / (e) is a function of the
strain in the wire along which the current is passing, then
putting b=f (e\ where e denotes a strain in the wire, we
T. D. 8
1 14 DYNAMICS.
see by equation (128) that at each point of the wire there
is the electromotive force
acting along the wire, ds being an element of its length.
Now if we have a closed circuit made of one metal, in
which <r x may vary with the temperature and state of strain,
then the integral of the expression taken round the circuit
will vanish if either or e is constant all along the circuit
but will not in general vanish if both and e vary round the
circuit. So that we cannot produce currents in a wire
whose temperature is constant by any variation in the strain,
nor in a wire where the strain is constant by any variation
in the temperature, while on the other hand we should
expect to get currents if both the strain and the tempera-
ture were variable. All these results agree exactly with
experiment, and hence we are led to conclude that b is a
function of the strain.
If this is so then communicating a volume distribution
of electricity to an unequally heated rod must tend to strain
it.
For let us suppose that the strain e is an extension of a
wire, then if a be the displacement of a point along the
wire
_da
~Js'
If a be increased by Sa, the coefficient of <k in the
change in the Lagrangian function is, when the medium is
isotropic
and therefore, by Hamilton's principle, the effects due to
PELTIER EFFECT. 1 15
this term are the same as would be produced by an external
force equal per unit length to (133) tending to strain the
wire.
Thus when an unequally heated wire has electricity dis-
tributed throughout the volume there will be stresses tending
to strain the wire.
If we consider twist instead of elongation we can show in
a similar way that an unequally heated wire will be twisted
when electricity is distributed through it.
56. The electromotive force in a thermoelectric circuit
is generally calculated from the heat developed in various
parts of the circuit by the passage of the current. The
amount of knowledge of the electromotive force which we
can derive from thermal considerations is however limited
in a way which I think is generally overlooked.
We see by 47 that when a coordinate x is increased
by #, the heat 8Q which must be supplied to the system
to prevent its temperature from changing is given by the
equation
where X is the force of type x acting on the system.
Now let X be an electromotive force in a thermoelectric
circuit and x a quantity of electricity, then we see by (114)
that from considerations about the heat developed we can
only derive information about the part of the electromotive
force which depends upon temperature and cannot tell
anything whatever about any other part.
As a particular application of this principle we see that
the Peltier effect can throw no light on the absolute differ-
ence of potential between two different metals and hence
there is nothing in the phenomena of thermoelectricity to
force us to attribute the large difference in potential ob-
82
1 1 6 DYNAMICS.
served by Volta between two different metals in contact
to chemical action between them and the surrounding
medium.
57. Electromotive forces produced by inequal-
ities of temperature in a magnetic field, v. Etting-
hausen and Nernst (Wiedemanri s ;47ma/en, xxxi. 737 and
760, 1887) have recently discovered an electromotive force
due to inequalities in temperature which is very analogous
to the Hall effect. They found that when heat is flowing
across a thin plate made of a substance which can conduct
electricity, electromotive forces are produced in the plate, if
it is placed in a magnetic field. The direction of the elec-
tromotive force is at right angles both to the magnetic force
and to the direction in which the temperature is changing
fastest. The magnitude of the electromotive force is pro-
portional to the product of the magnetic force into the rate
of increase of the temperature at right angles to the lines of
magnetic force.
This electromotive force is especially large in bismuth.
If represents the temperature, and a, ft, y the com-
ponents of magnetic force parallel to the axes of x, y, z
respectively, the components of the electromotive force due
to this effect will if the laws we have just quoted are true be
given by the expressions
where Q is a quantity which has very different values in
different substances. The results of Nernst's determinations
ELECTROMOTIVE FORCE IN MAGNETIC FIELD. 1 1/
of this quantity (Wied. Ann. xxxi. 775), are given in the
following table
Bismuth
- 'IS 2
Antimony
-00887
Nickel
- -00861
Cobalt
'00224
Iron
+ '00156
Steel
+ -000706
Copper
+ -000090
Zinc
+ '000054
Silver + '000046
We shall now proceed to see what term in the Lagran
gian function would give rise to forces of this kind.
Let us consider the term
III 6 {T X
where / g, h are the components of the electric displace-
ment parallel to the axes of x, y, z respectively.
The variation of this term when /is increased by 8/is
JJS/Q 6 ($n - ym) dS- jjj S/Q (fl J - y |) d x dy tk,
where /, m, n are the direction cosines drawn outwards of
the normal to the surface enclosing the volume through
which the integrals are extended.
By Hamilton's principle the term in the surface integral
indicates that if we draw any circuit in the field then when
this circuit crosses the boundary of two media there is an
Il8 DYNAMICS.
electromotive force whose components parallel to the axes
of x, }', z are respectively
where corresponding quantities in the media (i) and (2) are
indicated by affixing the suffixes i and 2 respectively to the
symbol representing the quantity, and /, m, n are the direc-
tion cosines of the normal drawn from medium (2) to
medium (i).
By the same principle the terms in the volume integral
indicate the existence of an electromotive force throughout
the body whose components per unit length parallel to the
axes of x, y, z are
dB
d&_ t_
these are the expressions for the components of the electro-
motive force discovered by v, Ettinghausen and Nernst.
These forces do not satisfy the solenoidal condition ; they
will therefore produce a distribution of electricity throughout
the substance whose volume density p is given by the equa-
tion
-*L\lLo( d A_a d ^\ d oi dB --
d ( dB i
where K is the specific inductive capacity of the substance,,
thus,
THERMAL EFFECTS. 119
KQ$d6 (dft dy\ d6 fdy da\ dO (da. dfi^
or neglecting Q 2
dO dO d
where ?/, z/, w are the components of the current.
58. Thermal phenomena arising from this term.
We can see by equation (113) that H the heat required by
unit volume to prevent the temperature from changing
when a quantity of electricity Sf passes through it parallel to
the axis of x is given by the equation
8ff = (that part of the electromotive force which
arises from the part of the energy corre-
sponding to the sensible heat) 8f;
thus the part of $7? which arises from this term is given by
so that when quantities of electricity Bf t $g, S/z pass parallel
to the axes of x, y, z respectively then
*TT nil d Q dff \*f ( d& d \* ^fn d6
-f= Q {\y-j- ft -r)f + \ a ^r ~y^~ %" + (P7r~^r
^\\ r dy ^ dzJ \ dz r dxj 6 \ dx dy
or, if u, v, w are the components of current parallel to the
axes of x, y, z respectively and 8t the time the displacement
takes, then since
S/=z/S/; $g=vto; U = w&,
we have
120 DYNAMICS.
If X, Y, Z are the mechanical forces acting on unit
volume of the conductor arising from the action of the
magnetic field on the currents flowing through the volume
X=n{yv-Pw}
Y= JJL {aw yu]
Z=p{pu-av},
where /x is the magnetic permeability, combining these
equations with (135) we see that
dx dy
So that if the action of the mechanical force on the
current tends to make the substance conveying the current
move in the direction in which heat is flowing, then when
Q is negative, heat must be abstracted from the substance to
keep its temperature constant when currents of electricity
flow through it. And the heat which has to be supplied in
unit time to unit volume to prevent the temperature from
changing is given by the equation
SZT= - Q {resultant mechanical force on
/*
unit volume x flow of heat x
cosine of the angle between
these two quantities}.
Heating effects in a magnetic field have been detected
by v. Ettinghausen (Wiedemann's Annalen, xxx. pp. 737
760, 1887).
59. Magnetic effects of this term. If we apply to
the term
dxdydz
MAGNETIC EFFECTS.
121
the same method as the one which in 43 we applied to
the term corresponding to Hall's effect, we shall see that it
involves the existence at a point , 17, where there are no
electric displacements of a magnetic force whose components
parallel to the axes of x, y, z are respectively
where
dg dh\ d i
dlC, '
dh df\ d i
df
+ -f
dy
de\ d i~l ,
- -/- -T- - dxdydz,
dx) dz r J
where r is the distance between the points x, y, z and
ft ^ fc
If ^, 77, is a point at which there is an electric dis-
placement, then as before the components of the magnetic
force parallel to x, y, z respectively are
(136).
We see that the equation
-
dc,
is no longer true, but that now
4?r _ (dO fdf d? dh\
47TU Q < ( + -* + __ ) _
3 ^ \d\d drj dJ
122 DYNAMICS.
It follows from equation (136) that if dielectrics as well
as conductors exhibit the phenomenon discovered by v.
Ettinghausen and Nernst then a steady electric displacement
through a heated dielectric may produce magnetic forces.
A numerical calculation similar to that in 43 will show
however that these forces are exceedingly small.
60. Thermal effects accompanying changes
in magnetization, arising from this term. Since
the magnetic forces expressed by equation (136) arise from
that part of the kinetic energy which corresponds to the
sensible heat, changes in the intensity of magnetization
must by equation (114) be accompanied by reversible
thermal effects. If the intensities parallel to the axes of
j?, y, z be increased by 8/4, SZ?, BC respectively, then by
equation (114) SZ7 the mechanical equivalent of the heat
which must be supplied to unit volume to prevent its
temperature from changing is given by the equation
61. Rotation of the plane of polarization pro-
duced by the flow of heat. Rowland has shown that
if Hall's effect exists in dielectrics, then, according to Max-
well's Electromagnetic Theory of Light, the plane of polariza-
tion of plane polarized light will be rotated when the
dielectric through which the light is passing is placed in
a magnetic field the lines of force of which are more or
less parallel to the direction of propagation of the light.
We shall now proceed to investigate whether the existence
of v. Ettinghausen's and Nernst's phenomenon will produce
EFFECT ON VELOCITY OF LIGHT. 123
a rotation of the plane of polarization when a ray of plane
polarized light passes through a dielectric through which
heat is flowing.
Let us suppose that we have a circularly polarized ray
of light travelling parallel to the axis of z through a
dielectric in which there is a uniform flow of heat also
parallel to z.
Let / and g be the electric displacements parallel to
the axes of x and y respectively, ^and G the components
of the vector potential parallel to x and y respectively,
and X and Y the components of the electromotive force
parallel to these axes, then since dQjdx, dOjdy both vanish
dF d6
v dG
Y= -
where a and /3 are the components of the magnetic force
parallel to x and y respectively.
Hence if K be the specific inductive capacity
47r f _dF _ O ndQ\
47r ^ dG dO
Differentiating the first of these equations with respect
to / we get
~K~dt^~~df ~^Wdz ( I37 )
but in a dielectric
df = _d*F
dt dz 2 '
124 DYNAMICS.
and in the small term
^ dt dz '
if we neglect Q 2 we may put
dj^ _d*F
^~dt~ dtdz '
where /A is the magnetic permeability of the dielectric.
Substituting these values in equation (137) we get
j_d*F _d z F QdQd 2 F
H.K dz 2 ~ df + ~fJi dz dtdz '
For a circularly polarized ray we may put
4 27J " / .
F= A sin -r- (vt - 2)
where Fis the velocity of the light, and X its wave length.
Substituting this value for F in equation (138) we get
J-=,*-e^,
and thus the velocity of the ray is greater than if the
temperature had been uniform by
1 QdO
2 ^ dz'
The velocity of propagation of a ray circularly polarized
in the opposite sense will also be increased by the same
amount. So that regarding a plane polarized ray as made
up of two rays circularly polarized in opposite senses we
see that when such a ray passes through a medium in which
there is a steady flow of heat, the plane of rotation will
not be rotated, but the velocity will be increased by
1 Q <M
2 dz'
EFFECT ON VELOCITY OF LIGHT. 12$
Thus even if we had a transparent substance for which
Q was as great as for bismuth, viz. '13 and a fall of tempe-
rature of iooC. per millimetre, the change in the velocity
would only amount to
l x -13 x io 3
or 65,
this change is only about 2' 2 x io~ 7 per cent, of the velocity
of light, and violet light would have to traverse about
20.000 c.m. to gain or lose a wave length. This effect
therefore is much too small to be detected experimentally.
We saw by equation (135) that when electric displace-
ments take place in a field in which the temperature is not
uniform, heat is absorbed or evolved, so that we should
expect thermal changes to accompany the propagation of a
ray of light through a medium the temperature of which
was not uniform.
By equation (135) the heat SZf which must be supplied
in unit time to unit volume of the medium to prevent the
temperature changing, is if the heat is flowing along the
axis of z, given by the equation
Let us take the case of a plane polarized ray for which
approximately
g=0
a = o
cos (vt - z),
A
126 DYNAMICS.
thus
. ^ <x . A CIV 27T . 27T . . x
tH(2 x - -v- COS-T- (vt z) sm^ (vt-z).
A <z A A
So that the propagation of light along an unequally
heated medium would if this theory is correct be accom-
panied by periodic emission and absorption of heat, analogous
to that which accompanies the propagation of sound according
to Laplace's theory. According to Maxwell (Electricity and
Magnetism, Vol. n. p. 402) the maximum value of /3 for
strong sunlight is '193 so that
^TrvA - '193,
and therefore
let us take X as 3-9 x io~ 5 the wave length of the violet ray
H in air
H= 5 x io 2 x Q x cos (vt- z) sin-^ (vt-z).
(tz A A
If Q were as large as it is for bismuth, i.e. '132, and
there was a fall of 100 C. in one centimetre, then the
maximum amount of heat absorbed or emitted would be
3-3 x io 3 ;
this would correspond to changes of temperature of not
more than one ten thousandth of a degree centigrade, if
the specific heat of the substance were as great as that of
water.
62. Longitudinal effect, v. Ettinghausen and
1 Ernst found that in addition to the transversal electromotive
force there was a longitudinal one along the lines of flow of
the heat, which was not reversed when the magnetizing force
was reversed, and which was proportional to the square
of the magnetizing force as long as this was small. This
THOMSON EFFECT. I2/
shows that the quantity o- which we considered when we were
discussing the Thomson effect in 51 is a function of the
square of the magnetic force. If we consider the effect of
this term magnetically we shall see that it indicates that the
magnetic permeability of a magnet will be affected by the
proximity of a conductor throughout which electricity is
distributed.
63. It is interesting, because suggestive of new physical
phenomena, to trace the consequence of the existence in the
Lagrangian function of terms, which are symmetrical func-
tions of /, g, /i, a, (3, y and their differential coefficients,
such as terms proportional to
da. d(3 dy
* dx + *dy + l ~dz'
df O dg dh
a jr + P -T~ + y-T)
dx dy ' dz
, fdft dy\ /dy da\ /da dft\
* \dz~~dy) +g \Ix~ dz) + \dy~~dx)'
/tf73 _ ^ 7 \ fdg dh\ /dy da\ fdh df\
\dz dy) \dz dy) \dx dz) \dx dz)
dfi\ /df dg\
<~7x)\4y~dx)'
The reader however who is interested in this will have
no difficulty in tracing the consequences of these terms by
the methods already given.
CHAPTER VIII.
ON "RESIDUAL" EFFECTS.
64. THERE are a great many cases in which the appli-
cation of forces to a body seems to produce a change in
it, from which it does not recover for some time after the
forces have been removed.
Thus, for example, if we keep a metal wire or glass
fibre twisted for some time, it will not when the twisting
couple is removed at once vibrate symmetrically about its
original position of equilibrium, but will oscillate about a
new zero which gradually approaches the old one, the
maximum difference between the temporary and the true
zero and the time which elapses before these coincide
increasing within certain limits with the duration of the
original twisting couple.
Phenomena of this kind are called in German treatises
" elastiche nachwirkung." This peculiar effect of torsion
does not seem to have received a name in this country,
but the analogous cases in electricity and magnetism are
called respectively "residual charge" and "residual mag-
netism." This latter effect is only partly analogous to that
of the twisted wire, as they differ in one very important
respect, that of permanence. In the case of the twisted
RESIDUAL EFFECTS. I2Q
wire the effect of the previous torsion will disappear if
time be given to it, but soft iron if kept free from dis-
turbance seems to be able to retain its magnetism for any
length of time.
We shall now endeavour to find a dynamical analogue
to the case of the twisted wire. Let us suppose that we
have a frictionless machine whose configuration is fixed
by one coordinate x and that this is connected with another
machine fixed by the coordinate y, the motion of this
machine being resisted by a frictional force proportional
to the velocity. We shall suppose at first that the mass of
the second machine is so small that its inertia may be
neglected, and that the connexion between the two machines
is expressed by the existence in their Lagrangian function
of a term/(jcj) which involves both x and y, but not their
differential coefficients with respect to the time. Then if
the force X acting on the first machine is the only external
force acting on the system, the equations of motion will be
of the form
72 7
If x and y are small we may put
where a, /?, y are constants.
Making these substitutions we have
T. D.
130 DYNAMICS.
A -jjz + (p-a.)x-py--=X (i42) r
dy
*4 + (*-y)y-P*=* (143)-
The solution of (143) is
7 = 7 c b xdt'
D J o
substituting this value of y in equation (142) we get
We see by this equation that the effect on the first system
of its connexion with the second is to make the forces called
into play at any time by the displacement of the system
from its position of equilibrium depend not merely upon the
displacement of the system at that time but also upon the
previous displacements, and that a displacement x lasting for
a short time r produces after a time T a force TX$ (T)
where
02 _ (<L-V) T
T
Neesen ("Elastiche Nachwirkung bei Torsion," Berlin
Monatsberichte, Feb. 12, 1874, p. 141) has shown that the
assumption that \l/ (T) is proportional to C kT agrees with his
experiments on the twisting of wires. Boltzmann (Sitz. der
k. Akad. zu Wien, 70, p. 275, 1874) works out a theory
where \jr (T) is proportional to i IT.
In many cases we are given the forces at the time f and
not the displacement, and in these, equation (145) is not
convenient. If as is generally the case the motion is so slow
that we may neglect the effects of inertia, then we have
RESIDUAL EFFECTS. 131
so that
and therefore
where = ~
and thus when the external force is removed
/3 2
/3 2 ft
*=,, v. I
*0*- a ) Jo
If the primary machine had been connected with several
secondaries instead of with only one, we should have, if the
displacements are given
2 ~ y
and if the forces are given
where TJT is written for /3 2 / (/A a) 2 , and the sum taken
for all the secondary systems. This is the general expression
for the residual effect in terms of the forces acting on the
system when it was under constraint.
If we cannot neglect the inertia of the secondary system
we must introduce the term BcPyldf into equation (143) so
that that equation becomes
92
132 DYNAMICS.
of which the solution corresponding to (144) is
where \ and \ a are the roots of the equation
jB\ 2 + b\ + a - y = o.
Thus the introduction of inertia into the secondary
system does not change the form of the solution, it only
introduces fresh terms of the same type as those which
previously existed, and the general solution is of the form
where c is a constant which depends on the constitution of
the secondary system but not upon x. This is the general
expression for the residual effect in terms of the initial
displacements.
65. In those cases in which residual effects occur we
may suppose that the secondary systems which are affected
by the changes in the primary are the molecules of the body
which is the seat of the phenomenon or a portion of such
molecules. For example in the case of the residual charge
of the Leyden jar we may look upon the electrical system
as the primary system and the system consisting of the
molecules of the glass as the secondary system, and may
suppose that during the actions of the electromotive force
on the glass, the arrangement of the molecules of the glass
suffers gradual changes which react upon the electric dis-
placement.
MAXWELL ON THE " CONSTITUTION OF BODIES." 1-33
The following extract from Clerk-Maxwell's article on
the "Constitution of Bodies" in the Encyclopedia Britannica
is most instructive on this point.
"We know that the molecules of all bodies are in motion.
In gases and liquids the motion is such that there is nothing
to prevent any molecule from passing from any part of the
mass to any other part ; but in solids we must suppose that
some, at least, of the molecules merely oscillate about a
certain mean position, so that if we consider a certain group
of molecules, its configuration is never very different from a
certain stable configuration about which it oscillates.
"This will be the case even when the solid is in a state of
strain provided the amplitude of the oscillations does not
exceed a certain limit, but if it exceeds this limit the group
does not tend to return to its former configuration but
begins to oscillate about a new configuration of stability,
the strain in which is either zero or at least less than in the
original configuration.
"The condition of this breaking up of a configuration
must depend partly on the amplitude of the oscillations and
partly on the amount of strain in the original configuration \
and we may suppose that ditferent groups of molecules,
even in a homogeneous solid, are not in similar circumstances
in this respect.
"Thus we may suppose that in a certain number of
groups the ordinary agitation of the molecules is liable to
accumulate so much that every now and then the configura-
tion of one of the groups breaks up, and this whether
it is in a state of strain or not. We may in this case assume
that in every second a certain proportion of these groups
break up and assume configurations corresponding to a
strain uniform in all directions.
134 DYNAMICS.
" If all the groups were of this kind, the medium would
be a viscous fluid.
" But we may suppose that there are other groups, the
configuration of which is so stable that they will not break
up under the ordinary agitation of the molecules unless the
average strain exceeds a certain limit, and this limit may be
different for different systems of these groups.
"Now if such groups of greater stability are disseminated
through the substance in such abundance as to build up a
solid framework, the substance will be a solid which will
not be permanently deformed except by a stress greater
than a certain given stress.
"But if the solid also contains groups of smaller stability
and also groups of the first kind which break up of them-
selves, then when a strain is applied the resistance to it will
gradually diminish as the groups of the first kind break up,
and this will go on till the stress is reduced to that due to
the more permanent groups. If the body is now left to
itself, it will not at once return to its original form but will
only do so when the groups of the first kind have broken up
so often as to get back to their original state of strain.
" This view of the constitution of a solid, as consisting of
groups of molecules some of which are in different circum-
stances from others, also helps to explain the state of the
solid after a permanent deformation has been given to it.
In this case some of the less stable groups have broken up
and assumed new configurations, but it is possible that others
more stable may retain their original configurations, so that
the form of the body is determined by the equilibrium
between these two sets of groups ; but if on account of rise
of temperature, increase of moisture, violent vibration, or
any other cause, the breaking up of the less stable groups is
facilitated, the more stable groups may again assert their sway
RESIDUAL EFFECTS. S 135
and tend to restore the body to the shape it had before its
deformation."
66. Let us now apply equation (146) to a definite case.
Let us suppose that the force X l acts on the system from
/= o to t= T^ and that from /= 2* to /= T 2 , the force - X 9
acts, then we have by equation (146)
= w
J
- k V-^ Xdf
*/ l Tl
If the primary system is connected with several seconda-
ries instead of one then we have
{-^-n)- -*(*- 21)}.. .(148).
,
We see from equation (147) that if we have only one
secondary x will never change sign, but that the system will
return slowly to its position of equilibrium and never get
beyond it whatever may have been its previous history. We
know however that in the case of residual torsion of glass
fibres and the residual charge of a Leyden jar the residual
effects may be made to change sign. Thus if we give a fibre
a strong twist in the positive direction for some considerable
time and then a twist in the negative direction for a short
time, the residual torsion after the twisting couple is taken
off may be first in the negative and then in the positive
direction. This is sometimes expressed by saying that the
residual charges come out in the inverse order to that in
which they went in.
136 DYNAMICS.
If there are only two values of /, k v and / 2 , then since
x = 6~ V f* mXcWdf +-*** ('mX&Jdt (149)
Jo Jo
and since after the external force is removed
/:
is not a function of /, we see that the sign of the residual
effect can only change once however complicated the
alternations in the signs of the twists or electrifications
previously applied to the system may have been.
Dr John Hopkinson represented the residual charge of
a Leyden jar by a formula of the same type as (149) (see
Chrystal's art. Electricity, Encyclopedia Britannica, p. 40)
and he showed that for the formula to agree with his
experiments on the residual charge in glass it was necessary
to take more than two values of k. Now when we included
the effects of the inertia of the secondary system we got two,
but only two values of k for each secondary, so that as we
have to introduce more terms than two to represent the
residual effect in glass we must have more than one second-
ary system. This is an indication that glass is not a homo-
geneous substance but a mixture of different silicates.
According to Neesen (loc. cit) the residual effects of
torsion in silk and guttapercha fibres can be represented
by a single term of the form c*~ k *.
67. We shall now investigate another effect due to the
same cause as the residual effect but of a different kind.
This is the effect of the secondary system on the way in
which the free vibrations of the primary die away.
Using the same notation as before and neglecting the
inertia of the secondary system we have for the free vibra-
tions the equation
RESIDUAL EFFECTS. 137
or say
eliminating y we have
As this is a linear equation let us assume that x varies
as e^, then/ is given by the equation
or A ? JF < f = ip^t-
The right hand side of this equation is small, and if the
residual effect does not produce a large change in the period
of vibration we may on the right hand side of the equation
substitute for/ its value when there is no secondary system,
i.e. i {p-/A}* and for ', a ; making these substitutions we have
A A a 2 +
or p~i\^-\
So that if /5 2 be small
. I/*'
- ' G
138 DYNAMICS.
And thus the real part of p is approximately
i W
2 Aa 2 + nl> 2 '
the amplitude of the vibrations of x are thus given by the
expression
where exp (x) = e x .
So that the ratio of the amplitudes of two successive
swings is
where T is the time of a complete oscillation, and is
given by the equation,
T 27r{A/fj} 2 approximately.
Substituting this value of T in (151) we get for the ratio
of the two amplitudes the expression
exp
r
/
(
\
~j~ .
Aa 2 + fib 2 )
Now if the motion of x were resisted by a frictional
force proportional to the velocity, the equation for x would
be
d 2 x dx
the solution of which is
_\t , , A 2 \2
>S \V4~4^Tv
where C and c are constants.
The ratio of the amplitudes of two successive swings
in this case is
' \T\
2AJ '
RESIDUAL EFFECTS. 139
or approximately
When the decrease in the amplitude is due to the
connexion with the secondary system, the ratio of two
successive amplitudes is
:so that the logarithmic decrement when the resistance is
frictional varies as
when it is due to the secondary system it varies as
Hence we see that if the mass of the vibrating body is
altered, the variation of the logarithmic decrement will be
less in this case than it would if the decay in the oscillations
were due to friction. This agrees with the results of Sir
William Thomson's experiments on the decay of the tor-
sional vibrations of wire, as he found that the loss was
greater with the longer periods than that calculated ac-
cording to the law of square roots from its amount in the
experiment with shorter periods. In fact if A were much
smaller than /^ 2 /# 2 the rate of decay would be increased
instead of diminished by increasing the vibrating mass, as
ihe rate of decay has its maximum value when A
CHAPTER IX.
INTRODUCTORY TO THE STUDY OF REVERSIBLE
SCALAR PHENOMENA.
68. So far we have been dealing with phenomena in
which as in ordinary dynamics the quantities concerned
were mainly of a vector character. We shall now how-
ever go on to consider phenomena when the quantities-
we have to deal with are chiefly scalar, such as the
phenomena of evaporation, dissociation, chemical combi-
nation, etc. where the quantities which have to be considered
are such things as temperature, vapour density, or the num-
ber of molecules in a particular state. The chief difference
between these cases and those we have been considering is
that in these we have as in the kinetic theory of gases to-
deal chiefly with the average values of certain quantities and
cannot attempt to follow the variations of the individual
members which make up the average, while in the previous
cases we have been able to follow in all detail the changes
in most of the quantities introduced. In these new
cases all that we can get by the application of the Hamil-
tonian principle are relations between the averages of a series
of quantities ; as however these averages are all that we can
SCALAR PHENOMENA. 141
observe in these cases, this limitation is not serious from a
practical point of view.
The relations we shall deduce are those which exist
when the body is in a steady state.
69. The systems we shall have to consider are portions
of matter in the solid, liquid or gaseous state, and consist,
according to the molecular theory of bodies, of a very large
number of secondary systems or molecules. Now we can
control a primary system in many ways, we can fix its geo-
metrical position, we can within certain limits strain it in any
way we please, we may establish electric currents or electric
displacements through it, and if the body is magnetic we
can magnetize it within the limits of saturation : so that the
coordinates fixing the geometrical, the strain, the electric
and the magnetic configurations are under our control and
have therefore been called ( 46) controllable coordinates.
The coordinates fixing the positions of the several
secondary systems are not however within our control and
we have not the power of altering any one of them ; v/e
have called these unconstrainable coordinates.
70. When we say that a system consisting of a great
number of molecules is in a steady state we mean that the
state is steady with respect to the controllable coordinates
and make no supposition as to whether it is so or not with
respect to the unconstrainable ones, all that we shall assume
is that the mean values which we can observe and which
depend upon the unconstrainable coordinates are steady.
Thus when the system is in a steady state the velocity of
each controllable coordinate must be constant, and if the
coordinate enters explicitly into the expression for either
the kinetic or potential energy, that is if the coordinate is
not a " kinosthenic " or speed one, the velocity must vanish.
142 DYNAMICS.
71. We shall now proceed to prove that when a system
consisting of a great number of molecules is in a steady state
the mean value of the Lagrangian function has a stationary
value so long as the velocities of the controllable coordinates
are not altered.
Let us denote the controllable speed coordinates by the
symbol q^ the controllable positional coordinates by q a and
the unconstrainable coordinates by ^ 3 , then we have by the
Calculus of Variation
f
I
Jt
/</Z d dL\ ^{ h ( dL d d
( -- -r -7T- }oq 2 dt + 2, I ( - -- -r; -p
t,\dq 2 dtdqj 4 Wa dt dq
Remembering Lagrange's Equations we see that this
equation reduces to
Let us suppose that the symbol of variation refers to a
disturbed motion in which the values of the controllable
coordinates are slightly altered while the velocities of the
speed coordinates remain unaltered and constant during the
disturbed as well as the undisturbed motion.
We shall for the sake of greater clearness consider the
three terms on the right-hand side of equation (152)
separately, as the considerations which apply to them are
different in each case.
72. Let us first take the term
SCALAR PHENOMENA. 143
Since we suppose that in the disturbed motion the velocities
of the speed coordinates are unaltered q } is always zero,,
and thus the term we are considering vanishes.
73. We shall now show that the term
also vanishes. Since q 2 is not a speed coordinate it must
enter explicitly into the expression Z, so that when the
motion is steady the velocities of coordinates of this class,
vanish. The terms in Z which contain the velocities of
positional coordinates always vanish when the motion is
steady. They do not therefore contribute anything to the
mean value of Z, and so we may without loss of generality
suppose that the coefficients of terms in the kinetic energy
involving the velocities of positional coordinates are all zero
and that therefore - may be put equal to zero. In this
way we may see that the terms we are considering in the
expression for the variation of the mean value of the La-
grangian function vanish.
74. To show that the terms
vanish we must use the reasoning given by Clausius in his
paper "On the Second Axiom of the Mechanical Theory of
Heat," Phil. Mag. XLII. p. 178.
Let us in the first place consider what the coordinates
denoted by q 3 are. They are coordinates fixing the
position of the molecules of the system and may be
144 DYNAMICS.
divided into two classes : firstly, coordinates fixing the
position of the centres of mass of the molecules, and
secondly, coordinates fixing the position of the molecules
relatively to their centres of mass. The motion of the latter
coordinates will be periodic, while that of the former will not
be so ; in consequence however of the frequent collisions
between the molecules their direction of motion will be
continually reversed, so that if the position of a molecule be
arbitrarily changed the distance between the disturbed and
the undisturbed positions will not increase indefinitely with
the time, the difference will sometimes be positive, some
times negative, but will fluctuate between limits which do
not increase with the time. Thus if q^ is a coordinate of
this kind
will fluctuate between positive and negative values which do
not increase with the interval / - t r
The same reasoning will apply with still greater force to
those coordinates which fix the configuration of a molecule
relatively to its centre of mass, for these coordinates will
oscillate and therefore the part of
depending upon these coordinates will fluctuate between
limits which do not increase as the time /, - / increases.
Now any change which we have the power to produce
in any of the coordinates fixing the system will, since the
motion is steady, produce a change in each term of
MODIFIED LAGRANGIAN FUNCTION. 145
which will increase proportionately to the increase in the
interval t l -t Q \ and thus if we integrate over a sufficiently
long interval we may neglect any terms on the right hand
side of equation (152) which fluctuate between fixed values
and therefore as far as coordinates of the kind q z are
concerned put
when the interval / t - / is sufficiently long.
We have seen however that this is also true as far as the
variations of the other coordinates q^ q 2 are concerned,
so that when the motion is steady we have
where L denotes the mean value of L taken over unit time,
e.g. one second, and where the variations are such as could
be produced by slightly altering the values of the coordi-
nates. We may conclude that one second is a sufficiently
long interval over which to integrate since according to the
molecular theory of gases there are both a great many
collisions and a great many vibrations in this period.
75. In the above investigation we have supposed
that the Lagrangian function Z is expressed in terms
of the velocities of the coordinates and the proof is only
valid when it is so expressed and does not hold when the
velocities corresponding to some coordinates are elimi-
nated and the momenta corresponding to them introduced
instead.
We can prove however in this case that the modified
Lagrangian function ( n) is stationary when the system is
in a state of steady motion.
For let L' be the modified Lagrangian function and q a
T. D. 10
146 DYNAMICS.
coordinate whose velocity has not been eliminated, then L
is a function of q, q... and the momenta corresponding to
the other coordinates, and since
d_ d^_ _ dL =
dt dq dq
we have by the Calculus of Variations
where p is the momentum corresponding to one of the
eliminated coordinates. We can prove exactly as before
that the right hand side of equation (154) vanishes for all
variations in which the momenta corresponding to the
eliminated coordinates remain unaltered.
Thus we have in all cases an equation of the form
8Z=o,
where L is the mean value of the ordinary Lagrangian
function or its modified form according as it does not or
does contain the momenta corresponding to some of the
coordinates.
76. In the physical applications of this principle it
would sometimes be difficult to tell whether a symbol
occurring in L represented a momentum or a velocity.
Fortunately however this knowledge is unnecessary if we
calculate the Lagrangian function from the forces required
to preserve equilibrium. For when the system is in a
steady state, X the force of type x which must be applied
to maintain equilibrium is given by
dL
X= ~-d X '
where L is the Lagrangian function or its modified form
according as the kinetic energy does not or does contain the
momenta corresponding to some of the coordinates. So
LAGRANGIAN FUNCTION. 147
that by what we have just proved
-ZSJXdxdt .................. (155),
the sum being taken for all the coordinates and X expressed
in terms of them, is the expression for the terms depending
on the controllable coordinates in a function which possesses
the property of having a stationary value when the system
to which it refers is in a steady state.
77. Thus to take an example let us consider the case
of a heavy particle whose mass is m attached to a fixed
point by a string whose length is /, and moving so that
the string makes a constant angle & with the vertical. The
kinetic energy of the system is
where < is the angle which the plane containing the string
and a vertical line makes with some fixed plane. The couple
which must act on the system to keep & constant is
- mP sin & cos $(j> 2 .
When the system is acted on by gravity the potential
energy is mgl cos S so that the Lagrangian function is
\ml 2 sin 2 3< 2 + mgl cos 3
which may be written
- $dO + mgl cos S
and this possesses the property of being stationary.
If however the Lagrangian function is expressed in
terms of 3> the momentum corresponding to < and given by
the equation
$ = mP sin 2 $<j>
the Lagrangian function becomes
^ 2 < 2 cosec 2 3- + mgl cos &,
and this expression is not stationary.
10 2
148 DYNAMICS.
The function which possesses this property is the
"modified" Lagrangian function
' - <1> 2 cosec 2 S + mgl cos S.
2ml 2
Since however when expressed in terms of 6 and <
equals
I 2 COS S
~^? sin 2 3'
we see that the " modified " Lagrangian function again
equals
- \dO + mgl cos S.
Thus the expression
is stationary however may be expressed, whether in terms
of <j> or <.
This example illustrates the principle that if we calculate
the Lagrangian function from the forces necessary to pre-
serve equilibrium we need not consider whether it is ex-
pressed in terms of velocities or momenta.
78. If we consider the proof by which the equation
was established we shall notice one point which we must
continually bear in mind when we are calculating the value
of the potential energy.
By the Calculus of Variations
dL
PI T ,, f'i (dL d dL} (dL
Ldt=\ <-j -T.-JT-I fydt + 2 ( -=- .
It, !t, \dq dt dq } y \dq
LAGRANGIAN FUNCTION. 149
and thus if equation (156) holds we must have
dl^_d_ dL_
dq dt dq
Now in ordinary Rigid Dynamics perhaps the most
usual form of Lagrange's equation is
d_ dJL_dJ^_
dt dq dq ~ V>
where Q is the external force of type q tending to increase
this coordinate. In this case L = T - V where V is the
potential energy when the coordinates have their assigned
value and the system is free from the action of external
forces. If however we are to use equation (153) we must
put L = T V where
that is we must add to the potential energy we are con-
sidering the potential energy of the system which produces
the external forces.
Lagrange's equation may now be written
and equation (153) is true.
Thus to take an example, in Electrostatics we often
assume that the potential energy V of unit volume of a
dielectric whose specific inductive capacity is K and through
which the electric displacements parallel to the axes of x, y.
z are respectively /j , h is
f {/+*+*}
and that the equations of equilibrium are
_ __
df ' dg ' dh~ '
150 DYNAMICS.
where X, Y, Zare the components of the electromotive force
parallel to the axes of x, y, z respectively.
If however we wish to apply the theorem we are now
considering we must put
for then the equations of equilibrium are
dV _dV___dV_
~df~~4i~J/i~ '
The necessity of choosing V so that the equations of
motion are of the form
d dT d
IfTj-T^-^^'
is one to which we must always be alive in dealing with this
subject.
CHAPTER X.
THE CALCULATION OF THE MEAN LAGRANGIAN
FUNCTION.
79. SINCE we can observe and regulate the forces of the
types of the "controllable" coordinates we can determine
how they depend upon the values of these coordinates and
then by means of the expression (155) calculate all those
terms in the mean Lagrangian function which involve such
coordinates. There may however be some terms in the
Lagrangian function which do not involve these quantities
and if we require these we must determine them by other
considerations ; a large number of problems can however be
solved even though we do not know the values of these
terms.
To get some idea of the different kinds of terms which
may exist in the Lagrangian function let us consider the
energy of a system consisting of a large number of molecules.
In the expression for the energy we can calculate all the
terms involving the coordinates which fix the electric,
magnetic or elastic configuration of the system, and in the
terms depending upon the strain coordinates we may include
those terms which involve the average distance between the
molecules. There may however be some terms left which
152 DYNAMICS.
each molecule contributes independently of its neighbours
and which do not involve any of the controllable coordinates.
The sum of these contributions will be proportional to the
number of the molecules and must also be a function of the
temperature, because the mean state of the system is fixed
by the controllable coordinates and the temperature, and the
mean kinetic energy must therefore be a function of these
quantities. By hypothesis the terms we are considering do
not involve the controllable coordinates, so that the only
quantity they can depend upon is the temperature. The
potential energy of the molecules may also contribute terms
to the Lagrangian function which do not involve the con-
trollable coordinates and which therefore we cannot calculate
by equation (155)- For the purposes for which we use the
Lagrangian function all that we require to know about it is
the change in its value when the system is changed in some
definite way. Now if we measure the amount of heat
absorbed or evolved when the change takes place and
know the change, if any, which takes place in the kinetic
energy, we can calculate the alteration in the part of the
potential energy which is independent of the controllable
coordinates.
The methods of calculating the mean value of the
Lagrangian function will be best illustrated by working out
some particular cases. Let us begin with that of a perfect
gas.
Mean value of the Lagrangian function for a perfect gas.
80. Let us suppose that unit mass of the gas is enclosed
in a cylinder furnished with a piston, whose distance from
the base of the cylinder is represented by the coordinate x,
then since the pressure of the gas is a force tending to alter
LAGRANGIAN FUNCTION FOR A GAS. 153
the value of x, the mean Lagrangian function for the system
of molecules forming the gas must involve the coordinate x.
If H denotes the mean value of the Lagrangian function
of the system, the mean value of the force of type x pro-
duced by the system when in a steady state is by Lagrange's
equations
dH
dx'
Since there is equilibrium between the pressure due to
the gas and the external pressure
dH
^c= A *>
where / is the pressure of the gas and A the area of the
piston.
But if the gas obeys Boyle's law
Re
* m T'
where v is the volume of unit mass of the gas, the absolute
temperature and J? a constant such that RQ equals the
square of the velocity of sound in the gas.
Now
so that
dff_RB dv_
dx v dx'
Integrating this equation we have in so far as H depends
upon v and 0,
(157),
where v is an arbitrary constant and /(0) an arbitrary func-
tion of 0, which does not involve x. It corresponds to the
154 DYNAMICS.
part of the kinetic energy which depends entirely upon un-
constrainable coordinates. We shall find in the course of
this work that a great many problems can be solved without
a knowledge of the value of / (0). As far as f (0} is linear
it may be included in the first term, as we may regard v as
quite arbitrary.
The expression (157) will give the value of the mean
Lagrangian function so far as it involves x, it also includes
that part of the kinetic energy which is expressed entirely in
terms of unconstrainable coordinates, for this can be included
in the term f(6) ; to complete its value we must subtract
from it w the potential energy of unit mass of the gas when
its particles are infinitely distant from each other, as this is
the part of the potential energy which depends upon uncon-
trollable coordinates.
Thus for unit mass of the gas
or if p be the density of the gas
H=RQ log Po
p
We shall see later on, when we consider the phenomenon
of evaporation, that f(0) is of the form
AO + 0\ogO .................. (158).
The value of H for a mass m of gas whose density is p
is given by the equation
ff=m0log^ + mf(6)-mw ......... (159).
P
This is the Lagrangian function for the gas itself; when
an external pressure acts upon it we must add to this value
the mean Lagrangian function of the system producing the
pressure. We may suppose that this system is a weight
LAGRANGIAN FUNCTION FOR A SOLID. 155
placed upon the piston, the variable part of the potential
energy of this is, if V be the volume occupied by the gas
pv.
So that its mean Lagrangian function is
-pv
and the Lagrangian function of the two systems is therefore
mRQ log'' + mf(Q)-mw-pV ......... (160).
P
Mean value of the Lagrangian function for a liquid or solid.
81. We must now proceed to find the mean value If of
the Lagrangian function for a liquid or solid. Let us suppose
that we have a piston whose distance from a fixed plane is
x pressing upon a bar of the substance.
Then we have by Lagrange's equations when the motion
is steady
dH
= mean force tending to increase x. produced by
the substance,
so that
dp
where p is the pressure required to balance this force and a
the area of the cross section of the bar. The differential
coefficient dpIdQ is obtained on the supposition that the
volume is constant.
Since adx = dv,
fdp\
we have ^ ,- = ( -^ )
d6dv \dO) v constant
156 DYNAMICS.
Thus
dv
v constant
where ft is the mean value of (dpIdO) between zero and 0.
Thus H=
= Oy + S l (0) say.
Where / (0) is an arbitrary function of the temperature,
it .is unnecessary to add an arbitrary function of v on
integration as this will be included in the potential energy
due to strain.
If the mass of the substance is unity
a
where o- is the density, so that in the expression for H
for unit mass of the substance there are the terms
From this we must subtract w' the potential energy
of unit mass of the substance. Thus in the Lagrangian
function for a mass m of the substance there are the terms
f " B
- mO I da- + ;/// (0) - mw .
"
If there is any external pressure we must add to this
the expression for the mean value of the Lagrangian
function of the system producing this pressure. This, as
in the case of the gas, will be
-pV,
where p is the external pressure and V the volume of the
solid or liquid. Adding this term we get
-pr' (161).
LAGRANGIAN FUNCTION FOR A SOLID. 157
It must be remembered that we have only calculated
the value of the Lagrangian function in the simplest case
when the body is in a steady state, when it is free from all
strain except that inseparable from the body at the tempe-
rature we are considering, and when it is neither electrified
nor magnetized. The change in the Lagrangian function
due to any additional strain or to electrification or mag-
netization can be at once determined by finding the energy
required to establish this particular condition. For example,
the change in H produced by statical electrification equals
minus the potential energy of the electrical distribution, the
change due to any system of electric currents flowing through
solids or liquids is the kinetic energy due to this distribution
of currents, and can be calculated by the ordinary formulae
of electrokinetics.
82. The problems which we shall now proceed to
solve, making use of the principle that the mean value of
the Lagrangian function is stationary, are those which can
often be solved on thermodynamical principles by using
the condition that the value of the entropy of the system
is stationary. The value of H must therefore be closely
connected with that of the entropy, and in fact we see
from its value for a perfect gas in equilibrium under
external pressure that, with the exception of the term p V,
those terms in H which depend upon the controllable
coordinates occur also in the expression for the entropy. It
seems however preferable to use the function H which has a
direct dynamical significance, rather than the entropy which
depends upon other than purely dynamical considerations.
CHAPTER XI.
EVAPORATION.
83. WE shall now go on to apply the principle that the
value of H is stationary to solve some special problems in
Physics. The first problem we shall consider is that _of
finding the state of equilibrium when a given mass of some
liquid is placed in a closed vessel from which the air has
been exhausted; some of the liquid will be vaporized and
we wish to find how far the vaporization will proceed
before equilibrium is obtained. This of course is equivalent
to finding the density of a vapour when in equilibrium in
presence of the liquid.
Let v, v' be the volumes occupied by the vapour and
liquid respectively, the mass of the vapour, 17 that of the
liquid, the rest of the notation being the same as that used
in 80 and 81.
Then assuming that the vapour obeys Boyle's law we
see from equation (158) that the vapour contributes to the
expression for H for the whole system the terms
io g + &(0) - &> (162),
since p the density of the vapour equals /z/.
EVAPORATION. 159
From equation (161) we see that the liquid when it is free
from surface tension, electrification and the like, furnishes
to the same expression the terms
-r)0 I -. 2 da- + r;/, (Q) - rjw' ( J 63).
Thus H the mean value of the Lagrangian function for
both the liquid and vapour is the sum of (162) and (163) so
that we have
H = IRQ log ^ + / (0) + ^ (6i)
- ^0 I *-td<T %w vfw r (164).
When there is equilibrium the value of H has by the
Hamiltonian principle (75) a stationary value, so that in
this state no small change can affect the value of the right
hand side of equation (164).
The small change which we shall suppose to take place
is that which occurs when the mass of the vapour is
increased by a small amount S while the mass of the liquid
is diminished by the same amount. The change in H is
dH
%*
so that when there is equilibrium we have by the Hamil-
tonian principle
dH
^r-
When 8 of the liquid is vaporized the volume of the
liquid diminishes by 8/<r so that we have
dif i
160 DYNAMICS.
and since the volume of the vapour and liquid remains con-
stant
-(z, + z/) = o,
and therefore
dv i
Now
t + y - W + W
where for brevity y is written instead of
Substituting for dvjd^. its value we have
= ./?$ lO2f + C ~~ \W "W ) 4
dc, Vo"
where \f/ (0) is written for
a quantity which does not involve Since /z> = p we may
write (165) as
^-JWtog& + *-( -0+W
Since dHld vanishes in the state of equilibrium we have
then
J?0 log & = - ^ + (w-w] - if/ (0) ...(166),
P (T
or p ~ p f. a e ^
since p/cr is very small we may write this as
p=*0)- * (l6 7 ),
where <^> (0) is some function of 9.
EVAPORATION. l6l
Bertrand (Thermodynamique, p. 93) has shown that the
results of Regnault's experiments on the vapour pressures
of different liquids can be represented by the following
expressions, / being the pressure in millimetres of mercury :
water; log/ = 17-44324 - 2795/0 - 3' 8682 log 0>
ether; log/ = 13 -42311 - 1729/0 - 1-9787 log 0,
alcohol; log/ = 21-44686 - 2743/0 - 4-2248 log 0,
chloroform; log/ = 19-29792 - 2179/0 - 3*91583 log 0,
bisulphide of carbon Iog/ = i2'58852 -1684/0- 1*7689 log 0.
This form of expression was originally used by Dupre.
(Theorie Mecanique de la Chaleur, p. 97.)
The coefficient of i/0 in each of these expressions is nearly
\\R, where X is the latent heat of the substance at the
absolute zero of temperature. This is the term (w - w')j6
in our expression (166) and w-w' is the latent heat at
absolute zero, hence by comparing the other terms in
these expressions we see that/(0) must be of the form
A6 + BQ log 0.
84. We can by the aid of the preceding formulae very
easily determine the effect upon the vapour pressure of any
slight change in the physical condition of the liquid or
vapour.
Let us suppose that the physical conditions are so
changed that the mean Lagrangian function exceeds the
value we have hitherto assumed for it by \. Then instead of
equation (167) we have evidently
RO log + R6 + \l/ (0) (w w") + - = o,
p cr dc,
so that Sp the change in the vapour density due to the cause
T. D. II
162 DYNAMICS.
which produced the change x i n tne mean Lagrangian
function is given by the equation
or
so that if x increases with the vapour pressure in the state
of equilibrium is increased, while if x diminishes as
increases the equilibrium vapour pressure is diminished.
This very important principle is a particular case of the
more general one that ; when the physical environment of a
system is slightly changed and the consequent change in the mean
Lagrangian function increases as any physical process goes on,
then this process will have to go on further in the changed
system before equilibrium is reached than in the unchanged one,
while if the change in the mean Lagrangian function diminishes
as the process goes on it will not have to proceed so far. We
shall have numerous examples of this principle in the
course of the following pages.
85. Let us now consider the effect of surface tension
upon the vapour pressure. In order to take a definite case
let us suppose that the liquid is a spherical drop. It will
possess in consequence of surface tension potential energy
proportional to its area, and as the area of the drop
diminishes as the water evaporates the energy due to the
surface tension changes, and since anything which causes
the energy to change as evaporation goes on alters the state
of equilibrium, the vapour pressures when there is equi-
librium in this case cannot be the same as when evaporation
produces no change in the area and therefore no change in
the energy due to surface tension.
EVAPORATION. 163
If a be the radius of the drop and T the energy per unit
area due to surface tension then in the expression for the
potential energy of the liquid there will be in addition to
the terms we have already considered the term
and therefore in the mean Lagrangian function for the liquid
and vapour the additional term
So that with our previous notation
x = -47ra 2 T.
i da ii dv'
NOW - -jr = -- , TT ,
ad 3 z'' d%
and therefore = --- -,- ,
da i
hence ^> = ~ 2"
df 4iKr<r
and therefore - = - .
at, aa-
So that if Sp be - the change in the vapour density pro-
duced by the surface tension we have by equation (168)
20 T I
and if 8^ be the change in the vapour pressure, since
8/
we have by (170)
cr-p a
This agrees with the formula given by Sir William
Thomson in the Proceedings of the Royal Society of
II - 2
1 64 DYNAMICS.
Edinburgh, Feb. 7, 1870, and quoted in Maxwell's Theory
of Heat, 5th edit. p. 290.
If we take the case of a drop of water -^ of a millimetre
in radius, then if the temperature is about ioC. we have
by (170), since RB for water vapour is about 1*3 x io 9
fy> = 200 ^ T
p ~ 1-3 x io 9
since T '- 81 we have
-- = 1*2 X I0~ 5 .
P
We see that the energy due to surface tension makes
the Lagrangian function increase as evaporation goes on,
so that by the principle given at the end of 84, the effect
of it will be to make evaporation go on further than it
otherwise would.
If we have the water in narrow capillary tubes then
when it evaporates the area of the surface of contact of the
tube with water is diminished but that of the surface of
contact of the tube with air is increased. Since the surface
tension of the surface of separation of the tube and air is
greater than that of the tube and water, the potential energy
due to surface tension increases as evaporation goes on,
thus the mean Lagrangian function diminishes as the liquid
evaporates, so that by the principle of 84 the effect of
surface tension in this case will be to stop evaporation
and promote condensation. We can easily shew that if a is
the radius of the tube then in this case
<r p a
86. Effect of a charge of electricity on the
vapour pressure. We can by the use of formula (168)
EVAPORATION. 165
find the effect on the vapour density of electrifying the liquid.
If the charge e is given to the liquid which we shall suppose
spherical and of radius a, the potential energy is increased
by
I _ 2
*K~a'
where K is the specific inductive capacity of the surrounding
medium.
The mean Lagrangian function of the liquid and its
vapour is diminished by this amount. Blake's experiments
on the evaporation of electrified liquids (Wiedemann's
Ekktricitat) iv. p. 1212) show that e remains constant as the
liquid evaporates, in other words that the vapour proceeding
from the electrified liquid is not electrified. Thus the new
term e 2 J2Ka in the mean Lagrangian function diminishes
as the liquid evaporates and therefore by the principle of
84 evaporation will not go on so far as before, that is
the vapour density when there is equilibrium will be
diminished by electrifying the liquid.
We can easily calculate by equation (168) the amount of
this diminution.
In this case
i e^
X ~ 2K a'
, d\ i e 2 da
and therefore jj=-,j&#->
substituting the value of dald given by (169) we have
dx i e 2
d 87rA" a* a- '
so that if Sp be the change in the vapour density produced
by the electrification we have by (168)
1 66 DYNAMICS.
To calculate the magnitude of this effect let us suppose
K= i ; then e/a 2 is the electric force just outside the sphere,
and this cannot exceed a certain value, otherwise the insu-
lating power of the air would break down and the electricity
escape. The maximum value of e/a 2 when the sphere is
surrounded by air at the atmospheric pressure is about 120
in electrostatic measure : and as o- is unity, p a small fraction,
the maximum alteration in the vapour density will be given
by the equation
So i
'
now RB for water vapour at ioC. is about 1*3 x io 9 , so
that
this value will be independent of the size of the drop.
Comparing equations (170) and (172) we see that the
maximum effect due to electrification is about equal in
magnitude though opposite in sign to that due to a curvature
of 1/4 of a centimetre.
The effect of electrification is to diminish the vapour
density when there is equilibrium between the liquid and
the vapour, it therefore increases the tendency of the
vapour to deposit on the liquid. We should therefore
expect an electrified drop of rain to be larger than an
unelectrified one, so that this effect may help to produce
the large drops of rain which fall in thunderstorms.
87. Effect of an electric field upon the vapour
pressure. Electricity also produces an effect upon the
vapour pressure when the drop is not charged but merely
placed in an electric field. Let us suppose that the
field is due to a charge of electricity e collected at a point
EVAPORATION. l6/
P, let the radius of the drop of water which we shall
suppose spherical be a and let/ be the distance of the centre
from P. Then (Maxwell's Electricity and Magnetism, Vol. I.
p. 232), the potential energy due to the mutual action of the
electrified point and the drop of water is
so that the increase in the mean Lagrangian function is
and therefore by (168) the change Sp in the density of
the vapour when in a state of equilibrium is given by the
equation
Sp i (r i j 3 ^a 2 e 2 ^ "~> \da
7 = RV o--p ~K \ 2f 2 (f 2 ~a 2 ) + f*(J*-a*f I d '
Substituting for dajdt; from (169) we have
3p_ i / (3 i a 2 I i
if # be small compared with / then approximately
p
Now ^/Ay 2 is the force at the centre of the drop due to
the electrified point, calling this F and remembering that <r
is large compared with p we have
so that the effect of electrification on a neighbouring body
is again to diminish the vapour density in the state of
equilibrium. The formula (173) will evidently hold even
though the field is not due to an electrified point provided
1 68 DYNAMICS.
P the force at the drop does not vary much in a distance
comparable with the radius of the drop.
88. Effect of strain upon vapour pressure.
We shall now investigate how the vapour pressure depends
upon the state of compression of the liquid. Let us
suppose that the pressure p acts upon the liquid, then if k
be the modulus of resistance to compression, the potential
energy possessed by the liquid in virtue of this strain is
1/77
2 k <r'
so that
and therefore by equation (168)
p 2 R cr-p
or approximately
So i p 2
j-i&tx. ............ : ..... (I75)
for water k= 2-2 x io 10 , so that the effect on the vapour
pressure of the compression due to the pressure of icoo
atmospheres is at the temperature of i5C. given by
for ether the effect would be given by
= - approximately.
P 5
In the next paragraph we shall consider another effect
due to pressure which except for exceedingly large pressures
is larger than the one we have just been considering.
EVAPORATION. 169
89. Effect of the presence of a gas having no
chemical action on the water vapour on the
equilibrium vapour pressure. It is generally believed
that the equilibrium vapour pressure of water depends only
upon the temperature of the water and not upon the pressure
produced by an indifferent gas, that it is for example the
same in a vacuum as under atmospheric pressure. If however
we remember that when a portion of the liquid evaporates
the air above it must expand and do work we shall see that
this cannot be the case, but that since evaporation is
accompanied by a diminution in the density of the air, and
therefore by an increase in its mean Lagrangian function,
it must by the principle of 84 go on further when air is
present than in a vacuum, so that the vapour density will be
increased by the presence of the air. We shall now go on
to investigate the magnitude of this increase and shall
consider two cases. In the first case we shall suppose that
the air and liquid are placed in a closed vessel whose
volume remains constant.
Let , 77, be the masses of the vapour, water, and air
respectively ; w, w lt w 2 , the mean potential energy of unit
mass of each of these substances respectively.
Then the mean Lagrangian function for the water vapour
is
** log^ +/(*) -ft*,
where v is the volume of the vessel above the liquid.
The mean Lagrangian function for the water is by
equation (161)
vyO + T//I W - w ;
and the mean Lagrangian function for the air is
I/O DYNAMICS.
Thus when a quantity d of the liquid evaporates, since
dv I d= i/o-, the condition that dH '/ d% = o gives
RB log ^
Now if 8/0 be the change in p due to the presence of the
air, 8w 1 the corresponding change in w l9 we have by this
equation
ROZp ROSp 6
------ " + - +Sw. +2_j_ = ...... (176).
p a- (TV
The presence of the air will increase the pressure and so
cause the liquid to be more compressed than it would be if
the air were away, so that w 1 will be increased by the
presence of the air. If &? be the compression due to the
pressure /' of the air, and p the pressure due to the water
vapour, Sze/j will be proportional to (p +/') Be and unless the
pressure due to the air amounts to many thousands of
atmospheres this term will be very small compared with
(TV
which is equal to/'/o-.
Hence we may write equation (176) as
MSp ROZp p'
r 4. r + . =
p (T (T
or, since a is very large compared with p,
p <r
Now if S/ be the change in the pressure of the water
vapour due to the presence of the air
so that equation (177) becomes
8/_/
p o-
EVAPORATION. I /I
But if p is the density of steam when the pressure is /'
i I
~ P ~ p"
so that equation (177) becomes
= p;
>" a-
and thus the alteration in the vapour pressure produced by an
external pressure of an atmosphere is given by the equation
& _ density of steam at atmospheric pressure
p density of water
= atoC.
1200
So that for each atmosphere of pressure the vapour
pressure of water is increased by about one part in twelve
hundred. For ether the increase would be about one part in
220.
90. The other case we shall consider is when the
pressure acting on the system remains constant. We shall
use the same notation as before. The only change we shall
have to make in the mean Lagrangian function is to add to
it that of the system producing the steady external pressure.
We may suppose in order to fix our ideas that this system is
a quantity of mercury placed on the piston, which may
be supposed to move vertically up and down, then if P be
the steady pressure per unit area the potential energy is
equal to
so that the mean Lagrangian function of this system is
-P(v+v'\
where if is the volume of the liquid.
I7 2 DYNAMICS.
^ dH
\ he condition - = o
gives
M log - ^ +
Now
and i=/'
where/ and/' are the pressures due to the water vapour and
air respectively.
Since />=/ +p'
and 'J' = - 1
tff 07-
the above equation reduces to
RQ log ^-RQ +/(#) _/ x (0) -yO-(w- Wj + ^=0,
or if S/> be the change produced by the external pressure,
=
P o-
a similar result to the one we obtained before.
We see from this result that (apart from any other cause)
rain drops will form more easily when the barometer is low
than when it is high.
Regnault's experiments seem to show that the vapour
pressure in a vacuum is greater by nearly 5 per cent, than
when there is air at atmospheric pressure above the liquid
(Wullner's Lehrbuch der Physik, in. p. 703), but he attributed
this difference to the condensation of the liquid on the sides
EVAPORATION. 1/3
of the vessel ; the absorption of the air by the liquid might also-
tend to produce an effect in this direction, though, as the
following investigation will show, to nothing like the extent
of 5 per cent.
91. Effect of absorbed air on the vapour
pressure. When the liquid contains some gas diffused
through its volume which remains behind when it evaporates,
the evaporation of the liquid will cause the volume occupied
by the gas to diminish and its density to increase. Thus by
(157) the mean value of its Lagrangian function will diminish
as evaporation goes on, so that by 84 the presence of the
gas dissolved throughout the volume will diminish the
equilibrium vapour pressure.
Let e be the mass of the dissolved gas, v' the volume of
the liquid in which it is dissolved, then the Lagrangian
function of the gas is
e^log^-ew'-K/W (178),
where w' is the intrinsic potential energy of unit mass of the
dissolved gas.
The expression (178) is the quantity we denoted by x i n
84. The only variable in x which involves is v and
dv'jd = - i/cr, so that we have
d x _ R'Oe
d v'cr
and therefore by (168)
,-Sp i It'Qe , .
6-?- = - r (i79)-
p a- p V
If 8/ be the increase in the vapour pressure caused by the
dissolved gas, P the pressure this gas would produce
if it were free from the liquid and filled the volume z/, then
since
1/4 DYNAMICS.
R'Qt
and - = P.
v
equation (179) may be written
tr-p
So that since p is very small compared with a- we have
approximately
or if/ be the vapour pressure, and p the density of steam at
the atmospheric pressure w, equation (180) may be written
ty = _pP
p a- TT
And since p' I <r is about i / 1200 we see that
/ I200
where (/>) is the pressure P expressed in atmospheres.
The volumes of the various gases which one volume of
water will absorb at oC. under the pressure of 760 milli-
metres of mercury were determined by Bunsen and are given
in the following table :
Hydrogen '019
Nitrogen -0203
Air '0247
Carbonic Acid 179
Chlorine 3*0361
so that according to equation (180) the vapour pressure
of water saturated with air will be lowered by about one
part in 50,000, when saturated with carbonic acid by about i
part in 660 and when with chlorine by about i part in 400.
In this investigation we have assumed that the properties
of the liquid are not altered by the presence of the gas ;
if they are, then we must regard y and w' as functions of e,
and this would lead to the introduction of several additional
EVAPORATION. 1/5
terms into- the equation for S/. We have assumed too that
the gas remains behind as the liquid evaporates, as in this
case the diminution of the vapour pressure is greater than if
some of the gas were to be set free when the liquid evapo-
rates.
92. The effect of dissolved salt on the vapour
pressure. Van J t Hoff (" L'equilibre chimique dans les
systemes gazeux ou dissous a Petat dilue," Archives Neer-
landais, xx. p. 239, 1886) has pointed out that Pfeffer's
experiments on the osmotic pressure produced by salts
dissolved in water (PfefTer, Osmotische Untersuchungen,
Leip/ig, 1877) and Raoult's experiments on the effect of
dissolved salts on the freezing point of solutions (Annales de
Chimie, 6 me serie, iv. p. 401). show that the molecules of a
salt in a dilute solution exert the same pressure as they
would exert if they were in the gaseous state at the same
temperature and occupying a volume equal to that of the
liquid in which the salt is dissolved, and that the pressure
exerted by these molecules obeys Boyle's and Gay Lussac's
law. This being so, the mean Lagrangian function for the
salt dissolved in the liquid is the same as that of an equal
mass of the salt in the gaseous state filling the volume
occupied by the liquid. Thus if the properties of the liquid
are not altered by the presence of the salt the results of the
preceding section will apply, and we shall have, supposing
that the salt remains behind when the liquid evaporates,
where cr is the density of the liquid, p the density of its
vapour at the atmospheric pressure, (P) the pressure in
atmospheres which would be exerted by the dissolved salt
if it were in the gaseous state.
Thus, for example, suppose that we have n grammes of
i;6 DYNAMICS.
salt in a litre of the solvent, where n is the molecular
weight of the salt. This strength of solution is often called
for brevity a strength of one equivalent per litre. This
quantity of salt will by Avogadro's law produce the same
pressure as 2 grammes of hydrogen per litre, that is about 22
atmospheres ; if this quantity of salt were dissolved in water
it would by equation (181) since p/o- is about 1/1200
diminish the vapour pressure by about i part in 55, if it
were dissolved in ether, C 4 H 10 O, where p/o- is about 1/2 20,.
then the vapour pressure would be diminished by about i
part in 10, if it were dissolved in alcohol, C 2 H 6 O, where p/o-
is about 1/380, the vapour pressure would be reduced by
about one part in 17. We see from equation (181) that the
diminution in the vapour pressure is proportional to the
quantity of salt dissolved. We can also express the result of
this equation as follows. If P is the pressure due to one
equivalent in grammes of the salt dissolved in a kilogramme
of the solvent, then P/o; where o- is the density of the
solvent, is the pressure in atmospheres due to one equi-
valent of the salt dissolved in a litre of the solvent. Hence
P 2
- = J(density of hydrogen at atmospheric pressure.) So-
o- 10
that we may write equation (181) as
= (molecular weight of solvent) x i x io~ 3 ,
where p is the diminution in vapour pressure when one
equivalent of the salt is dissolved in a kilogramme of the
solvent. Another way of expressing the same thing is that
when one equivalent of the salt is dissolved in 1000
equivalents of the solvent, i.e. in 1000 m grammes where
m is the molecular weight of the solvent, the diminution in
the vapour pressure amounts to i part in 1000, whatever
be the nature of the salt or solvent.
VAPOUR PRESSURE.
*P P P
Since - ,
or TT
and since P is directly, while p' is inversely, proportional to
the absolute temperature, we see that the ratio Bp/p ought to
be nearly independent of the temperature since o- only
varies very slowly with it.
93. In the preceding investigation we have assumed
that the properties of the solvent were unaltered by the
presence of the salt, and that all the solvent did was to
enable the salt to exist in a condition in which the
molecules were very far apart.
If however the properties of the solvent are altered by
the presence of the salt, then we must regard w as a function
of the quantity of salt dissolved.
In this case instead of equation (179) we shall have
K" " 8 "'
where W is the change in it/ produced by the presence of
the salt.
Now if s be the strength of the solution, i.e. the quantity
of salt in unit volume of the solvent,
dw' dw'
77 ~T S j ,
arj aS
so that equation (182) becomes
dw' P
- s =- + ~ = o.
as o-
If the change in w is proportional to the strength of the
solution then
8w' s : o.
as
T. D. 12
178 DYNAMICS.
In the general case the lowest power of s which occurs
in the expression
s , du*
OW S 7
ds
is the second, so that the effect produced by the alteration
of the properties of the solvent depends upon the squares
and higher powers of the concentration, while the effect we
investigated in the preceding section was proportional to
the first power, and therefore when the solution is dilute is
relatively the more important.
Raoult, Comptes Rendus 104, p. 1433, has recently found
that when one equivalent of a substance is dissolved in 100
equivalents of a solvent the vapour pressure is reduced by
1-05 parts in 100, which agrees very well with the results
we have just obtained.
CHAPTER XII.
PROPERTIES OF DILUTE SOLUTIONS.
94. THE effect produced on the vapour pressure of any
solvent by dissolving other substances in it has been discussed
in the last chapter ; in this chapter we shall consider some
other properties possessed by dilute solutions.
Absorption of gases by liquids. Let us suppose
that we have a closed cylinder containing a gas and a liquid
and that we wish to find how much of the gas will be ab-
sorbed by the liquid. In this case we have four substances
to consider,
1. The liquid.
2. The vapour of the liquid.
3. The free gas.
4. The gas dissolved in the liquid.
The variation which we shall suppose to take place, and
which will not by the Hamiltonian principle alter the value
of H when the system is in equilibrium, is that corre-
sponding to the escape of a small quantity of gas from the
liquid. This will not affect the value of the mean Lagrangian
function of the vapour of the liquid, so that we may leave this
12 2
l8o DYNAMICS.
out of account in solving this problem. Let the mass of the
liquid be 17, that of the free gas , and that of the absorbed
gas . Then using the same notation as we have hitherto
employed, the mean Lagrangian function for the liquid is
............... ( l8 3),
for the free gas
&0 log?* + tf (&)-&, ............ (184),
for the dissolved gas
W ............ (185),
where v is the volume occupied by the free gas, and v' the
volume of the liquid or that occupied by the dissolved gas,
and where w and f'(6) are the quantities for the dissolved
gas which correspond to w and/(#) for the free gas. If we
denote the sum of the expressions (183), (184) and (185) by
If, then by the Hamiltonian principle H is stationary when
the system is in equilibrium, so that if we suppose a small
variation to be caused by a quantity of gas S escaping from
the liquid then we must have for equilibrium
dH
this is equivalent to
RO log -?- - RO +f(0) -w-RO log ->- + .0 -/' (0) + a-'
o (186).
The last term when the amount of gas absorbed is
not large will be very nearly independent of the quantity
of gas dissolved. Equation (186) may be written
DIFFUSION OF SALTS. l8l
RB log c^ = w-w' +f (0) -f(6)
where ^ is a constant and p arid p are the densities of the
free and dissolved gas respectively. Since the temperature
is constant, we see from this equation that p'/p is constant,
that is, the quantity of gas in unit volume of the liquid is
proportional to the density of the free gas. This is Henry's
law of the absorption of gases by liquids and it has been
verified by the researches of Bunsen and others. Bunsen's
experiments showed that the value of the ratio p'/p depends
upon the temperature, hence we see from equation (187)
that w-0-w'-'0 + r& + n0-r7ii cannot
be zero, otherwise p'/p would be the same at all tempera-
tures. Thus either the properties of the free gas can not
be quite the same as those of the dissolved gas, or else
the properties of the water are altered by the gas dissolved
in it.
95. A similar investigation will apply to the case of
a solid or gas which can dissolve in two fluids which do
not mix. We can prove in this way that when there is
equilibrium when the fluids are shaken up together then,
provided the solutions are dilute, the amount dissolved in
unit volume of one fluid will bear a constant ratio to that
dissolved in the same volume of the other (see Ostwald's
Lehrbueh der allgemeinen Chemie, I. p. 401).
96. The diffusion of salts through the solvent, a process
which goes on until the solution acquires a definite state,
can be explained by the same principles. In the following
investigation of this problem we include the consideration
of the effect of gravity upon diffusion. Let us suppose that
182 DYNAMICS.
we have a shallow vessel whose volume is v and that this is
connected by a capillary tube of fine bore with another
shallow vessel whose volume is v' situated at a height h
above the lower vessel. Let the two vessels be filled with
water containing a certain quantity of salt dissolved in it,
then we wish to find how the salt is divided between the
vessels when equilibrium is established. Let and f\ be
the quantities of salt in the lower and upper vessels
respectively, then if h' be the height of the lower vessel
above some fixed plane, the potential energy of the salt in
the lower vessel may be taken to be g/i', so that there is
the term - gh f in the expression for the mean Lagrangian
function of this salt, similarly there is the term rjg(/i + h')
in the expression for the mean Lagrangian function of the
salt dissolved in the upper vessel.
Thus using the same notation as before the expression
for the mean Lagrangian function of the salt dissolved in
the lower vessel is
(188),
the mean Lagrangian function of the salt dissolved in the
upper vessel is
+ if (6) - ifw - ng (h' + h) . . . ( 1 89).
"
Let us' suppose that a quantity 8rj of salt goes from the
lower to the upper vessel, then if there is equilibrium
this change must not alter the value of H^ the mean
Lagrangian function of the salt and solvent in the two
vessels. If the solutions are dilute the only part of H
which varies is the sum of the expressions (188) and (189),
and the condition
dH
COMPRESSIBILITY OF SALT SOLUTIONS. 183
leads to the equation
or if ', rj f are the masses of salt in unit volume in the
lower and upper vessels respectively
-,-#4 = 0,
' _**
or | = * ..................... (190).
So that the concentration of the solution when there is
equilibrium varies in the same way with the height as the
density of a gas under the action of gravity.
97. A large number of experiments have been made
on the effect of dissolved salts on the coefficients of com-
pressibility of various solutions (see Schumann " Compressi-
bilitat von Chlorid Losungen," Wied. Ann. xxxi. p. 14,
1887 and Rontgen and Schneider, Wied. Ann. xxix. p.
165, 1886), we shall therefore investigate an expression
for this effect and see what information can be gained
by comparing it with the results of the above-mentioned
experiments.
Let us suppose that the solution whose original volume
is v is subjected to a hydrostatic pressure p which reduces
its volume to v, and that k' is its coefficient of compressi-
bility. Then the mean Lagrangian function of the solution
and the system producing the pressure is
the mean Lagrangian function of the dissolved salt is, using
the same notation as hitherto,
184 DYNAMICS.
where is the mass of the salt.
If H is the sum of these expressions then by the
Hamiltonian principle H must be stationary when there is
equilibrium. Let us suppose that the volume is increased
by dv, then since H is stationary we must have
dH
now %RQ\v is the pressure due to the molecules of the salt,
let us call this P. If p be increased by 8/, the correspond-
ing diminution in volume v is by (191) given by the equation
i Sv -
or since v is very nearly equal to v we may write this
equation in the form
So that the apparent coefficient of compressibility is
K
i^Pk''
thus the pressure due to the molecules of the dissolved salt
produces a decrease in the coefficient of compressibility.
Let us see what the magnitude of this effect would be if
the pressure of the molecules were the only way in which
the dissolved salt affected the resistance to compression.
If we make this assumption k' 1/2*2 x io 10 , this being the
COMPRESSIBILITY OF SALT SOLUTIONS. 185
value when measured in c. G. s. units of this constant for
pure water at i5C. If there is one equivalent of salt in a
litre of water, P is 22 atmospheres or in absolute measure
2-2 x io 7 . Since the reduction in the coefficient of com-
pressibility is very nearly equal to
or to one part in i /-/%', we see that when the strength of
solution is one equivalent per litre the reduction in the
coefficient of compressibility ought to amount to one part in
2'2 X IO 7
that is to one part in 1000.
The following table taken from Rontgen's and Schneider's
paper will show that the effect of dissolved salts is some-
times more than a hundredfold that calculated on the
above assumptions, and hence we conclude that in addition
to producing a pressure in the solvent the dissolved salt
must directly alter its elastic properties.
Reduction in com-
Names of salt
Strength of
solution in
pressibility found
by Rontgen and
or acid.
equivalents per
Schneider reckoned
litre.
in parts
per thousand.
HNO 3
49
42
HBr
49
40
HC1
52
52
H 2 S0 4
48
79
NH 3 I
49
90
NH 3 NO 3
5
94
NH 3 Br
5
90
NH 8 C1
Na 2 C0 3
t
'97
1 86 DYNAMICS.
98. The pressure due to the molecules of the dissolved
salt will explain many of the phenomena exhibited by
solutions. The molecules of the salt may be regarded
as confined within a limited volume by the solvent, and
they will take any opportunity of expanding even though
they may have to do work to enable them to do so. Thus
if the solution was contained in a vessel provided with a
bottom pervious to water but impervious to the substance
dissolved in it, then if the vessel is placed in water with its
top above the surface water will flow up into the vessel
through the bottom, the work required to lift the water
being supplied by the expansion of the molecules of the
dissolved salt. This constitutes the well-known phenomenon
of osmosis.
A diaphragm which is said to be impervious to all salts
though it allows water to pass through it can be made by
allowing weak solutions of sulphate of copper and ferrocyanide
of potassium to diffuse into a porous plate from opposite
sides, these solutions when they meet form a membrane
of the kind desired. . Detailed instructions for making these
membranes are given in Pfeffer's Osmotische Untersuchungen,
Leipzig, 1877. By following his directions I have succeeded
in making such membranes though the number of failures
was very large compared with the number of successes.
Mr Adie, who is making some investigations on this subject
at the Cavendish Laboratory, finds that the membranes are
formed more readily if ferric chloride is used instead of
copper sulphate.
We shall now attempt to find by means of Hamilton's
principle the height to which the fluid will rise in the
osmometer. Let us suppose that the osmometer is a long
tube with a diaphragm of the kind we have been describing
at the bottom, and that it contains water and salt. Let
OSMOSIS. IS/
be the mass of the salt, rj that of the water inside the tube,
that of the water outside, and let v be the volume of
the tube occupied by the solution. Then, using the same
notation as hitherto, the mean Lagrangian function for the
salt is
*# log ^ + /(*) -*(, + *) (192),
where z is the height of the centre of gravity of the salt
molecules above some fixed plane.
The mean Lagrangian function for the liquid in the tube
is
rjy'O + -nfl (0) - rj (wj +&) (193),
and for the liquid outside the tube
y<9 + t/;W-K + ^) (194),
where y is the height of the centre of gravity of the water
outside the osmometer : the quantities for the liquid inside
the tube are denoted by affixing dashes to the symbols
denoting the corresponding quantities for the water outside
the tube.
By the Hamiltonian principle the value of H, the sum
of (192), (193) and (194), is stationary when there is equili-
brium. Let us suppose that a quantity of water Srj flows
into the osmometer.
Then since if there is no contraction
dv dz i
di\ d-rj 20. '
where a is the area of the cross section of the osmometer,
and
1 88 DYNAMICS.
where h is the height of the top of the fluid in the osmometer
above the level of that outside : the condition
dH
^ =
leads to the equation
if the properties of the solution are not altered by the pre-
sence of the salt then
and equation (195) becomes
where/ is the pressure due to the molecules of the dissolved
salt, and h' the height of a column of water whose mass is
the same as that of the salt dissolved in the osmometer. If
the strength of the solution in the osmometer is one equiva-
lent per litre, / is about 22 atmospheres, so that in this case
h + \ti is about 660 feet ; that is the water would flow into
the osmometer until the height of the liquid in the tube is
nearly an eighth of a mile above the level of the water
outside.
If the liquid is not allowed to expand but confined
in a constant volume we can easily prove in a similar way
that if the properties of the solvent are not changed by the
addition of the salt then when there is equilibrium the
pressure exerted by the fluid in the osmometer must be
the same as that due to the molecules of the salt. This
result is given by Van t' Hoff {L'equilibre chimique, Archiv
Neerlandais t. 20, p. 239).
Pfeffer (Osmotische Untersuchungen, p. 12) gives as the
OSMOSIS. 189
pressure for a i % solution of potassium sulphate that due to
192-6 centimetres of mercury and for a i / solution of
potassium nitrate that due to 178-4 centimetres. The
pressure calculated on the above principles for potassium
sulphate is 97 centimetres if we assume that the molecule
is K 2 SO 4 and 194 if the molecule is J (K 2 SO 4 ), for potassium
nitrate it is 167 if the molecule is KNO 3 .
We see as in 90 that the terms in (195) depending
upon the alteration of the properties of the solvent by the
addition of the salt do not contain any powers of the strength
of the solution below the second.
A measurement of the osmotic pressure produced by any
salt solution will on the above assumptions give the same
information about the structure of the molecule of the salt
in the solution as a vapour density determination does about
the structure of the gas whose vapour density is determined,
for it enables us to find the number of molecules in a given
mass of the gas. Thus Pfeffer's measurement of the os-
motic pressure due to potassium sulphate suggests that the
relation between the composition of the molecule of this
salt and that of potassium nitrate is represented by ^K 2 SO 4
and KNO 8 , and not by K 2 SO 4 and KNO 3 .
Even if we do not assume that the molecules of a salt
produce a pressure analogous to that of a gas, it would still
follow from the Hamiltonian principle that there would be
a rise in the osmometer if the increase in the mean Lagran-
gian function of the liquid inside the osmometer caused by
the addition of unit mass of water is greater than the
diminution in the mean Lagrangian function in the water
outside the osmometer caused by the abstraction of unit
mass of water.
Anything that causes a change of this kind will increase
the height to which the fluid will rise in the osmometer ;
1 90 DYNAMICS.
thus, if the addition of water to the solution inside the os-
mometer is attended by an evolution of heat, the solution
will rise higher in the osmometer than one of similar strength
in which no heat was evolved in dilution. On this account
the indications of the osmometer are somewhat ambiguous,
and before coming to any definite conclusion as to the
structure of the molecule of the salt it would be necessary
to use several solvents and to show that the osmotic height
varied as the absolute temperature.
99. Surface Tension of Solutions. The experi-
ments of Rontgen and Schneider already alluded to have
proved that for most solutions the product of the height to
which the solution rises in a capillary tube into the density
of the solution is greater for a solution of a salt than for pure
water, and that for dilute solutions of most (though not all)
substances this product increases with the strength of the
solution. It follows from this that the tension of the surface
of contact of the solution with air increases with the strength
of the solution, while the tension of the surface of contact
of the solution with glass or any other solid body diminishes
as the solution gets stronger.
The variation of the surface tension with the strength of
solution may cause the strength of the solution to vary near
the surface.
To investigate the magnitude of this effect let us suppose
that we have a thin film whose area is S and surface tension
T, connected with the bulk of the liquid by a capillary
thread. Let be the mass of salt in the thin film, f] that in
the rest of the liquid ; then if is the mass of water in the
film, that of the rest of the water, the mean Lagrangian
function of the liquid and salt in the film is, using the
same notation as before,
SURFACE TENSION,
RO log - + tf(0) - fa + 7 + /; (0) -
where v is the volume of the film.
The mean Lagrangian function for the rest of the liquid
is
t]RB log VP + ij/(0) - 77?^ + eyO + e/;(0) - ?/,.
Let us suppose that a mass of salt S goes into the film,
the change in the mean Lagrangian function is
and this by the Hamiltonian principle must vanish; thus if
p, p are the masses of salt in unit volume of the film and
liquid respectively, we get
P
or if T is the increase in the surface tension when the mass
of salt in unit volume is increased by unity
, zT'
p r e >
where t is the thickness of the film. Thus if the surface
tension is increased by the addition of the salt there will be
less salt per unit volume in the film than in the liquid in
bulk, while if the surface tension is diminished by the addi-
tion of salt there will be more salt in unit volume of the film
than in unit volume of the rest of the liquid. We saw that
the surface tension of a solution in contact with a solid di-
minished as the strength of the solution increased, thus if we
had a film in contact with a solid there would be more salt
in unit volume of the film than in unit volume of the bulk of
I Q2 DYNAMICS.
the liquid ; if we dipped for example a piece of filter paper
in such a solution, the solution in the filter paper would be
stronger than the rest. Or, again, if such a solution were to
flow through a capillary tube the salt would have a tendency
to flow to the sides, so that the more quickly moving fluid
at the centre would get weaker and weaker. Many experi-
mental illustrations of this could be given ; one of these is
an experiment tried by Dr Monckman and myself at the
Cavendish Laboratory, in which a deep coloured solution of
potassium permanganate emerged almost colourless after
trickling through finely divided silica. Again, if a piece of
filter paper be dipped into a coloured solution of a salt such
as potassium permanganate, unless the salt has a very strong
affinity for the water the solution after rising some height
in the filter paper becomes colourless.
If a small quantity of paraffin oil be mixed with water
the surface tension of the solution against a solid is greater
than that of water, and such a solution will increase in
strength when it flows through finely divided silica.
CHAPTER XIII.
DISSOCIATION.
100. THE Hamiltonian method can t>e used for the pur-
pose of obtaining the laws which govern the phenomena of
dissociation, i.e. the splitting up of a molecule into its atoms,
such as the iodine molecule I 2 into the atoms I and I ; or
of a complex molecule into simpler ones, as in the case of
nitrogen tetroxide, where the molecule N 2 O 4 splits up into
two molecules of NO 2 , or when the molecule of chloride of
ammonium splits up into ammonia and hydrochloric acid.
This phenomenon has some analogy with that of
evaporation ; as in the latter case we have equilibrium
between portions of matter in two different states, the
gaseous and the liquid, matter being able to pass from the
one state to the other by evaporation and condensation, so
in dissociation we have also equilibrium between portions of
the same substance in two different conditions, both in the
gaseous state, the molecules in the one condition being
more complex than those in the other, and matter being
able to pass from one condition into the other by the more
complex molecules splitting up, " dissociating" as it is called
into the simpler ones, while on the other hand some of the
simpler ones combine and form the more complex molecules.
Equilibrium is attained when the number of the complex
T. D. 13
IQ4 DYNAMICS.
molecules which split up in any time is the same as the
number formed in the same time.
Let . us first investigate the case when the complex
molecules contain two of the simpler ones ; this is the case
when, as in iodine, the more complex systems are di-atomic
molecules and the simpler ones atoms, as well as in such
cases as the dissociation of N 2 O 4 .
Let us suppose that the system is contained in a closed
vessel and that is the mass of the complex molecules, 77
that of the simpler ones. We shall for the present assume
that both gases obey Boyle's law and that the fundamental
equation for the complex gas is
f = J? lP 0,
and for the simpler gas
/ = *,P0,
where / is the pressure, p the density and the absolute
temperature.
Since the molecules of the complex gas consist of two of
those of the simpler gas, the density of the simpler gas will
at the same pressure and temperature be half that of the
complex gas and therefore
^ 2 =2^.
The mean Lagrangian function of the complex gas is
fl^log^a+S/iW-frr, (196),
where v is the volume of the vessel in which the gas is
contained and w l the potential energy of unit mass of the
complex gas.
The mean Lagrangian function for the simpler gas is
^ 2 01og^W 2 W-W 2 (197),
where w 2 is the potential energy of unit mass of this gas.
DISSOCIATION. 195
The mean Lagrangian function H for the two gases,
assuming that the properties of each are not modified by
the presence of the other, is the sum of the expressions
(196) and (197). By the Hamiltonian principle the value
of H is stationary when the system is in equilibrium. Let
us suppose that the state of equilibrium is disturbed by
a mass S of the simpler molecules combining to form
complex ones. Then since the value of H is stationary
we must have
dH_
Since the mass of the gas is constant
and the condition
~=o
is equivalent to
log - xf +/ (0) -w,- R 2 e log p
or since
we have
vp
2 = o ...... (198),
$ log
This can be written
2 Wj-TOg
^=*(). *. .................. (199),
where <jf> (0) is a function of 6 but not of , rj or v.
In experiments on dissociation the quantity usually
measured is the vapour density of the mixture at some
standard pressure TT.
132
196 DYNAMICS.
Let A be the density of the mixture of the two gases at
this pressure, D that of the complex or undissociated gas at
the same Dressure.
the same pressure.
The pressure in the vessel is
or since R z - 2^,
the pressure equals Rfl.
The density of the gas at this pressure is
v '
so that A the density at the pressure TT is given by the
equation
A= ll _^_ .
the density D of the complex gas at this pressure is given
by the equation
so that
u r
and therefore
+2rj~ D
and fi^T 4H
since PV ( + 2rf) Rfl,
pv Z>-A
we have *) -^-^ j^~
pv 2^-D
DISSOCIATION. 197
so that equation (199) becomes
where
10 1. Before discussing this equation we shall investigate
the way in which it must be modified if the gas does not
obey Boyle's law.
Formulae connecting the pressure and volume in such
gases have been given by Van der Waals (Die Continuitat
des gasformigen una flilssigen Zustandes] and Clausius (Wied.
Ann. ix. p. 337).
Van der Waals' formula, which is rather the simpler of
the two, is
RQ a
where R is the value of pfpO for a perfect gas of the same
specific gravity, and b and a constants depending upon the
nature of the gas.
Clausius' formula is
RB__ *_
P ~V-a 6(v + p)*'
where R is the same as in Van der Waals' formula for the
same gas, and a, /?, K are small constants depending upon
the nature of the gas. We shall now investigate the differ-
ence produced in the state of equilibrium of a dissociable
gas if it and the components into which it is decomposed
obey Van der Waals' law instead of Boyle's.
198 DYNAMICS.
Let the fundamental equations of the complex and
simple gases be respectively
and
where
as before.
Then we can easily prove that instead of the term
P
in the mean Lagrangian function, we have the term
where p is the density of the complex gas; with a corre-
sponding term in the expression for the Lagrangian function
of the simpler gas.
The condition
dH
=0
will now lead to the equation
log (I - b lP ) + log P - -
P I -
- R 2 | log ( i - b 2 p') + log ^ - ^-^1 -f
+f l (0)~f 2 (6)-(w l -w a ) = o .......... (201)
and not to the equation (198).
DISSOCIATION. 199
Equation (201) may be written, since J? a =2jRj,
= / - w
Now if we suppose that the deviations from Boyle's Law
are slight, so that b l and b z are so small that their squares
may be neglected, we may write this equation as
=<M0) VR e Rio ....... (202);
since ^vR-fl is approximately equal to p 2 // and a l and p
are both small fractions while /= io 6 if the pressure is one
atmosphere, equation (202) may be written as
*'
an equation of the same form as when the gases obeyed
Boyle's law. The connexion between the masses of the
complex and simple gases and the vapour density of the
mixture will not however be the same as when the gases
obeyed Boyle's law, and so the relation between the vapour
density, the pressure and the temperature may be different
although equation (202) shows that the relation between the
masses of the dissociated and undissociated gases is the
same.
It would be an interesting problem to find an expression
for the vapour density of the mixture in terms of the masses
of the two gases in this case, we shall not however stop to
200 DYNAMICS.
investigate it as it would not be of any use for the purpose
of connecting theory with experiment, for in determining
the vapour density from the experiments Boyle's law was no
doubt assumed.
102. Formulae corresponding to equation (200) deduced
from thermodynamical considerations have been given by
Willard Gibbs (Equilibrium of Heterogeneous Substances, p.
239) and Boltzmann (Wied. Ann. xxn. p. 39, 1884).
Thus according to Gibbs
this agrees with (200) if fa (6) is constant.
According to Boltzmann
and this agrees with (200) if fa (6) is proportional to 0.
Gibbs in his paper (" On the vapour densities of per-
oxide of nitrogen, formic acid, acetic acid and perchloride of
phosphorus," American Journal of Science and Art, xvm. p.
277, 1879), discusses the results of experiments on the vapour
densities of these substances at different temperatures and
pressures and has found that they agree fairly well with the
results calculated by formula (203). Quite recently however
E. and L. Natanson have made a most elaborate investigation
of the vapour density of nitrogen tetroxide at various tempe-
ratures and pressures (Wied. Ann. xxvu. p. 306). They
found that so long as the temperature remains constant the
vapour density of nitrogen tetroxide at different pressures is
given with great accuracy by the formula (200) but that if the
DISSOCIATION. 201
temperature changes the difference between the observed
results and those calculated from either Gibbs' or Boltz-
mann's formula, assuming that the quantity a which occurs
in it is constant, is greater than can be accounted for
by errors of experiment. Part of this difference may arise
because the N 2 O 4 does not obey Boyle's law. The differ-
ences seem however to be too great to be explained
altogether in this way, and a value of fa (0) different from
that adopted by either Gibbs or Bolt/mann would probably
fit in better with the observations.
103. In the Philosophical Magazine for October, 1884,
I considered the question of dissociation from the point of
view of the kinetic theory of gases, supposing that the
complex molecules are continually being broken up while
the simpler ones are continually combining, and that the gas
attains a steady state when the number of complex molecules
broken up in the unit time is the same as the number formed
in that time. It is shown that, using the same notation as
in 99, these conditions lead to the equation
if, and only if, the average time a complex molecule lasts
without splitting up into simpler ones, is independent of the
number of molecules of the gas in unit volume. This will
evidently not be the case if the breaking up of the complex
molecules is due to their collision with other molecules, for in
this case the greater the number of molecules the greater the
number of collisions, and therefore the shorter the time the
complex molecule lasts. Since the results of a large number
of experiments prove that equation (200) holds when the
temperature is constant we conclude that the dissociation of
202 DYNAMICS.
the complex molecules is not due to the collision with other
molecules. We have however deduced (200) from mechani-
cal principles which hold whenever the two gases obey Avo-
gadro's law and whenever the pressure produced by a mixture
of gases is the sum of the pressures which would be produced
by each of the gases separately if the other were removed.
Hence we conclude that when we have a gas some of whose
molecules are complex and keep breaking up into simpler
molecules which after a time recombine to form the complex
molecules, then if the splitting up of the complex mole-
cules is due to their striking against other molecules, the
pressure due to the gas will not be the sum of the pressures
which the dissociated and undissociated gases would produce
if each were by itself in the vessel.
104. We shall now consider how external influences
may modify the amount of dissociation which takes place in
some given gas at a given temperature and pressure.
If we denote rfjv by A. and use X as a measure of the
amount of dissociation, then if the Lagrangian function from
some external cause is increased by x we see by equation
(198) that SA the change in X is given by the equation
Thus if x increases as diminishes that is as dissoci-
ation goes on SA will be positive, that is dissociation will go
on further than it did in the undisturbed state. This is
another illustration of the general principle stated in 84
that any slight alteration in the conditions under which a
system is placed which increases the rate of increase of the
mean Lagrangian function with any change in the system,
will cause that change to go on further before equilibrium is
DISSOCIATION. 203
attained than it had to do in the undisturbed system and
vice versa.
We shall now consider the effects on dissociation of such
things as surface tension, electrification, the presence of
other gases, corresponding to those we considered in the
analogous case of evaporation.
105. Effect of surface tension upon dissocia-
tion. Though the effects, of surface tension are not nearly
so prominent in gases^as in liquids, still, since there is perfect
continuity from the liquid to the gaseous state, we should
expect that the outer layer of molecules of a gas which was
not in the "perfect" condition would like the outer layer in a
liquid be under different conditions from the other molecules,
and would therefore not possess the same amount of energy
as the same number of molecules in the midst of the gas.
In Van der Waals' theory of the relation between the
pressure and volume in an imperfect gas, the result of which
is expressed by the relation
the term ajv 2 is due to the action of the surface tension of
the gas (Van der Waals, Die Continiiitat ties gasformigen
undfliissigen Zustands, p. 34).
Though it is much more difficult to detect the existence
of the action of surface tension experimentally in gases than
in liquids there is still some evidence of its existence from
experiments such as those of Bosscha on the forms of
clouds of fog and tobacco smoke.
There must therefore be a term in the expression for the
potential energy of a gas proportional to its surface. We
shall write this term
TS,
204 DYNAMICS.
where T is the quantity corresponding to the surface tension
and S is the area of the surface of the gas. Thus the change
X in the Lagrangian function, 104, is
-TS,
so that by (205)
o ............ (206).
Thus, if the surface tension diminishes as dissociation
goes on, in which case dTld is positive, the dissociation will
be greater the larger the surface of the gas. We should
expect a priori that the surface tension of the dissociated
gas would be smaller than that of the undissociated, for in
most cases the dissociated gas approaches more nearly than
the other to the state of a perfect gas : thus in most cases
dTld% will be positive, so that dissociation will be facilitated
by increasing the surface of the gas.
Let us now endeavour to form a rough estimate of the
magnitude of this effect. According to Van der Waals the
energy of unit area of surface of gas is measured by
xa
7'
where x is a distance comparable with molecular distances.
Now for a cubic centimetre of ether vapour at oC. and
under atmospheric pressure a!v 2 is a pressure of about
324 x io~ 4 atmospheres, or in absolute measure 3-24 x io 4 .
If we take the molecular distance x as io~ 7 , we have
rr, oca _ 7
T = -j-= TO 7 x 3-24 x io 4
= '24x io~ 3 .
DISSOCIATION. 205
Now by equation (206)
SA - ' * (ST)
~X~Rfidk^
In order to form a rough estimate of the value of dT\d,
let us suppose that the complex gas possesses surface
tension but that the simpler one does not; this is an
approach to the truth, as the value of a and therefore of
the surface tension is very much greater for complex gases
than for simple ones. Let p be the density of the complex
gas, v the volume in which it is contained, then
= vp.
Since the surface tension varies as a/z? t it is proportional
to the square of the density, so that
i dT _ 2 dp
and thus
dT 2T
so that we have
8X 2 ST
(2 7) '
Now at the atmospheric pressure, at which we reckoned 7J
jRflp = io 6 ,
and substituting for T its value, we have
8\ 28 3
-T- = - 9 approximately.
A V I O
206 DYNAMICS.
If the gas be supposed to be a film of thickness /, then
8A. _ 1.2
T ~ /X TO 8 '
so that if the thickness of the film were comparable with
molecular dimension, say if /=io~ 7 , then the surface
tension would produce very large effects.
This example may be sufficient to show that if we have
the gas in thin films surface tension may produce a very con-
siderable effect ; such films occur adhering to glass fibres or
to matter in a fine state of division, such as spongy platinum
or charcoal. The value of T given above is only part of the
surface tension of the surface of contact of the gas and the
solid. The surface tension of the surfaces separating A and
B is due to the energy of thin layers of A and B next
their junction differing by a finite amount from the energy
possessed by equally thin layers in their interior. The ab-
normal energy of these layers is due to the want of symmetry
of the action on the two sides. In the preceding investigation
we have calculated the part of the energy of the layer of one
of these substances arising from the effects produced by its
own molecules, in addition to this there is the energy arising
from the action of the glass on the gas as well as the energy
in the thin film of glass. Thus the value of the surface
tension may be much greater than that given above and the
effects due to it may therefore be greater than our estimate.
1 06. The value of T may depend upon the substance
to which the film adheres, and thus the nature of the walls
of vessels used for chemical experiments may affect the
chemical combination which goes on inside them. Van 't
HofT has described some experiments which seem to show
that effects of this kind do exist. He shows (Etudes de
Dynamique Chimique, p. 56) that the rate at which the
DISSOCIATION. 207
polymerization of cyanic acid goes on is increased by
increasing the area of the walls of the vessel in which it is
contained, the volume being kept constant. Thus when
the area of the walls was increased six times, the rate of
polymerization was increased in the ratio of 4 to 3. He
also found that when the walls of the vessel were covered
with a deposit of cyamelide the rate of polymerization of
cyanic acid was increased threefold. Victor Meyer too
found that the decomposition of carbonic acid takes place in
a porcelain vessel at a temperature several hundred degrees
lower than in a platinum vessel. When the effects produced
are of this magnitude, it is doubtful whether they can be
due to the effect of surface tension, but it is probable that
in the case of many catalytic actions, where we have thin
films of gas, the effects observed might be explained by
considerations of this kind.
107. Effect of Electricity upon Dissociation.
When there is no electric discharge electrification will not
produce any effect upon the final state of the system, unless
the specific inductive capacity of the gas changes as disso-
ciation goes on. As all the specific inductive capacities
of gases which have been determined are very nearly equal,
the effect of electrification on dissociation must be very
small, and we shall not stop to determine it.
1 08. Effect of a neutral gas. If the properties of the
neutral gas are not affected in any way by the presence of the
gas which is dissociating, the value of the mean Lagrangian
function of the neutral gas will not change as dissociation
goes on. The presence of this gas will therefore not affect
the maximum amount of dissociation. The presence of a
foreign gas certainly alters the rate of dissociation, and
208 DYNAMICS.
in some cases the experiments seem to show that it does
alter the maximum amount of dissociation. This is contrary
to the result we have just arrived at, and the only way of
reconciling the two is to suppose that the gas is not per-
fectly neutra! but has its properties affected to some extent
by the presence of the other gases. If the dissociation
were at all catalytic, we might explain the action of the
neutral gas by supposing that by itself forming a film on
the surface of the vessel it prevented to some extent the
dissociating gas from doing so.
109. In the preceding investigations we have assumed
that the complex molecule splits up into two molecules or
atoms of the same kind. In some cases however the
constituents into which the molecule splits up are different,
as for example when PC1 5 splits up into PC1 3 and C1 2 .
We can easily modify the preceding investigation to
suit cases of this kind.
Let us take the dissociation of phosphorus pentachloride
as a typical case, and let , rj, be the masses of PC1 5 , PC1 3 ,
and C1 2 respectively.
Then the mean Lagrangian functions for these gases are
respectively
Now ifVj, c# ^ 3 are the molecular weights of these gases
respectively, then since the increase in the number of mole-
DISSOCIATION. 209
cules of PC1 3 is the same as that in the number of C1 2 and
to the decrease in the number of PC1 5 , we have
where </, dv\> dt, are the alterations in the masses of PC1 8 ,
PC1 3 , and C1 2 respectively ; hence, remembering that
we see that the condition
dH
leads to the equation
it
(208),
and thus ifQ^v is constant as long as the temperature is
constant.
Let us suppose that the values of , -rj, before dissoci-
ation commenced were , iy , and that the mass c^p of
PC1 5 gets decomposed, then we have
and the equation to find / is
(% + ^ 2 ^) (^o + 'a/) ^ ^ tfo - 'i/) ...... ( 2 9),
where ^ is a function of the temperature.
We shall now discuss the effect upon / of alterations in
the values of v, r] and .
Differentiating (209) we get, writing y for
T. D. 14
210 DYNAMICS.
dp I
........................ (210),
1 dv v
dp_ _ i ,
= *
We see from (210) that dpjdv is positive, so that dissociation
will be promoted by increasing the volume in which a given
quantity of gas is confined. From equations (211) and (212)
we see that both dpjd-q^ and dpjd^ are negative, so that the
presence of free PC1 3 and C1 2 tends to stop the dissociation.
Wiirtz proved experimentally that there was very little
dissociation of PC1 5 when it was placed in an atmosphere of
PC1 3 . We can also see from general principles that this must
be so, for as soon as the molecule PC1 5 breaks up the free
chlorine will be surrounded by such a multitude of mole-
cules of PC1 3 that most of it will recombine and form PC1 5 ,
and in this way stop the dissociation.
In this case, as in the former, theory indicates that if
there is no catalytic action the presence of a neutral gas
would not produce any effect.
In some cases, though the results of the dissociation
are in the gaseous state, the body which dissociates is in
the solid or liquid state instead of, as in the previous
instances, the gaseous. The dissociation of NH 5 S into
H 2 S and NH 3 is an example of this kind.
We have only to slightly modify the preceding work to
make it applicable to this case. Let as before be the mass
of the dissociating body, rj and those of the components
into which it is dissociated. Then the mean Lagrangian
function for the solid or liquid dissociating body is by (81)
DISSOCIATION. 211
The mean Lagrangian functions of the gases into which
it dissociates are respectively
+. -*.,
and
Then from the condition
dH
<*r>
we get, since d% = - (drj + d),
dv i
# = ~a'
and *-*,
', ','
where ^ 2 and <r 3 are the combining weights of the gases into
which the solid dissociates, and <r the density of the solid
or liquid
It follows from this equation that, as before, dissociation
is hindered by the presence in excess of either of the results
of the dissociation.
In this case the dissociation would be affected to a small
extent by the presence of a neutral gas, for if the system is
confined in a closed vessel the volume of the solid or liquid
diminishes as it evaporates, the neutral gas above it expands,
and its Lagrangian function therefore increases. Hence
we see by (84) that the presence of the neutral gas will
increase the dissociation.
142
212 DYNAMICS.
By an investigation similar to that in (89) we can
easily show that if K denotes the value of r]/v 2 , and 8/< the
change produced by the presence of the neutral gas, then
where e is the mass of the neutral gas, c its combining
weight and o- the density of the solid or liquid. Since e/wr
is the ratio of the mass of the gas to the mass of the same
volume of the dissociable solid, we see that the effect
produced by the neutral gas, unless its pressure amounts
to some hundreds of atmospheres, is extremely small. If
we take the case of sal-ammoniac, where o- is about 1-5,
we see that for a pressure of 100 atmospheres
SK
- = '3 approximately,
so that if the pressure were increased by about 3*3 atmo-
spheres the change in K would be about one per cent.
no. Dissociation of Salts in Solution. We have
seen 92 that Van 't Hoif has given reasons for believing
that the molecules of a salt in a dilute solution exert the
same pressure as they would if they were in the gaseous
state at the same temperature and volume : and that the
mean Lagrangian function of the molecules in the solution
is therefore the same as that of the same number of gaseous
molecules. We might therefore expect from analogy that
in some cases these molecules would be dissociated though
the effects of this dissociation might not be so recognisable
as in the case of gases. Many cases of the dissociation
of salts in solution have been observed, sodium sulphate
and the ammonium salts are well-known examples (Muir's
Principles of Chemistry, p. 367). Indeed the theory has
DISSOCIATION. 213
recently been started that in dilute aqueous solutions the
dissolved acid or salt is in most cases dissociated and that
to a very considerable extent ; thus it has been stated that in
dilute solutions of HC1 as much as 90 per cent, of the acid
is dissociated. The reasons given for this conclusion do not
seem to me to be very convincing, and the experimental
results on which they are based seem to admit of a differ-
ent interpretation. The supporters of this theory urge
that for the salt to produce the effect which in some cases
it does, it is necessary to suppose that the molecules of the
salt exert a greater pressure than they would if they
occupied the same volume at the same temperature when in
the gaseous condition. This reasoning is founded on the
assumption that all the effects due to the dissolved salt may
be completely explained merely by supposing the volume
occupied by the solvent to be filled with the molecules of
the salt in the gaseous condition. Now though we may
admit that the salt does produce the effects that would be
produced by this hypothetical distribution of gaseous mole-
cules, still it does not follow that these are the only effects
produced by the salt. The salt may change the properties
of the solvent and the effects attributed to the dissociation
of the molecules may in reality be due to this change. The
investigation in 97 proves that this must be so in some cases,
for we saw that the effects of the addition of salt on the
compressibility of the solution were much too large to be
explained by any amount of dissociation.
In the case of the dissociation of salt solutions the proper-
ties of the solution might alter as the dissociation progressed.
Thus the dissociation might alter the surface tension of the
solution, in which case the amount of dissociation would
depend upon the shape and volume of the solution ; or it
might alter the coefficient of compressibility or the volume
214 DYNAMICS.
of the solution, and then the amount of dissociation would
be influenced by external pressure. In fact the dissociation
of the dissolved salt would probably be much more sus-
ceptible to external physical influences than the dissociation
of a gas. We shall however discuss these as particular
cases of the next investigation, which deals with a much
more general case of chemical equilibrium between either
gases or dilute solutions.
CHAPTER XIV.
GENERAL CASE OF CHEMICAL EQUILIBRIUM.
in. THE case we shall consider in this chapter is the
equilibrium of four substances A, B, C, D, either gases or
in dilute solutions, such that A by its action on B pro-
duces C and Z>, while C by its action on D produces A
A well-known example of this kind of action is the case
in which the four substances A, B, C, D are respectively
nitric acid, sodium sulphate, sulphuric acid and sodium
nitrate : the nitric acid acts on the sodium sulphate and
forms sodium nitrate and sulphuric acid, while the sulphuric
acid acts on the sodium nitrate and forms sodium sulphate
and nitric acid.
The problem we have to discuss is to find, when any
quantities of four such substances are mixed together, the
quantity of each when there is equilibrium.
Let , -T), , be the masses of the substances A, B, C,
D respectively, w lt w 2 , w 3 , w 4 the mean potential energy of
unit mass of each of these substances, w the mean potential
energy of the mixture. Let us suppose that each of these
substances obeys Boyle's law; and p denoting the density
2l6 DYNAMICS.
.
of any one at the temperature 6 and pressure /, let the
fundamental equation of A be
that of B p =
that of C p =
and that of D p = R^B.
Then the mean Lagrangian functions of A, B, C and D
are respectively
where z; is the volume in which the substances are confined,
The above expressions represent the mean Lagrangian
functions equally well whether the substances A, B, C, D
are gases or dilute solutions, provided the solutions are so
dilute that the molecules of the substances dissolved in
them exercise the same pressure as they would if placed at
the same temperature in the same volume when empty.
If we are considering solutions we shall require the
mean Lagrangian function of the solvent, for the properties
of this may alter as chemical combination goes on. If TT is
the mass of the solvent, w 5 the potential energy of unit mass,
then its mean Lagrangian function will be of the form
Try 6 + 7r/ 5 (0) TTW & .
We must now investigate the relations between the
changes in , 17, , c as chemical action goes on.
CHEMICAL EQUILIBRIUM. 217
Let us denote by (A) the molecule of the substance A,
with a similar notation for the other molecules, and let the
chemical action which goes on between the four substances
be represented by
a(A) + b(B} = c(C) + d(D] ............ (213).
Thus, in the case of the mixture of sulphuric and nitric
acids, sodium nitrate and sodium sulphate, since the equation
which expresses the reaction is
2 HNO 3 + Na 2 SO 4 - H 2 SO 4 + 2 NaNO 3 ,
if the molecules of nitric acid, sodium sulphate, sulphuric
acid and sodium nitrate in the solution are represented
respectively by HNO 3 , Na 2 SO 4 , H 2 SO 4 and NaNO 3 , then
a = 2, b = i, c= i, d 2. If however the molecules of these
substances are represented by H N 2 O 6 , Na 2 SO 4 , H 2 SO 4 ,
Na 2 N 2 O 6 , then a = b = c = d=\.
Thus we see that it is necessary to know the structure of
the molecule as well as its relative composition.
From equation (214) we see that if a molecules of A
disappear it must be because they have combined with b of B
to produce c of C and d of Z>, so that b molecules of B have
also disappeared, while c of C and d of D have appeared.
Let 8 1} 8 2 , 8 3 , 8 4 represent the relative densities of A, B, C, D
at the same temperatures and pressures, then
If the masses of A, B, C, D are altered by d^ dv\, d^ de
respectively, then the alterations in the number of molecules
of A, B, C, D are respectively proportional to
dj_ drj d di
V V V V
218
DYNAMICS.
Thus, since the alterations in the number of molecules
are proportional to a, b,-c,-d respectively, we have
So that
d*
Now when the system is in equilibrium the value of the
Hamiltonian function must be stationary, so that if we
suppose the equilibrium displaced by the quantity d of A
combining with the proper quantity of B the change in
the Hamiltonian function must be zero, hence we must
have
dH . ,.
- . =o (216).
Let us take first the case when A, J3, C, D are gases,
then since H is the sum of the mean Lagrangian function
for these substances the condition (216) with the help of
equations (215) gives the equation
R$ log - a\R$ + b^R 2 log -
- c^Rf log p
- d^R log V ^
where w = w l + rpv a + w a + cw 4 .
Then by (214) we may write this equation in the form
CHEMICAL EQUILIBRIUM. 2IQ
a dw
# # ......... ( 2I y)
when <(0) is a function of & but does not involve , ;, r -
112. In the case of dilute solutions the equation corre-
sponding to (217) is easily seen to be
Y c d a ^ w a e Q
J_l = ^) 7J c+d-a-6 JWl*t R# d ..... ( 2I g)
where Q is the mean Lagrangian function of the solvent and
equals
The value of dQjd will be zero if the properties of the
solvent do not change as chemical action goes on; in any case
since the solutions are very dilute the properties of the
solvent may be assumed to be changed by an amount pro-
portional to the quantity of salt dissolved, Q will therefore
be a linear function of , rj, , e and dQfd^ will not involve
any of these quantities, and in this case as in the former one
we have
-* K* & ......... (219)
so that the equations of equilibrium for gases and dilute
solutions are of exactly the same form.
113. The value of dw\d% measures the increase in the
potential energy of the system when the mass of is increased
by unity. Now if heat is produced when C and D combine
to form A and B, the potential energy diminishes as in-
creases, and when the quantity of heat is large its mechanical
equivalent may be taken as a measure of the decrease in
the potential energy.
If the combination of C and D is accompanied by the
production of heat, dwjd^ is negative, and we see therefore
220 DYNAMICS.
that if 6 = o
or either or e must vanish, that is, the combination of C and
D will go on until one of these substances gives out, in other
words the reaction attended by the production of heat will at
the zero of absolute temperature go on as far as possible.
According to Berthelot's law of " Maximum Work " the
reaction accompanied by the formation of heat goes on as far
as possible at all temperatures, the equation (218) however
shows that this is strictly true only at the zero of temperature.
For substances which give out large quantities of heat
when they combine equation (218) shows that the com-
bination increases so rapidly as the temperature diminishes,
that if there is any combination at all at temperatures as
high as 1000 C., Berthelot's law will be practically true at
all ordinary temperatures. To illustrate this let us take
the case of hydrogen and oxygen, where the combination
is represented by the equation
2 H 2 + 2 =2H 2 0.
Let , y, be the quantities of hydrogen, oxygen and
water respectively, then a = 2, =i, c=2, d=o, and equa-
tion (218) becomes
c>2 2 (JV!
<--I*(0).**.
If we substitute for/(0) its value given on page 270 we
shall find that </> (0) in this case = C/0 3 ' 5 . For hydrogen at
oC. JK^O T'i x io 10 , and since in the combination of one
gramme of hydrogen with oxygen 34000 calories are given out,
dw 19
^-=1.43x10".
Let us suppose that equivalent quantities of hydrogen
and oxygen are mixed together, and that the number of
equivalents which combine to form water is to the whole
CHEMICAL EQUILIBRIUM. 221
number of equivalents of either oxygen or hydrogen present
initially as x to i, then x is given by the equation
2 fl 2 dlV
Suppose that at 1092 C. one half of the equivalents
combine, then the value of x at 546 C. is given by the
equation
3-5 /l, 30 _ 130\
= 2.1 1
thus approximately
i
= - x 10
5
So that at this temperature only about one in five hun-
dred thousand of the molecules will be left uncombined.
Thus in a case like this very considerable dissociation at one
temperature is compatible with almost complete combina-
tion at a temperature not very much lower.
114. The effect of pressure on chemical equi-
librium. We have by equation (219)
thus if a + b
the ratio gJjg'irf is independent of the volume, so that if we
mix given quantities of the four substances the amount of
chemical action which will go on will be independent of
the volume into which the substances are put. Since the
chemical reaction is such that when A acts on B, a molecules
of A and b of B disappear while c of C and d of JD are
produced, we see that if a + b = c+d the number of mole-
cules in the vessel does not change as the reaction goes on.
This is sometimes expressed by saying that the combination
222 DYNAMICS.
takes place without change of volume, and in this case, as
we have just seen, the amount of chemical combination is
not affected by the volume in which the combining sub-
stances are placed. If a + b is greater than c+d then the
larger the volume z/, the smaller will be the ratio of g^ to
t^rf. Now the action of C on D tends to diminish this ratio,
while that of A on B tends to increase it, and if a + b is
greater than c + d the number of molecules is increased
when C acts on D and diminished when A acts on B.
Thus we see from equation (219) that when chemical com-
bination alters the number of molecules the state of equi-
librium depends upon the volume within which the substances
are confined, and that the effect of increasing the volume is
to favour that reaction which is accompanied by an increase
in the number of molecules. In other words, the chemical
action which produces an increase in volume is hindered by
pressure, while that which produces a diminution is helped
by it. This is another example of the law stated in (84).
115. Let us now consider a little more closely some of
the results of equation (219), taking for the sake of simplicity
the case when a = b = cd i.
Let us suppose that the masses of the four substances
A, -B, C, D before combination begins are , iy , , e , and
that when they have reached the state of equilibrium a
quantity 8,/ of A has disappeared, then by equations (215)
we have
and equation (219) becomes
MASS ACTIONS. 223
Let us suppose that the quantities of the substances
mixed together initially were proportional to their combining
weights, i.e. that initially equivalent quantities of the four
substances were taken, then we may put
And equation (220) becomes
if we put
t+p
then * =
and is called the affinity coefficient of the reaction (Muir's
Principles .of Chemistry, p. 417). Thus we may write equa-
tion (220) in the form
where / is constant as long as the temperature remains
unchanged.
The effects due to what are called " mass actions," that
is the effects produced by varying the quantities of the four
substances initially present may be deduced at once from
this equation.
Let S/j be the increase in / when is increased by S ,
the quantities r/ , > e o remaining constant; and let S/ 2 , 8/ 3 ,
8/ 4 be the respective increases when 77 , ^ , e are increased
by 877,,, 8^ , 8c respectively. Then we get at once from
equation (221)
224 DYNAMICS.
where y is the positive quantity
or
We see from these equations that tyJ8 and fy>J$r) are
positive while S/ 3 /S , S/ 4 /^ are negative, so that any increase
in the quantity of A and B initially present increases the
amount of combination that goes on between these sub-
stances, while any increase in the quantities of C and D
initially present decreases the amount of combination, and
further that the effects of equal small changes in the masses
of A, B, C, D before combination takes place are inversely
proportional to the amount of these substances present in
the state of equilibrium.
In the more general case, where a, b, c, d are not each
put equal to unity, we may easily prove that
A- f ,
CHEMICAL COMBINATION. 22$
and that if 8/ be the change in p due to an increase 87'
in volume, everything else being constant,
, 2
where y = y + __*+ _a + _, .
here a&^p is the mass of A which has disappeared.
1 1 6. The expression (221) agrees with the formula ob-
tained by Guldberg and Waage from quite different principles
(see Muir's Principles of Chemistry, p. 407, and Lothar Meyer,
Modernen Theorien der Chemie, chap. xm). The case when
a = b = c = d\s the only one however in which the expression
deduced from Hamilton's principle agrees with that given by
Guldberg and Waage. According to their theory, as given
in the works we have just cited, the equation (221) is
always true, while according to the theory we have been
explaining it is only true when a-b = c = d. It would seem
however that the principles from which Guldberg and Waage
deduced their equations would when a, b, c and d are not
all equal lead to equation (219) rather than (221), for their
point of view seems to be as follows. Consider first the
case when a = b = c = d =- i, then in a certain proportion
of the collisions which occur between the molecules of A
and B, chemical combination between A and B will take
place. The number of collisions in unit time is propor-
tional to the product of the numbers of molecules of A
and B, and so is proportional to jjt\. The number of cases
in which combination takes place may be taken therefore
to be k&\ when k is a quantity which is independent of the
quantities of A, B, (7, D present. In other words, the
number of molecules which leave the A, B states and enter
those of C and D is k&\ ; in a similar way we can see that
the number of molecules of C and D which become A and
T. D. 15
226 DYNAMICS.
B is. 'e. Now when the system is in a steady state the
number of molecules of A and B formed must be the same
as the number which disappear, and therefore
V =
which is Guldberg's and Waage's equation. We can easily
see however that the above reasoning is only applicable
when chemical combination takes place between one
molecule of A and one of B, and again between one of C
and one of D, or in other words when a = b = c = d=i. If
on the other hand the equation which represents the chemi-
cal reaction is
2 (A) + (B) = (C) + 2 (D\
then chemical combination will take place when one mole-
cule of B is in collision with two of A simultaneously ; the
number of such combinations will be proportional to i/* and
not to rjg, and thus the number of molecules of A which
disappear owing to their combination with B molecules
may be represented by ktff \ similarly the number of
molecules of D which disappear and of A which appear by
the combination of C and D may be represented by k'^ 2 ;
and since in the state of equilibrium the number of molecules
of A which disappear must be the same as the number
which appear we must have
hff = Ktf,
which agrees with equation (219) but not with Guldberg and
Waage's equation.
117. As we noticed before in (107), there is some
ambiguity as to what the molecule of the dissolved salt or
acid really is. For example, take the case already mentioned
where the reaction is represented by the equation
H 2 SO 4 + 2NaNO :j = 2 HNO 3 + Na.SO,,
we do not know whether the molecule of sodium nitrate is
CHEMICAL COMBINATION.
227
represented by NaNO 3 or by Na 2 N 2 O 6 , or whether the
molecule of nitric acid is represented by HNO 3 or H 2 N 2 O 6 .
This point could probably be settled by experiments on
osmotic pressure, the lowering of the vapour pressure of the
solution and the effect of the salt or acid upon the freezing
point. If the molecules are represented by Na 2 N 2 O 6 , H 2 N 2 O 6
and not by NaNO 3 , HNO 3 , it would be necessary to dissolve
170 and 126 grammes of these substances in a litre of water,
instead of 85 and 63 to produce the effects observed in
solutions of one gramme equivalent per litre.
We can however use the formula (219) giving the amount
of chemical action between these substances to decide this
point. If the molecules are represented by HNO 3 , Na 2 SO 4 ,
H 2 SO 4 and NaNo 3 then by equation (219) e 2 /?7 2 is constant
provided the temperature remains unaltered, if however the
molecules are represented by H 2 N 2 O 6 , Na 2 SO 4 , H 2 SO 4 , and
Na 2 N 2 O 6 (or by HNO 3 , JNa a SO 4 , JH 2 SO 4 , NaNOJ then
is constant as long as the temperature is unaltered,
where , 17, , e are the masses of the sulphuric acid, sodium
nitrate, nitric acid and sodium sulphate respectively.
This reaction has been investigated by Thomsen (Thermo-
chemische Untersuchungen i. p. 121) and in the following table
n
ef/&
*W
8
4
2.07
2.6
40.5
33
2
I
2-5
3-3
13-05
8
J
4.1
3-2
*
4.1
I.O
the values of eg/grf, /??, calculated from his experiments
2
228 DYNAMICS.
for different proportions of the substances, are given, n is
the ratio of the number of equivalents of sodium sulphate
to the number of equivalents of nitric acid before chemical
combination commences.
It will be seen from this table that when there is only a
very small quantity of nitric acid present initially, formula
(219) seems to agree with the observations as well as (221),
but that it ceases to be any approximation when the solution
gets stronger, and that now equation (221) agrees better
with the experiments. From this we should conclude that
in very dilute solutions the molecules of nitric acid and
sodium nitrate rnay possibly be represented by HNO 3 ,
NaNO 3 , but that in stronger solutions either they are re-
presented by H 2 N 2 O 6 , Na 2 N 2 O ti , or else that the molecules
of sulphuric acid and sodium sulphate are represented by
1H 2 SO 4 , JNa 2 SO 4 . Pfeffer's determination (98) of the
osmotic pressure produced by a potassium sulphate solution
suggests that the molecule is represented by J (K 2 SO 4 ). We
ought not however to attach as much weight to the experi-
ments with dilute solutions as to those with strong, because
in the weak solutions a very small error in the determina-
tions will produce a considerable error in the value of
2 /W or eC/ft.
If there was any change of this kind in the constitution
of the molecules as the strength of the solution increased
it would probably show itself in the effect of the substance
on the osmotic pressure, on the vapour pressure, and on
the lowering of the freezing point, even though these effects
were complicated by the alteration in the properties of the
solvent produced by the addition of the salt.
118. In the case we have just been considering the
four substances A, B, C, D were supposed to be either
CHEMICAL COMBINATION. 22Q
gaseous or soluble. We must now see how the equations
have to be modified when one or more of the substances is
a solid, and if we are considering the case of solutions an
insoluble one.
Let us take first the case when only one of the substances
D is an insoluble solid, for example when the four bodies
are oxalic acid, calcium chloride, hydrochloric acid, and
calcium oxalate.
The mean Lagrangian function for D will now be of the
form
and the condition
dH
-rr
will lead to the equation
** ............ (222 ).
If two of the substances are insoluble solids, as for
example when A is potassium carbonate, B barium sulphate,
C potassium sulphate, D barium carbonate, then we can
easily prove that
| = ^-^(0)/i<> .............. (223).
We see from these equations that the amount of com-
bination which goes on does not depend on the masses of
the insoluble substances.
119. As an example of a case where the conditions
are rather more complicated than in those discussed in the
last paragraph, we shall consider a case investigated by
Horstmann ( Watts' Dictionary of Chemistry, 3rd Supple-
ment, p. 433) where hydrogen, carbonic oxide and water
were exploded, and water and carbonic acid produced.
230 DYNAMICS.
Here we have to consider five substances, hydrogen,
carbonic oxide, oxygen, water, and carbonic acid ; let , ?,
, c, TT, be the masses of these substances respectively, and
let c lt r 2 ,...<: 5 , be their molecular weights.
Let the relation between the pressure /, the density p,
and the absolute temperature be for hydrogen
p-Rfa
for carbonic acid
with a corresponding notation for the others.
Let the mean Lagrangian function for the hydrogen be
where v is the volume in which the gases are confined, and
o/j the mean potential energy of unit mass of hydrogen.
The mean Lagrangian function of the other gases will be
given by analogous expressions.
Now whatever changes go on among the various gases we
have since the quantity of hydrogen is constant
- + = a constant ;
fi ^
since the carbon is constant
-n TT
+ - = a constant :
', '5
since the oxygen is constant
1 f] i 7T
i-+ ~ + i-+-=a constant ;
2 ' 2 ^3 ^ ^
these are three equations between five unknown quantities,
so that if we give arbitrary variations to two of them the
variations of the others will be determinate.
CHEMICAL COMBINATION. 231
Let us choose and rj as the independent variables, then
when 1} is constant we have
and when is constant
When the system is in equilibrium the mean Lagrangian
function is stationary for all possible variations, so that we
must have
fdff\ = Q idff\
\ dc, / 17 constant \ drf /^constant
Remembering that
the first equation gives
e ,k - 1 ( d ^L\
-! = d> (6) g^Ki 6 ^ />? constant .......... (224),
8*
and the second
i i fdw\
' lA^V^ /Constant
where w is the mean potential energy of the mixture of
gases.
These are of the same form as the equations I obtained
from kinematical considerations alone in my paper on the
Chemical Combination of Gases already referred to.
If we divide (224) by (225) we get
> MKf )-(!")!
232 DYNAMICS.
so that, as long as the temperature is constant, the ratio of
the quantity of water formed to the quantity of carbonic
acid always bears a constant ratio to the ratio of the
quantity of free hydrogen to that of free carbonic oxide.
This was the result obtained by Horstmann in the experi-
ments before mentioned.
CHAPTER XV.
EFFECTS PRODUCED BY ALTERATIONS IN THE
PHYSICAL CONDITIONS ON THE COEFFICIENT
OF CHEMICAL COMBINATION.
120. SINCE the value of
*c
IV
is independent of the values of , rj, , e, and since when it
is known the amount of chemical combination can be
determined, it is convenient to have a name for it, we
shall therefore call it the coefficient of chemical combination
for A and B and denote it by k. The more intense the
chemical action between A and B the smaller the values
of , 77 in the state of equilibrium and therefore the larger
the value of k.
We have by equation (220)
a dQ a dw
k = <^> l (ff)e **# e*i* ^ (226).
The alterations which we shall suppose to take place
in the physical conditions can be represented by changes
8(2 and 8w in the values of Q and w, and we see from
234 DYNAMICS.
equation (226) that if k be the corresponding change
in k
a d&Q a
*" "
If the substances with which we have to deal are gases
we must put Q and &Q equal to zero. We considered when
we were discussing dissociation in chapter xiv. most of the
changes in the physical conditions which could influence the
state of chemical equilibrium in this case, and the results
obtained then will apply to the more general problem we
are discussing now. We see from (227) that any cause
producing a change in the potential energy which increases
as any chemical action goes on will tend to stop this action
which will not have to go on so far before attaining equili-
brium as it would if the disturbing cause had been absent
and vice versa.
We shall now go on to consider more particularly the
cases of dilute solutions and the effects produced upon
chemical equilibrium by changes in the properties of the
solvent arising from the progress of chemical change.
121. Effect of Surface Tension. The first effect
we shall consider is that due to the surface tension of the
solution. We know that the surface tension depends upon
the strength and the nature of the solution, so that since
the composition changes as chemical action goes on the
surface tension of the solvent and therefore its mean
Lagrangian function will change ; and therefore by the
principle we have just stated the conditions for equilibrium
will be altered by the surface tension.
Let A be the area of the surface of the solution, T the
surface tension, then the potential energy due to the surface
CHEMICAL COMBINATION. 235
tension is TA and there is therefore in the expression for
the mean Lagrangian function the term TA, so that by
equation (227) the effect of the surface tension on the
coefficient of chemical combination is given by the
equation -
-- d (AT*}
k ~ Rfd^
Let us endeavour to get some idea of the magnitude of
this effect. If c is the molecular weight of the substance
whose mass is , then since at o C.
iT x io,
we have, if for simplicity a be put equal to unity,
No\v cd (A T}jd is the increase in A T when the quantity
in the solution is increased by one gramme-equivalent.
If v be the volume of the vessel whose surface we shall
suppose to remain constant as combination goes on, then
where T' is the increase in T when the quantity is increased
by one gramme-equivalent per litre. Now the experiments
of Rontgen and Schneider (" Oberflachen Spannung von
Fliissigkeiten," Wied. Ann. xxix. 165) show that T' even in
the case of simple salts may be as much as 5 or 6 so that
8& . ? A
T is of the order -=- .
k 10' v
and if the solution be spread out in a film of thickness /,
236 DYNAMICS.
A/v = 2// so that
6
6/e . 01
T is of the order,-;
thus if the thickness of the film is i/ioooo of a centimetre
the value of k is altered by about '6 per cent. If the
thickness of the layer is comparable with molecular distance,
say about io~ 7 , then kjk might be as large as 6. This of
course implies that the conditions of equilibrium would
be completely altered. Thus in very thin films the in-
fluence of capillarity might be sufficient to modify com-
pletely the nature of chemical equilibrium, though we
should not expect it to do much in the body of a fluid.
If the surface tension increases as the chemical action
goes on the capillarity will tend to stop the action, while if
the surface tension diminishes as the action goes on, the
capillarity will tend to increase the action.
Thus the chemical action in a space such as a thin
film throughout which the forces producing capillary
phenomena are active might be very different from the
chemical action in the same substance in bulk when most
of it would be free from the action of such forces.
This point does not seem to have received as much
attention as it deserves, but there are some phenomena
which seem to point to the existence of such an effect. One
of these is that called by its discoverer Liebreich " the dead
space in chemical reactions," which is well illustrated by
the behaviour of an alkaline solution of chloral hydrate.
If the proportion of alkali to chloral is properly adjusted,
chloroform is slowly deposited as a white precipitate, and if
this solution is placed in a test-tube, then at the top of the
liquid there is a thin film which remains quite clear and free
from chloroform, showing that, unless this effect is due to
, T He
fNIVERSITl
CHEMICAL COMBINATION. iF^MN^V
some chemical action of the air, the alkali and chloral do
not combine, or if they do chloroform is not precipitated.
In fine capillary tubes too, no deposit seems to be formed.
This phenomenon could be explained on the above
principles if the surface tension of the alkaline solution
increases when the alkali combines with the chloral and
chloroform is deposited, for in this case the surface tension
would increase as chemical action went on, and would
therefore tend to stop this action. Dr Monckman made
some experiments in the Cavendish Laboratory on the
changes in the surface tension of the solution as the
reaction went on, and he found that it increased to a
very considerable extent, so that this case is in accordance
with our theory. The thickness of the dead space (from i
to 2 mm. in Liebreich's experiments) is somewhat greater
than we should have expected, but any want of uniformity
in the liquid such as that produced by the deposition of
chloroform itself would increase the thickness of the dead
space.
Some other effects produced by surface tension are
discussed by Prof. Liveing in his paper " On the Influence
of Capillary Action in some Chemical Decompositions "
(Proceedings Camb. Phil. Soc. vi. p. 66).
122. Effects due to pressure. Pressure can pro-
duce effects of two kinds upon chemical action. The first
is when the volume of the liquid under pressure alters as
chemical action goes on, the effect of pressure in this case
is proportional to the amount of the pressure: the second
effect is when the coefficient of compressibility of the liquid
changes as the chemical action goes on, the effect of
pressure due to this cause is proportional to the square of
the pressure.
Let us suppose that P is the external pressure, v
238 DYNAMICS.
the volume, we may regard the external pressure as
produced by an external system whose mean Lagrangian
function is
and we have by equation (227)
?*- a -
k ~'
dv
Thus if v increases with , k is positive, in other words
the value of Tg^l^if is increased and therefore and TJ are
less than they would be if there were no external pressure.
Thus the external pressure tends to stop that action which
is accompanied by an increase in volume, and vice versa.
Let us now endeavour to form some estimate of the
probable size of this effect. If the molecules of the
substance produce the same pressure as if they were in
the gaseous state, then at o C.
2-2
where c is the combining weight of the substance. Thus if
the volume increases by y cubic centimetres per gramme of
A formed we have by (228) if the pressure is x atmospheres,
cxya
k 2'2 X I0 4 '
The cases in which in general y will have the greatest
value are those in which we have some of the bodies in
solution while others are precipitated, if we suppose that
when a salt is dissolved the volume of the solvent is not
altered then y will in general be not greatly different from
unity, and in this case we have
CHEMICAL COMBINATION. 239
S/ cax
so that it would require a pressure of 22ofac atmospheres to
change the coefficient of combination by one per cent, thus
if the substances taking part in the reaction have large
combining weights, the reaction will be sensitive to the
influence of pressure.
Let us now consider the effect on the chemical equi-
librium when the coefficient of compressibility changes as
the chemical action goes on.
Let o- be the expansion or contraction of the solution,
K its bulk-modulus, v its volume, then in the expression for
the potential energy of the solvent there is the term
ir/o-V,
and therefore in the expression for the mean Lagrangian
function the term
If S/ be the change in the coefficient of combination due to
the change in K as the chemical action progresses we have
by equation (227)
Now if P be the external pressure
K<r = P.
Substituting for cr the value given by this equation we
get
k av P* dK
= * .................. (229) '
To get some idea of the magnitude of this effect let us
suppose that when the mass of A in the solution is in-
creased by one gramme-equivalent per litre the value of K
is increased in the proportion of y to i.
240 DYNAMICS.
i d* yx io 3
1 nen r. = ,
K d v x c '
where c is the molecular weight of , we have therefore by
equation (229)
&& ay P*
_ _ <S ____ __ V T f\
~
K
NOW CR 1 0=2'2 X I O 10 ,
and for water,
K= 2'2 X IO 10 .
So that if the pressure is x atmospheres
k ax*y
-T = =f- approximately.
From the results of Rontgen and Schneider's experi-
ments given in (97) we see that y will often be as large
as i/io, so that in this case, supposing a unity, the effect
of a pressure of 100 atmospheres would be to alter k by
i/io per cent, while a pressure of 1000 atmospheres would
alter it by io per cent.
If the bulk-modulus increases as increases then the
action of the pressure is to retard the chemical action by
which increases.
123. Effect of magnetism on chemical action.
The magnetic properties of solutions are generally so feeble
that we cannot expect magnetism to produce any effect
except upon those which contain iron. In some of the
chemical actions however in which iron is dissolved or
deposited magnetism does seem to affect the result. Thus
when a solution of copper sulphate is placed on an iron
plate copper is deposited and iron dissolved, and if this
plate be placed over the poles of a powerful electro-magnet
it is found that the copper deposit is thinnest over the poles,
the places where the magnetic force is the most powerful.
MAGNETISM AND CHEMICAL ACTION. 241
The effect of the magnetic force is easily found. Let /
be the intensity of magnetization of the solution, I' that
of the iron plate, H and H' the magnetic forces and k' and
k" the coefficients of magnetization of the solution and iron
plate respectively, v and v r the volumes of these substances,
then if k' and k" are constant there is by 34 the term
2 {k k
in the expression for the mean Lagrangian function.
Thus we have by equation (227)
k 2^,t
where o- is the density of the iron, and the quantity of
iron in the solution.
Since H = k'l,
we get
k a //' 2 J z dk' \
= ( ,, j v } .
k 2R$ \k V k' 2 d )
dk'
Since in practice f' 2 /k"o- is greater than I 2 -r- vfk f we
see that k will be positive and will increase with /', hence
since k = V/Y, tne quantity of iron dissolved will be
least where /' is greatest, that is where the magnetic field
is strongest, which agrees with the results of experiment.
We can show in a similar way that any chemical action
which produces an increase in the coefficient of magnetiza-
tion is hindered by the action of magnetic forces.
If we place a solution of an iron salt in a magnetic field
where the strength is not uniform the magnetic force will
cause the strength of the solution to be greater in those
T. D. 16
242 DYNAMICS.
parts of the field where the force is intense than in those
where it is weak.
To calculate the magnitude of the effect due to this
cause let us suppose that the solution is contained in two
vessels connected with each other by a tube of small bore,
and that one vessel is placed in a region where the magnetic
force vanishes, the other in one where it is constant and
equal to H. Then if and rj are the number of molecules
of the salt in unit volume of the first and second of these
vessels respectively, we can easily prove by equating to zero
the variation of the mean Lagrangian function for the liquid
in the two vessels that
i P dk
^i' 10 *^** 5 #'
where k' is the coefficient of magnetization of the solution.
Thus if the coefficient of magnetization increases with the
strength of the solution the magnetic force will tend to drive
the salt from the weak to the strong parts of the magnetic
field.
CHAPTER XVI.
CHANGE OF STATE FROM SOLID TO LIQUID.
124. THE cases we have hitherto considered have
been those in which gases and dilute solutions have been
chiefly concerned, in this chapter we shall consider the
phenomena of solution, fusion and solidification in which
liquids and solids play the chief part.
Solution.
125. Let us consider the case of a mixture of salt
and solvent in equilibrium, and endeavour to find how
the amount of salt dissolved depends upon various physical
circumstances.
Let be the mass of the salt, rj that of the solution.
Let us for brevity denote dpjdQ for unit volume of the salt
by co and the corresponding quantity for the solution
by u>'. Let a/,, w 2 be the potential energies of unit
masses of the salt and solution respectively.
Then the mean Lagrangian function for the salt is
-to,
16 2
244 DYNAMICS.
where / (0) is the part of the mean kinetic energy of unit
mass which does not depend upon the controllable coordi-
nates.
If v be the volume of the salt and we put
then the Lagrangian function for the salt may be written
The mean Lagrangian function for the solution is with a
similar notation
where f a (0) is the part of the kinetic energy of unit mass of
the liquid which does not depend upon the controllable co-
ordinates, and v is the volume of the solution. We must
remember that though O and w l do not depend upon the
values of and ij yet the values of O', w z and f a (9) may do
so as the properties of the solution may and generally do alter
when the amount of salt the solution contains is altered.
By the Hamiltonian principle the value of the mean
Lagrangian function of the salt and solution when in
equilibrium is stationary.
Let us suppose that when the system is in equilibrium
the conditions are disturbed by a mass 8$ of the salt
melting, then the change in the value of H is, if <r
be the density of the salt, p that of the solution,
Since the value of H is stationary this quantity must
SOLUTION. 245
vanish when there is equilibrium so that we get
0' d& , i d . /m O / (0)
f
if we knew how the quantities in this equation varied with
the amount of salt dissolved we could use it to determine
the amount of salt dissolved when the solution is saturated.
But though we have not this knowledge and therefore can-
not use this equation to determine the solubility of a salt
in a given solvent, we can still get a good deal of informa-
tion from it about the effect produced by various physical
circumstances on the solubility.
126. The first effect of this kind we shall consider
is that of pressure ; and, just as in the case of chemical
combination, pressure will produce two effects, one de-
pending on the change of volume which takes place on
solution, the other on the change produced in the co-
efficient of compressibility.
Let us consider first the effect due to the change in
volume.
We may suppose that the external pressure is produced
by a weight placed on a piston which presses on the fluid,
the mean Lagrangian function of this system is
-PV,
where V is the volume of the salt and solution ; the increase
in this when diminishes by S is
246 DYNAMICS.
so that in this case instead of (230) we have now
We shall endeavour to find the change in temperature
which would produce the same effect on the solubility as
the pressure/.
We may regard the expression
O
as a function of , say/(), then if be increased by 80 the
corresponding change S in is, by equation (230), approxi-
mately given by the equation
this equation is only approximate as we have neglected
the variations of Q, O', f } (0)/0 and f 2 (6)/0 with the tem-
perature.
If S x be the change produced by the pressure /, the
temperature remaining constant, we have by equation (231)
so that the change 80 in the temperature which would pro-
duce the same effect as the pressure/ is given by the equation
SOLUTION. 247
Now w + ri - w, is the increase in the potential
ay
energy when unit mass of the salt dissolves; this will be
measured by q the mechanical equivalent of the heat
absorbed in this process at zero temperature, or at any
temperature, if the specific heat of the system does not
change as the salt dissolves: making this substitution
equation (232) becomes
If the volume diminishes as the salt dissolves
is positive, so that if q be positive the effect of pressure
is the same as that of an increase in temperature, while
if the volume increases as the salt dissolves the effect of
pressure will be the same as that of a diminution in
temperature.
The effect of pressure upon the solubility of various
salts has been investigated by Sorby (Proceedings Royal
Society, xii. p. 538, 1863). The salts he examined were
sodium chloride, copper sulphate, and the ferri- and ferro-
cyanides of potassium. He found that when the volume
increased on solution the solubility was diminished by
pressure, while when the volume diminished on solution
the solubility was increased by the same means. This
agrees with the results of equation (233).
The results of his experiments are given in the following
table the first column of which gives the name of the salt
dissolved, the second the increase in volume when 100 c.c.
of the salt crystallizes out, the third the increase in the
salt dissolved when a pressure of 100 atmospheres is ap-
plied, and the fourth the value of this quantity calculated
by (233)
248
DYNAMICS.
13-57
419
56
4-83
3'i83
2-4
2-51
o-335
28
31-21
2-914
4'4
8-9
2-845
Sodium Chloride
Copper Sulphate
Potassium Ferricyanide
Potassium Sulphate
Potassium Ferrocyanide
The numbers required to calculate by the aid of (233)
the theoretical amount of the alteration in the solubility are
given below.
The heat absorbed when the salt dissolves depends upon
the strength of the solution and the temperature, the value
of q required for our purpose is that which corresponds to a
saturated solution at the zero of absolute temperature; as
the variations in the value of q with temperature are probably
due to changes in the specific heat the effect of these
changes will be smaller the lower the temperature, we shall
always therefore take the heat of dissolution for the lowest
temperature at which it has been observed, though when
the variation with temperature is rapid this can only be a
very rough approximation.
Sodium chloride.
q at o C. for a strong solution
51
58
x 4-1 x 10
(Ostwald's Lehrbuch der Allgemeinen Chemie, n. p. 170).
Specific gravity = 2-1 (Watts' Dictionary of Chemistry,
v. p. 335)-
According to Gay-Lussac (Annales de Chemie et de Phy-
sique, xi. p. 310, 1819) the increase in solubility for each
degree centigrade is
j!7_ o/
_, 1 J /O'
Sulphate of copper.
SOLUTION. 249
q at 15 C.? - -^- 2 x 4-1 x io 9 (Ostwald, Lehrbuch, n.
249 ' 3 p. 250).
Specific gravity = 2*2 (Watts' Dictionary of Chemistry, v.
590-
Increase of salt dissolved for a rise in temperature of
i C. = 1 7% (Watts' Dictionary of Chemistry, v. 591).
Ferricyanide of Potassium.
q at i5C.?= x 4-1 x 109 (Ostwald, Lehrbuch, n.
P- 352).
Specific gravity = i '8 (Watts' Dictionary of Chemistry ',
ii. 247).
Increase of salt dissolved for a rise in temperature of
i C. = 1-27% (Watts' Dictionary, n. 247).
Potassium sulphate.
#at i5C.?= - x 4-1 x io s (Ostwald, Lehrbuch, n. p. 162).
Specific gravity = 2 '6 (Watts' Dictionary, v. 607).
Increase of salt dissolved for a rise in temperature of
i C. = 2/ (Gay-Lussac, Annales de Chemie et de Physique,
XL p. 311, 1819).
I have not been able to find corresponding data for the
ferrocyanide of potassium.
As an example of the way in which the effects of pres-
sure can be calculated from these data let us take the case
of sodium chloride: since 13*57/100 is the increase in
volume when i c.c. of the salt crystallizes out, and 2'i is
the specific gravity of the salt,
dV 'i357
250 DYNAMICS.
When the pressure is 100 atmospheres and the temperature
15 C.
/=io 8 ,
(9=288,
so that by equation (233)
8tf = 288x58 x -1357 x io 8
5-6 x 4*1 x 2'i x io 9
80 = 4'4 C.,
and since the solubility increases '13% f r each degree, the
solubility is increased by the pressure by -56 parts in 100.
Considering the imperfect nature of the data at our
disposal the agreement between the theory and the experi-
ments seems as close as could have been expected.
So far we have neglected the effect of the difference
between the compressibility of the salt and the solution, but
as this may be very considerable it is necessary to investi-
gate this effect in order to see when it may legitimately be
omitted.
If the bulk modulus of the salt is k, and that of the
solvent k', then in the mean Lagrangian function of the
two there is the term
- - ktv - - k'e' 2 v,
2 2
where as before v and v' are the volumes of the salt and the
solution respectively, and e and e their contractions.
Taking this term into account we find that the condition
dH
r
leads to the equation
d . .
dw, dV i / i i f d
-- w --~ '
,\
!'
SOLUTION. 251
and if 8(9 is the increase in temperature which would
produce the same effect as the pressure
( dV i p 2 i i / d ,,, )
80 = -\p-jfT + - -7 - + - TT 2 -jj. (kv) \ .
q Y d 2 k o- 2 k 2 d v ' }
Since k' is of the order io 10 , we see that if the change
which takes place in the volume of the salt when it dissolves
amounts to one per cent, of its original volume the terms
involving p 2 are not so important as those involving p if
the pressure is not more than 100 atmospheres. For very
much larger pressures however the terms depending upon
p 2 will be the most important, and in this case the effect
of the pressure will be proportional to the square of the
pressure and not to its first power, as in the cases examined
by Sorby.
127. Effect of Surface Tension upon the
Solubility. Surface tension may affect the amount of
salt required to saturate a solution in several ways.
In the first place the surface tension of the solution may
change as the salt dissolves; secondly, the alteration which
takes place in the volume may change the area of the
surface in contact with the glass or the air, and again when
the salt dissolves or is deposited the surface of contact of
the salt and solution may change ; when the salt is pre-
cipitated as a fine powder this increase in surface may be
very considerable.
To find the effect of these changes on the solubility, let
S be a surface of the solvent, T its surface tension. Then
in the expression for the mean Lagrangian function of the
solvent there is the term
-ITS,
where the summation is extended over all the surfaces of
the solvent.
252 DYNAMICS.
Then we get by applying the same methods as before,
proceeding as in 126 we see that the increase 80 in the
temperature which would produce the same effect as the
surface tension is given by the equation
(234),
so that if TS increases as the salt dissolves the effect of the
surface tension will be to retard solution, while it will
increase the solubility if TS diminishes.
Let us take as an example the case when the fluid is in
spherical drops and consider the effect of the change in
volume which takes place as the salt dissolves. If a is the
radius of the drop and / the increase in volume when unit
mass of the salt dissolves, then
d ,,
so that if S be the surface
dS da
and therefore by equation (234)
*-.-*
qa
SOLUTION. 253
Let us take the case of potassium sulphate, for which
i 1 1 12 and ^= 1*5 x io 9 ,
80 2T
a 1-8 x io lo;
since Tis about 81, we have at the temperature of 27C.
80 = Q - approximately,
so that if the radius of the drops was i/ioooo of a milli-
metre
80= -2-,
10
and since the solubility increases by 2 / for each degree of
temperature the solubility of spray of this fineness would be
diminished by about -6 / .
In this case the effect of the surface tension is very
small, but if the salt were deposited from its solution as a
very fine powder the effect of the increase in the surface
might be much more considerable.
Let us suppose that the salt is deposited in the shape of
small spheres of radius a, then
and if T' be the surface tension of the salt and solution we
shall have
^ = A IL -
~ a-a q '
in some cases the particles in which the salt is deposited are
fine enough to scatter light, so that their diameter must be
much less than the wave length of the blue rays, we may
254 DYNAMICS.
therefore put a = io~ 6 ; we do not know the value of T' but
it is probably greater than T for the surface of contact
of air and the solution; even though it were no greater
we should have with these numbers for potassium sulphate
80 i
y = approximately,
so that at 2jC. the solubility would be changed by about
60 per cent. This effect would help the salt to dissolve
and prevent its deposition from the solution. If the salt
before solution was not in a very finely divided condition
the diminution in the surface caused by the solution would
be much less than the increase in the surface due to the
deposition of the salt, so that surface tension would be much
more efficacious in preventing deposition from the solution
than in helping the salt to dissolve, it would thus tend to
promote something analogous to super-saturation.
Let us now consider the effect due to the alteration in
the surface tension of the solution with the quantity of salt
dissolved. We have as before
80 SdT
According to Rontgen and Schneider (Wied. Annalen,
xxix. p. 209, 1886) the surface tension of an 8 / solution of
potassium sulphate is about 3 / greater than that of pure
water, for this substance we have therefore, approximately,
dT _ 81x3
4 Sv '
where v is the volume of solution ; substituting this value for
dTjdt; in equation (235) and putting q i -5 x io 9 ,
SOLUTION. 255
80 81x3 S
x
6 8 x i -5 x io 9 v
= a approximately.
io 8 v
If the solution is in a cylindrical tube of radius a, S/v = 2 /a,
and therefore
SO 4
-Q 8 a approximately.
The sign is changed because if the angle of contact
vanishes the increase in the surface tension of the surface
separating the solution and air is equal to the diminution
in that separating the solution and the walls of the tube.
If we suppose that these cylindrical tubes are of the dimen-
sions of the pores in such substances as meerschaum or
graphite, then since we know by the laws of diffusion of
gases through these substances that the diameter of the
pores must be comparable with the mean free path of a
molecule of the gas we may assume that a is of the order
10 ~ 6 . In this case
W _^_
i oo '
so that at 27 C. the value of W would be about 12 C,
which in the case of potassium sulphate is equivalent to
an increase in the solubility by nearly 25 / . In most cases
the surface tension of the surface separating a solution
from air increases with the amount of salt in it, so that
the salt will be more soluble in liquid in capillary spaces
than in liquid in bulk.
Liquefaction.
128. Under this head we shall consider the influence
of changes in the physical condition on the passage of a
substance from the solid to the liquid state and vice versa.
256 DYNAMICS.
This problem has much in common with that of solution,
but since in this case the liquid and solid are the same
substances in different states, the properties of the liquid
will not, as in that case, change as the solid melts.
Let be the mass of the solid, by 81 its mean Lagran-
gian function is
where w l is the potential energy of unit mass of the
substance in the solid state.
Since the Lagrangian function is proportional to the
volume, we may put
(
J
5*-
where v is the volume of the solid.
If rj l is the mass of the liquid, v' its volume, W 2 the
potential energy of unit mass, the Lagrangian function of
the liquid is
where O' is denned by the equation
The terms /,(#), / 2 (#) are the parts of the Lagrangian
function which do not depend upon strain &c., that is, they
do not involve the controllable coordinates. They are
therefore independent of the arrangement of the molecules
and depend merely upon the number of the molecules and
the kinetic energy possessed by each. We should therefore
expect that so long as the temperature remains constant
these terms would not alter much however the arrangement
of the molecules might change, provided the molecules were
not decomposed.
LIQUEFACTION. 257
If there is no external pressure the change in the mean
Lagrangian function of the solid and liquid when the mass
8 of the liquid freezes is
?- 7)
where o- and p are the densities of the solid and liquid
respectively,
This change must vanish by the Hamiltonian principle
when the system is in equilibrium, so that in this case we
have
We may regard the left-hand side of this equation as a
function of 6, say <f> (0), which when equated to zero gives
the temperature at which melting takes place.
Let us now consider the effect of a slight change in the
physical conditions. If this change increases the Lagrangian
function by x and does not affect appreciably the values of
O/cr, O'/p, we have if the melting point is now 6 + 30,
or since <f> (0) = o
Let us consider the effect of pressure upon the freezing
point. If the external pressure is / then
x = -p(v + i/) 9
and since
T. D.
258 DYNAMICS.
so that
But from equation (116) we have
S C = <?S, (%)
^ \dOj
V constant
and if the heat supplied is just sufficient to melt unit mass
of ice, 8<2 = A, the latent heat of liquefaction and v = i/p - i/<r,
hence
whence if 80 be the increase in caused by the pressure /
comparing this with (237) we see that
afo (0) _ X
~dO~ ~0'
and equation (237) becomes
So that if the Lagrangian function increases when the liquid
freezes, the temperature at which freezing takes place is
raised, in other words freezing is facilitated. This is
another example of the principle of 84.
We see from equation (238) that if the body expands
on solidification 80 is negative or the melting point is
lowered by pressure, if the body contracts on solidification
W is positive and the melting point is raised by pressure.
This is the well-known effect predicted by Prof. James
Thomson and verified by the experiments of Sir William
Thomson.
This however is not the only effect produced by pressure
LIQUEFACTION. 259
on the melting point, there is another effect arising from the
difference between the energy due to strains produced by
the pressure in unit mass before and after solidification.
This energy is proportional to the square of the pressure, so
that the lowering of the freezing point from this cause will
also be proportional to the square of the pressure.
Let as before / be the pressure per unit area acting on
the solid and liquid, let k be the modulus of compression
of the solid, k' that of the liquid, the potential energy due
to the strain in the solid and liquid is
so that in the mean Lagrangian function of the solid and
liquid there is the term
-/ \v a (i - 8) + ,' (i - a-)} - j *ja? - j.*8",
where 8 and 8' are the compressions and v and z> ' the
volumes, <T O and p the densities of the solid and liquid when
free from pressure.
If 80 be the rise in the melting point due to this cause
Ave see from (239) that
Po 2 O'o 2 PC
I
So that unless k<r = k'p Q the freezing point will be altered
by an amount proportional to the square of the pressure.
Let us find the magnitude of this effect in the case of ice
and water. The only constant of elasticity for ice which
has been determined is Young's modulus, which Bevan de-
termined by flexure experiments to be about 6 x io 10 , the
modulus of compression k is therefore not likely to be less
172
260 DYNAMICS.
than 4 x io 10 . The value of this quantity for water is about
2 x io 10 . Substituting these values we get
3(9 i
<r = -8^' -
roughly.
This acts in the same direction as the effect due to
the alteration in volume on solidification. Comparing this
expression with equation (238) we see that for pressures less
than about 9000 atmospheres the effect depending on the
change in volume is the more important, while for pressures
greater than this the effect we have, just been investigating
is the larger. If kv is greater than k'p then this effect in
the case of substances which contract when they solidify
is in the opposite direction to that which is proportional to
the first power of the pressure, so that in these cases the
effect of pressure upon the freezing point is reversed when
the pressure exceeds a critical value.
129. Effect of torsion upon the freezing point.
Let us suppose that we have a cylindrical bar of ice twisted
with a uniform twist about its axis ; it will possess energy
in virtue of the strain, but if it melts (suppose on the
outside) the water will be free from strain, and will not
therefore possess any energy corresponding to that possessed
by the twisted ice. Thus the potential energy will diminish,
and the Lagrangian function therefore increase as the ice
melts, so that by the principle stated in 84 the torsion
will facilitate the melting of the ice, that is, it will lower the
freezing point.
We can easily calculate the magnitude of this effect.
Let us take the case of a thin cylindrical tube of ice, since
in this case the strain is uniform, and let a and b be
respectively the external and internal radii of the tube,
LIQUEFACTION. 26l
/ its length, n the coefficient of rigidity of ice, <j> tn e
uniform twist produced by a force P acting at an arm ,
then in the mean Lagrangian function of the tube there are
the terms
- - Itfn (a 4 - 0*)
4
where v is the volume of the ice.
So that if 80 be the rise in the freezing point produced
by the torsion we have
= i [Pbl- }*** (a' + If)] - J**
If the sides melt equally we have since a and b are
approximately equal
so that
20-X
since
dv _i
d^ IT'
To get some estimate of the magnitude of this effect let
us suppose that the cylinder is i centimetre in radius, and
that < is 1/40. Since Young's modulus for ice is 6 x io 10 ,
n is probably about 2 -4 x io 10 , substituting these values we
262 DYNAMICS.
find
80 i
-5- = approximately,
so that 80 = - -68 C.
So that in this case the ice on the surface would melt
unless the temperature was lower than - -68 C.
130. Effect of surface tension on the freezing
point. If a portion of a drop of water freezes, the form-
ation of the ice will cause a diminution in the surface of
separation of the water and air if the ice rises to the
surface of the drop, to balance this however we have two
fresh surfaces formed where the ice meets the water and air;
the diminution in the first surface would tend to promote
freezing, the formation of the other two would tend to pre-
vent it, but as we do not know the surface tension between
ice and water and between ice and air we cannot calculate
which of these tendencies would have the upper hand.
131. The effect of dissolved salt on the freezing
point. When a salt solution freezes the salt appears to
remain behind, and the ice from such a solution is identical
with that from pure water. Thus when a portion of a salt
solution freezes, the particles of salt are brought closer to-
gether, and work has therefore to be done upon them, the
Lagrangian function therefore diminishes, and we see by
equation (239) that the presence of the salt will tend to>
prevent the water from freezing. To calculate the magni-
tude of this effect, let be the mass of the salt, then using
the same notation as before, the mean Lagrangian function
for the salt if the solution is dilute is
LIQUEFACTION. 263
where w 3 is the mean potential energy of unit mass of the
salt. When the mass of ice is increased by 8 the only
quantity which changes in the expression is if which dimin-
ishes by S//o.
Thus equation (239) becomes
W i (Oft /Q'
- - H
where 8 (Q'/p) and B/ a (6) are the changes in O'/p and f a (6)
due to the salt. If zzr be the pressure due to the molecules
of salt in the solution,
so that
If we suppose that the salt does not alter the properties
of the solvent we have
Let us first suppose that the solvent is water; if we
consider solutions whose strength is such that a number of
grammes equal to the formula weight is dissolved in one
litre of water, then w is about 22 atmospheres, or in absolute
measure about 2 2 x io 7 , X = 80 x 4 2 x io 7 , = 273, and
p is unity ; substituting these values we get
86 = -i-8C.
Raoult, Annales de Chimie et de Physique, v. in. p. 324, 1886,
found that solutions of this strength of many substances,
chiefly organic salts, froze at --1*9, but that the freezing
points of solutions of salts and acids were generally lower
than this ; he attributed the increased effect to the dis-
264 DYNAMICS.
sociation of the molecules; it might however, as in the
analogous cases we considered before, be due to the altera-
tion of the properties of the solvent by the addition of the
salt. It would also take place if there were any chemical
action between the salt and solvent of such a nature that
heat is evolved when the solution is diluted.
When the solvent is acetic acid, X 44*34 x 4-2 x io 7
(Landholt and Bornstein Tabellen) p = i-o5 and $ = 290;
substituting these values we get for the lowering of the
freezing point of any solution of the same strength as before
In this case Raoult found 80 = 3*9.
When the solvent is benzine, A=29x4'2xio 7 , p = '9
and 0= 275, so that the lowering of the freezing point of a
solution of the same strength as before is
80= - 5-4 C.
Raoult found in this case that 80 was - 4*9 C.
Raoult found that the effect of dissolved salts on the
freezing points of acetic acid and benzine was much more
regular than their effect on the freezing point of water.
CHAPTER XVII.
THE CONNEXION BETWEEN ELECTROMOTIVE FORCE
AND CHEMICAL CHANGE.
132. THE principle that when a system is in equilibrium
the Hamiltonian function is stationary can be applied to de-
termine the connexion between the electromotive force of a
battery and the nature of the chemical combination which
takes place when an electric current flows through it.
We shall begin by considering Grove's gas battery, as
this is the case where the chemical changes seem on the
whole to be the least complex. In this battery the two
electrodes are covered with finely divided platinum, the
upper half of one is surrounded by some gas, say hydrogen,
while the lower half dips into acidulated water ; the upper
half of the other electrode is surrounded by some other
gas, say oxygen, the lower half again dipping into acidu-
lated water. The two electrodes are well coated with
hydrogen and oxygen respectively. If the electrodes are
connected a current will flow through the battery and the
hydrogen and oxygen above the electrodes will gradually
disappear, while the water will increase during the passage
of the current.
To investigate the electromotive force of a battery of
this kind let us suppose that the electrodes have got into a
266 DYNAMICS.
permanent condition, so that the gases attached to them are
not altered during the passage of the current, let us also
suppose that the electrodes are connected with the plates of
a condenser whose capacity is C, these plates being made
of the same material. Then if unit quantity of positive
electricity flows from the plate of the condenser which
is connected with the hydrogen electrode through the cell to
the other plate, by Faraday's Law an electrochemical equiva-
lent of hydrogen will appear at the electrode covered with
oxygen and one of oxygen at the electrode covered with
hydrogen; the hydrogen and the oxygen will combine and
the result of the passage of the unit of electricity will be that
an electrochemical equivalent of hydrogen and one of oxygen
will disappear and an electrochemical equivalent of water
will appear. The systems whose mean Lagrangian functions
change during this process are (i) the condenser, (2) the
hydrogen above one electrode, (3) the oxygen above the
other, and (4) the water.
Let Q be the quantity of positive electricity on the plate
of the condenser connected with the oxygen electrode, and
let , rj, be the masses of the hydrogen and oxygen above
the electrodes and of the acidulated water respectively.
The mean Lagrangian function for the condenser is
The mean Lagrangian function for the hydrogen is
where using the same notation as hitherto,
The mean Lagrangian function for the oxygen is
where
Za = *,0 log ^
ELECTROMOTIVE FORCE. 267
and for the acidulated water Z W where
Now .when unit of electricity passes from the one plate to
the other of the condenser, the electrochemical equivalent
of hydrogen is carried to the oxygen and there combines with
it at one electrode, while the electrochemical equivalent of
oxygen is carried to and combines with the hydrogen at the
other electrode. Thus if c l and <= 2 are the electrochemical
equivalents of hydrogen and oxygen, the net result of
the process is that Q has increased by unity, and t]
diminished by e 1 and e 2 respectively, while has increased
by (ej + 2 ). Hence by the principle that the Hamiltonian
function is stationary when there is equilibrium we must
have
but Q/C is the amount by which the potential of the plate
connected to the oxygen electrode exceeds that of the one
connected to the hydrogen electrode, in other words it is
the electromotive force of the battery, which we shall call /,
hence
If LW> be the mean Lagrangian function of unit mass of
aqueous vapour above the acidulated water and in equilibrium
with it, we have by 83
where is the mass of the aqueous vapour, and
Z W ^ 8 01og^Va'W-<.
Substituting these values in (240) we get
268 DYNAMICS.
= t&O log ^ + t 2 X 2 log^ - (c, + c,) ^ log
{ l ^ + V? 2 - (c, + c
But
and by (83)
-
is of the form
Lastly e^/j + e 2 o/ 2 - (, + 2 ) w^
is the loss of potential energy which occurs when an electro-
chemical equivalent of hydrogen combines with one of
oxygen and may be measured by the quantity of heat
developed by the combination of an electrochemical equiva-
lent of hydrogen at the zero of absolute temperature; we
shall denote it by e^, making these substitutions we see
/= ^0 lOg ^ + ^0 + ^0 lOg +,?... (241)^
/>
where A - A' + e l J? 1 log -^,
PoPo"
hence we have 0* - - 6 -^ +p = t.q ............. (242).
u-u au
Thus if we know the way in which/ depends upon 6 we can
determine g, so that by measurements of the electromotive
force of a cell and the variations of this force with the tem-
perature we can calculate the mechanical equivalent of the
heat developed in the combination which takes place in the
cell.
133. Equations (241) and (242) are not confined to the
case of the Gas Battery. We can prove in a similar way that
if p is the electromotive force of any battery where the
solutions used are dilute, then
ELECTROMOTIVE FORCE. 269
where e l is the electrochemical equivalent of hydrogen, }
the value of jR for this gas, /o 1? /o 2 ... the masses in unit volume
of those substances which disappear as the chemical action
which produces the current goes on, while o^, o- 2 ... are
the masses in unit volume of those which appear, a, b, ...c, d, ...
are the ratios of the electrochemical equivalents of the sub-
stances to that of hydrogen, divided by the molecular weight
of the substance, f.q is the mechanical equivalent of the
heat which would be evolved at the absolute zero of tempera-
ture by the chemical action which takes place when unit of
electricity passes through the cell.
From this equation we get as before
*-+'-* ............... (*>
By v. Helmholtz's principle (48) OdpjdB is the heat
which must be supplied to the cell in order to keep the
temperature constant when the unit of electricity passes
through the cell, or in other words BdpjdO is the mechanical
equivalent of the heat which is "reversibly generated when
unit of electricity passes through the cell. Now/ the work
done in driving this quantity of electricity through the cell
plus OdpfdO the heat reversibly generated must be equal to
fw the heat equivalent of the chemical action which takes
place in the cell, hence by (243) we have
2
Now W and tq are the mechanical equivalents of the
heat developed by the same combination when it takes place
at the temperatures and absolute zero respectively, and
the difference between these quantities must be the differ-
ence between the mechanical equivalents of the quantities of
heat required to raise them from zero to 6 degrees in their
2/0 DYNAMICS.
combined and uncombined states. If we consider the
case when two gases A and B combine to form two others
C and D y then if c lt c z , c# <r 4 are the mechanical equivalents
of the specific heats at constant volume of these gases,
15 c 2 , 3 , 4 their electrochemical equivalents, if we start
with A and B at zero and raise them to degrees and then
let them combine, we shall spend (e^ + 6 2 c 2 ) 6 units of work
in raising the temperature and gain ew by their combination,
so that the net result in our favour will be
W-(^ + vv)0.
If we let them combine at zero temperature and then
raised them to we should gain tq and spend (e 3 ^ 3 + e/ 4 )
units of work, hence since the balance of work in our favour
must be the same in both cases, we have
? ~ ( V 3 + / 4 ) = ew - (c^,
and therefore by (245)
But by (241)
so that B = / 3 + e/ 4 - e^ - / 2 ............ (246).
If the combination is attended by the production of an
amount of heat comparable with that which occurs when
hydrogen and oxygen combine, then 6 2 d 2 fl/d6 2 , which is com-
parable with the heat required to raise the temperature of
the substances degrees and is therefore at the most a few
hundred calories per gramme of substance, will be small
compared with q, which is measured by thousands of calories,
so that when the combination is attended by the evolution
of a large quantity of heat we may at ordinary temperature
neglect 6*d*pld6 2 and write
ELECTROMOTIVE FORCE.
Since by Dulong's and Petit's law c lt c^ c^ c 4 are in-
versely proportional to the combining weights of the gases
A, I>, C, D, we see that whenever the combination leaves
the number of molecules unaltered B will vanish and the
equation
will be rigorously true. We see by this equation that when
the electromotive force increases as the temperature in-
creases the electromotive force is greater, while when the
electromotive force diminishes as the temperature increases
it is less than that calculated from the formula p = c^, which
is often employed.
If k be the coefficient of the chemical combination (115)
which goes on in the cell, i.e. the value of
when the densities of the gases or solutions have the values
they possess when in chemical equilibrium with each other,
then since any small change cannot alter the value of the
mean Lagrangian function of the gases or dilute solutions
when in equilibrium, we get if we suppose the change is that
which would take place if unit of electricity were to pass
through the solutions
o = e l R,P\Qgk + AO + 0\o%0 + cq ...... (247);
combining this with (245) we get
or l og = l og ............ (248) .
2/2 DYNAMICS.
This equation affords a very easy method of finding the
coefficient of any chemical combination if we can make a
cell in which this combination takes place, for then if we
measure the electromotive force and the densities of the
solutions, equation (247) will at once give k. Thus the
Daniell's cell enables us to calculate the coefficient of the
combination
Zn + H 2 SO 4 + CuSO 4 = ZnSO 4 + H 2 SO 4 + Cu.
Here if p and o- are the masses per unit volume of the
CuSO 4 and ZnSO 4 respectively when there is chemical
equilibrium
so that if p' and o-' are the densities .of the CuSO 4 and the
ZnSO 4 when the electromotive force is p we have
Now at o C. t^Rfi is nearly io 6 and/ is about io 8 so that
log k = ^ log - 100,
or approximately since for ordinary strengths of solution
log/// "' is small compared with 100
log e - = -200,
hence we see that in this case when there is equilibrium
practically all the sulphuric acid goes to the zinc.
If we determined the electromotive force of a battery
when lead wire dipped respectively into acid solutions of
lead nitrate and lead chloride, we should be able by equation
(247) to determine the coefficient of the action
2 HC1 + Pb(NO d ) 2 = 2HNO 3 + Pb C1 2 ,
ELECTROMOTIVE FORCE. 273
and so determine the way in which lead divides itself
between hydrochloric and nitric acids.
If we return now to the hydrogen and oxygen gas
battery, equation (247) is for this case
(249).
We can easily deduce from this equation the way in
which the electromotive force of a gas battery depends
upon the pressure of the gases in the vessels above the
electrodes. If p l is the electromotive force of the battery
when the densities of the hydrogen and oxygen are p, p
respectively, / 2 the electromotive force when the densities
are a- and <r', then we have by (249)
If the densities of the oxygen and hydrogen were
diminished one thousand times then at the temperature
o C. since e = io~
JT3 f\ ------ .~ oe --
~i'i4xio 7 approximately,
so that the electromotive force is diminished by rather less
than the ninth of a volt. By making the densities of the
gases above the electrodes sufficiently small we could
reverse the electromotive force, though in the case when
the gases are oxygen and hydrogen the rarefaction required
would be more than could practically be obtained.
The diminution in the electromotive force caused by
rarefaction does not however depend upon the magnitude
of the electromotive force of the battery, so that in the case
T. D. 18
274 DYNAMICS.
of gas batteries with small electromotive forces this reversal
might be practicable.
The condensation which accompanies the combination
of oxygen and hydrogen diminishes the effect of rarefaction ;
if the combination were to take place without condensation,
the diminution in the electromotive force caused by
diminishing the density one thousand times would be about
one-seventh of a volt.
We see too from equation (249) that the electromotive
force in all cases tends to produce a current the chemical
action of which would make the densities of the gases
or dilute solutions approach the values they have when in
chemical equilibrium with each other. When they have
these values the electromotive force of the battery is zero,
and the electromotive force is in one direction or the oppo-
site according as there is more or less of some substance
present than there would be if the mixture of gases or dilute
solutions were in chemical equilibrium.
Experiments on the electromotive force of gas batteries
charged with various gases have been made by Pierce
(Wiedemands Annalen, vui. p. 98). The following table
taken from his paper gives the electromotive force of a large
number of batteries at 15 C. and of a few at 75 80 C.
It will be seen from this table that the effect of an
increase in temperature on the electromotive force of gas
batteries is very variable, for of the five batteries whose
electromotive forces were determined at different tempera-
tures, the, electromotive forces of three were less and of two
greater at the high temperature than the low.
ELECTROMOTIVE FORCE.
275
ELECTROMOTIVE FORCE AT i8C.
Gases.
Fluid between the
electrodes.
Ratio of elec-
tromotive
force to that
of a Daniell.
Gases.
Fluid.
Electro-
motive
force.
HandO
water
874
I and Br
water
'335
H and N 2 O
water
790
H and Br
NaBr + water
I'252
H and CO 2
water
9 8l
H and Br
KBr + water
i' 2 53
H and NO
water
'933
O and Br
KBr + water
'5
H and air
water
80 7
O and I
KI + water
057
H and H 2 O
water
807
Hand I
KI + water
86 r
H and CO
water
404
H and NO
HC1 + water
765
HandO
H 2 SO 4 + water
926
HandO
HC1 + water
855
H and CO 2
H 2 SO 4 + water
892
H and Cl
HC1 + water
I'/
H and NO
H 2 SO 4 + water
768
H and Cl
KC1 + water
i'39
H and O
Na 2 SO 4 + water
698
H and Cl
NaCl + water
i '39
H and O
K 2 SO 4 + water
698
H andO
NaCl + water
766
HandO
ZnSO 4 + water
771
H and CO 2
NaCl + water
8 4 6
H and CO 2
ZnSO 4 + water
820
H and NO
NaCl + water
750
H and NO
ZnSO 4 + water
860
H and O
H and NO
H and CO 2
ELECTROMOTIVE FORCE AT 7 5C-8oC.
water '828 I H and N 2 O i water
water '945 I H and H 2 O water
water '875
780
"954
The electromotive force of the hydrogen and oxygen gas
battery where iq = 3*4 x 4-2 x io 7 is less than that given by
the formula (246) even when the variation of the electro-
motive force with temperature is taken into account. This
seems most probably to arise from the arrangements being
such that the complete combination of the hydrogen and
oxygen contemplated in the preceding theory would not
take place, for we see from the table that the substitution
of acidulated water for water between the terminals increases
the electromotive force, this change favours the production
of ozone instead of oxygen when the current passes and so
increases the chance of complete combination.
1 8 2
2/6 DYNAMICS.
In the case of the hydrogen and chlorine battery with a
solution of hydrochloric acid between the electrodes, q in
formula (244) will be the heat in mechanical units given
out in the combination of one gramme of hydrogen with
chlorine plus the heat given out when 36-5 grammes of
hydrochloric acid are dissolved in a large quantity of water.
The gas battery will work even if we have the same
gas (say hydrogen) above the electrodes provided it is at
different pressures. In this case on closing the circuit there
will be no change in the volume of the liquid between the
terminals, but when the unit of electricity passes through the
battery an electrochemical equivalent of hydrogen will be
transferred from the vessel where the pressure is high to
the one where it is low. The electromotive force in this
case is easily seen to be
6,^0 log p/cr,
where p and or are the densities of the hydrogen (or oxygen)
in the two vessels, at o C. this equals
i o 6 log p/cr approximately,
so that if the density in one vessel is e times that in the
other the electromotive force will be one-hundredth of a
volt.
In fact when we have any arrangement in which the
passage of an electric current in a certain direction increases
the Lagrangian function of the system, there will be an
electromotive force tending to produce a current in this
direction and equal to the increase in the mean Lagrangian
function produced by the passage of unit of electricity.
134. We can sometimes transform equation (244) by
means of the following considerations. If we have a mixture
of chemical reagents in various proportions we can in many
cases though not in all find a temperature at which they
ELECTROMOTIVE FORCE. 277
would be in equilibrium if mixed in these proportions. Let
us suppose that it is possible to find a temperature at
which the reagents constituting the battery would be in
equilibrium in the proportion in which they exist in the
battery at the temperature 6.
Then by (243)
and by (246)
a b
Subtracting these equations we get
(250)-
If a considerable quantity of heat is given out by the
combination which takes place when unit of electricity
passes through the cell, then at ordinary temperatures the
last term on the right-hand side of this equation will be
small compared with the first and we may write equation
(250) in the form
. ..
An equation identical in form with this is given by Professor
Willard Gibbs in a letter to the Electrolysis Committee of
the British Association (British Association Report, 1886,
p. 388). According to Prof. Gibbs is the highest
temperature at which the radicles can combine with evolu-
tion of heat, while according to our view it is the tempera-
ture at which the .chemical system forming the battery
would be in equilibrium, and as it is not always possible
to find such a temperature the formula is not of universal
application. We see that for all cells to which the formula
2/8 DYNAMICS.
can be applied the temperature coefficient of the electro-
motive force must be negative, and therefore by v. Helm-
holtz's principle, the passage of the current through the cell
must be attended by the evolution of heat. When the
temperature exists it is given by the equation
= - approximately.
dO
We shall now investigate under what conditions it is
possible to find a temperature at which the system would
be in equilibrium. We shall consider the case of the
equilibrium of four substances (A), (B\ (C), (D).
If Pl , p 2 , <r v <r 2 are the densities of (A), (B), (C\ (D)
when there is equilibrium we have by equation (246)
o = e, j^fl log ^-^ + AO + BO log + tq.
Now if e e a , e# e^ are the electrochemical equivalents
and f lt <; 2 , <r 3 , <r 4 the specific heats at constant volume of the
substances (A\ (J3\ ( C), (D) respectively, then by equation
(246)
^ = '
so that
Now if M lt M 2 , M# M 4 are the molecular weights of the
substances
ELECTROMOTIVE FORCE. 279
and for gases by Dulong and Petit's Law
^ = M 2 c 2 = M./ 3 = M 4 c 4 = c say
p g
e^ is the heat given out when the quantity of (A) decomposed
by one unit of electricity combines with the equivalent
quantity of B. Let us first suppose that this quantity is
positive, and consider the following cases.
ist case. When a + b = c + d, i.e. when there is no change
in volume on combination. In this case as 6 increases from
zero to infinity pfpf/vfcrf ranges from zero to C, and there-
fore since it never exceeds C it is not always possible to find
a temperature which should be one of equilibrium for any
arbitrarily chosen set of values of p l9 p 2 , cr lt cr 2 .
2nd case. When a + b<c+d, i.e. when there is an in-
crease in volume after combination. In this case as 6 in-
creases from zero to infinity P 1 a p s *l<r 1 c <r a lt starts from zero then
reaches a maximum and decreases again to zero, so that
again as pfp*l<rfcr* never exceeds a certain maximum it is
not always possible to find a temperature which should be
one of equilibrium for any arbitrarily chosen set of values of
Pi P> V V
3rd case. When a + b > c + d, i.e. when there is a diminu-
tion in volume after combination. In this case as 6 increases
from zero to infinity p l a p a */cr ] c cr a rf also increases from zero to
infinity, so that in this case it is always possible to find
a temperature which should be one of equilibrium for any
arbitrarily chosen set of values of p lt p 2 , cr^ <r 2 .
We see too that when the combination is attended with an
absorption of heat it is in general only possible to find a tem-
perature which shall be one of equilibrium for any arbitrarily
280 DYNAMICS.
chosen set of values of p lt p 2 , o^, a- 2 when the combination is
attended by an increase of volume.
Summing up the results of this investigation we see that
equation (250) can only in general be applied to cases
where the reaction producing heat is accompanied by a
diminution in volume.
In these cases where pfpf/o-fo-* has a maximum value
at a finite temperature the mixture of gases after passing
this temperature will be in an unstable state, for any
increase in the temperature will promote combination
and produce an evolution of heat which will increase the
temperature still further.
CHAPTER XVIII.
IRREVERSIBLE EFFECTS.
135. WE have hitherto left out of consideration the
effect of such things as frictional and electrical resistances
which destroy the reversibility of any process in which they
play a part. If however we take the view that the properties
of matter in motion, as considered in abstract dynamics, are
sufficient to account for any physical phenomenon, then
irreversible processes must be capable of being explained as
the effect of changes all of which are reversible.
It would not be sufficient to explain these irreversible
effects by means of ordinary dynamical systems involving
friction, as friction itself ought, on this view, to be explained
by means of the action of frictionless systems.
But if every physical phenomenon can be explained by
means of frictionless dynamical systems each of which is
reversible, then it follows that if we could only control the
phenomenon in all its details, it would be reversible, so that
as was pointed out by Maxwell, the irreversibility of any
system is due to the limitation of our powers of manipulation.
The reason we can not reverse every process is because we
only possess the power of dealing with the molecules en masse
and not individually, while the reversal of some processes
282 DYNAMICS.
would require the reversal of the motion of each individual
molecule.
We are not only unable to manipulate very minute
portions of matter, but we are also unable to separate events
which follow one another with great rapidity. The finite
time our sensations last causes any phenomenon which
consists of events following each other in rapid succession
to present a blurred appearance, so that what we perceive
at any moment is not what is happening at that moment,
but merely an average effect which may be quite unlike the
actual effect at any particular instant. In consequence of
the finiteness of the time taken by our senses to act, we are
incapable of separating two events which happen within a
very short interval of each other, just as the finiteness of the
wave length of light prevents us from seeing any separation
between two points which are very close together. Thus if
we observe any effect we cannot tell by our senses whether
it represents a steady state of things or a state which is
rapidly changing, and whose mean is what we actually
observe. We are therefore at liberty, if it is more convenient
for the purposes of explanation, to look upon any effect as
the average of a series of rapidly changing effects of a
different kind.
Let us now consider the case of a system whose motion
is such that in order to represent it frictional terms
proportional to the velocity have to be introduced, and let
us assume at first that the motion is represented at each
instant by the equations with these terms in, so that the
dynamical equations are not equations which are merely
true on the average.
It might appear at first sight as if we could explain the
frictional terms in the equations of motion as arising from
the connexion of subsidiary systems with the original system
IRREVERSIBLE EFFECTS. 283
just as in ii we explained the "positional" forces as
due to changes in the motion of a system connected with
the original system. Let us suppose for a moment that
this is possible. Then if T is the kinetic energy of the
original system, and T that of the subsidiary system whose
motion is to explain the frictional forces, we have, by
Lagrange's equations,
d dT dT ddT dT dV
-j-. -p- - -j -j. -=r- - -j- + -3 = external force of type x ;
dt dx dx dt dx dx dx tff^
thus the term
CA\
must be equal to the "frictional term" which is proportional
to x. For this to be the case, it is evident that T must
involve x. The momentum of the system is, however,
d(T+ T')/dx, and this momentum must be the same as
that given by the ordinary expression in Rigid Dynamics,
viz. dT\dx. If these two expressions are identical, dT'/dx
must vanish for all values of x, that is, T cannot involve x,
which is inconsistent with the condition necessary in order
that the motion of the subsidiary system should give rise
to the "frictional" terms. Hence we conclude that the
frictional terms cannot be explained by supposing that any
subsidiary system with a finite number of degrees of freedom
is in connexion with the original system.
If we investigate the case of a vibrating piston in con-
nexion with an unlimited volume of air, we shall find that
the waves starting from the piston dissipate its energy just
as if it were resisted by a frictional force proportional to its
velocity j this, however, is only the case when the medium
surrounding the piston is unlimited, when it is bounded
by fixed obstacles the waves originated by the piston get
284 DYNAMICS.
reflected from the boundary, and thus the energy which
went from the piston to the air gets back again from the
air to the piston. Thus the frictional terms cannot be
explained by the dissipation of the energy by waves starting
from the system and propagated through a medium sur-
rounding it, for in this case it would be possible for energy
to flow from the subsidiary into the original system, while,
if the frictional terms are to be explained by a subsidiary
system in connexion with the original one, the connexion
must be such that energy can flow from the original into
the subsidiary system, but not from the subsidiary into the
original system.
Hence we conclude that the equations of motion, when
they contain frictional terms, represent the average motion
of the system, but not the motion at any particular instant.
Thus, to take an example, let us suppose that- we have a
body moving rapidly through a gas ; then, since the body
loses by its impacts with the molecules of a gas more
momentum than it gains from them, it will be constantly
losing momentum, and this might on the average be repre-
sented by the introduction of a term expressing a resistance
varying as some power of the velocity ; but the equations of
motion, with this term in, would not be true at any instant,
neither when the body was striking against a molecule of
the gas, nor when it was moving freely and not in collision
with any of the molecules. Again, if we take the resistance
to motion in a gas which arises from its own viscosity, the
kinetic theory of gases shows that the equations of motion
of the gas, with a term included expressing a resistance
proportional to the velocity, are not true at any particular
instant, but only when the average is taken over a time
which is large compared with the time a molecule takes to
traverse its own free path.
IRREVERSIBLE EFFECTS. 285
Since frictional forces cannot be explained by means of
a system in a uniform state, we shall consider the dynamics
of a system which is subject to the action of forces which
last only for a short time but which recur very frequently.
Let us suppose that in the expression for the Lagrangian
function of the system we are considering there is a term
L' which is intermittent. It has for some small time a
finite value, then vanishes, then springs into existence
again, then vanishes, and so on, repeating its value n times
in a second. We shall for brevity speak of each of the
epochs during which the function L' has a finite value as
a collision, and shall call n the number of collisions per
second. For example, in the case of a body moving through
a gas L' may be the part of the Lagrangian function which
represents the action of a molecule of the gas on the body,
when the body is in collision with a molecule L has a finite
value, when however the body is free from collision L' is so
small that it may be assumed to be zero without appreciable
error.
Lagrange's equation corresponding to the coordinate x
is, if L is the steady part of the Lagrangian function,
d dL dL d dL' dL'
Integrating this equation over a time T we get
dL dL>
Now unless the structure of the system is steadily chang-
ing \dL'ldx\* will either vanish or be exceedingly small, so
that in general we may neglect it and write
dL dL\ ( T dL' .
Tr~ -j- ) dt- I -j-dt
dt dx dx) J dx
286 DYNAMICS.
Let us choose T so that though a great many collisions
occur in this time, yet the values of x, x, x are not
changed in it by a finite amount.
Now if T be the time a collision lasts and if there are n
of them per second
if as in a numerous class of cases x may be supposed to
remain constant during the collision, we may write (252) as
,
dx dx
where X
Since
Jo \dtdx dx) dt T \dtdx~ dx}*
equation (251) becomes
dtdx dx dx'
Thus the effect produced by these intermittent forces is
the same as that which would be produced by a steady
force X of type x and given by the equation
X=n dx'
Similarly they would produce the same effect as a force
Y of type y where
Y=n ~dy'
If dxldx, dxldy do not involve the velocities x, y and
if n the number of collisions per second is a linear function
IRREVERSIBLE EFFECTS. 287
of these velocities, these forces will be of the character of
frictional forces.
If n does not involve the coordinates x, y explicitly then
we have -
dY dX
the consequences of this equation will be similar to those
developed in 44. Thus for example suppose we were to
find that the logarithmic decrement of the torsional vibra-
tions of a wire depended on the extension of the wire, then
it would follow from (252) that when the wire was vibrating
there would be a force tending to alter its length. If the
frictional resistance to the torsional vibrations were /x0,
where is the angular velocity of a pointer attached to the
wire, then if the above equation is true, there would be a
force X tending to lengthen the wire and given by the
equation
- = - 6 -_- , where x is the length of the wire.
au ax
Thus if -~ is constant we have
dx
x=-ei>f.
dx
Whence it follows that if the torsional vibrations were
periodic there would be a force tending to produce longi-
tudinal vibrations of half their period ; or again, if the
viscosity of an iron wire were altered by magnetization
there would be a periodic magnetizing force acting on a
vibrating wire whose period would be half that of the
torsional vibrations.
The relation (253) is only satisfied when n is in-
dependent of x and y, if n is a function of these quantities
we shall have the relation
288 DYNAMICS.
dY dX v d\o>gn
--- -j- = A j -- -
ax ay ay ax
instead, and if we consider forces of a third type z, the
two additional relations
d\ogn
_~
dz dx dx dz
and dY_ dtegn d\ogn
.dllvl t " ' -* T ~ ' -*
dy dz dz dy
so that
7 (dY dX\ ^(dX dZ\ v (dZ dY\
z(-_ j- }+y\~*-- j- ) + x (-j -j- )-o. ..(254).
\dx dy J \dz dx J \dy dz J
In these relations X, Y, Z are only those parts of the
forces of types x, y, z which are intermittent in their action.
If from the nature of the case we can see that the
number of collisions is independent of some one coordinate
x, then it follows from the above equations that
(i/dY dX\, [i/JZ dX\j
log n = l-=( j -- -=- ) dy + I -=, ( - ---- }dz.
) X\dx dy J ' ] X\dx dz J
If the viscous forces arise from collisions with several
distinct systems, instead of with one as we have hitherto
assumed, we shall have
= 1 ,-...
dy * ay
where n lt n a are the numbers of collisions per second with
the systems (i), (2)... respectively, and
where Z/ is the Lagrangian function of the rth system.
VISCOUS FORCES. 289
If ... are independent of x, y then as before
dX_dY
dy ~ doc '
but if j, , involve the coordinates x, y, then the relation
(253) must be replaced by one involving higher differential
coefficients.
The preceding considerations show that in those cases
where the viscous forces are due to "collisions" we have
several criteria the fulfilment or non-fulfilment of which will
afford us information about the constitution of the system.
Thus if (252) is not fulfilled we conclude that the number
of collisions depends upon the value of the coordinates, if
(253) is not fulfilled we conclude that the viscous forces are
due to collisions with more systems than one and so on.
There is a great dearth of experiments on the influence
of various physical conditions on viscous forces except
when these forces are those which resist the passage of
electricity through conductors. It does not seem probable
however that in this case the resistance can be due to a suc-
cession of impulses whose number is proportional to the
strength of the current ; for the case is not analogous to
that of a viscous force depending on the change of shape
or configuration of a system, where we might reasonably
expect the number of effective collisions to be propor-
tional to the velocity of the change.
In order to get some idea as to how discontinuous
forces can produce the effect of electric resistance, let us
consider some cases in which effects analogous to resistance
are produced by a succession of changes following one
another in quick succession. A very good example of a
case of this kind is the arrangement given by Maxwell
(Electricity and Magnetism, n. p. 385) for measuring in
T. D. 19
DYNAMICS.
electromagnetic measure the capacity of a condenser, in
which by means of a tuning fork interrupter the plates of a
condenser are alternately connected with the poles of a
battery and with each other. If the rate of discharge is
very rapid, this arrangement of condenser and tuning fork
produces the same effect as a resistance ijnC where C is
the capacity of the condenser and n the number of times it
is discharged per second. Thus in this case a combination
of induction and discharge produces the same effect as a
resistance. Another case in which the conditions are plainly
discontinuous but which produces the same effect as a
continuous current, if the rate of alternation is sufficiently
rapid, is when electricity passes through a closed glass tube
filled with air. If electrodes are fused into the tube and
connected to an electrical machine in action there will be
no discharge of electricity across the tube until the electro-
motive force gets large enough to break down the electric
strength of the air, when a spark will pass, an interval will
elapse before the second spark passes, during which the
electromotive force inside the tube will be increasing to the
value necessary to overcome the electric strength of the air.
If this interval is very short then the successive discharges
will produce the same effect as a continuous current through
the tube. The consideration of this case may also throw
some light on the mechanism by which the discharge is
effected, for there are many reasons for believing that in
this case the discharge is accomplished by the decomposi-
tion of the molecules of the gas, the energy required for
this decomposition coming from the electric field, and the
consequent exhaustion of the electric energy producing the
electric discharge. The reasons which lead us to this con-
clusion are as follows :
(i) Different gases differ much more in their electric
ELECTRIC RESISTANCE. 2QI
strengths than they do in other physical qualities, the
difference is much more comparable with the differences
between their chemical properties than their physical ones,
and the difference between a chemical and a physical pro-
cess seems to be that in the chemical process the mole-
cules are split up while in the physical one they are not.
(2) In many cases there is direct evidence from both
spectroscopic and chemical analysis that this decomposition
takes place, and again gases of complex composition
whose molecules are easily split up are also electrically very
weak.
(3) We can explain by this hypothesis in a general way
(Proc. Camb. Phil. Soc. v. 400) why the electric strength
should gradually diminish as the gas gets rarer and rarer,
until when the pressure is about that due to a millimetre
of mercury the electric strength is a minimum, when the
pressure falls below this value the electric strength increases
again until at the highest exhaustion which can be got by
the best modern air pumps the strength is so great that it
is almost impossible to get a spark through the gas.
(4) Dr Schuster has shown (Proc. Royal Society, xxxvu.
p. 318) that the electrical discharge through mercury vapour
which is supposed to be a monatomic gas presents a
peculiar appearance and passes with great difficulty, and
quite recently Hertz (Wied. Ann. xxxi. p. 983, 1887) has
shown that the electric discharge passes more easily through
a gas when it is exposed to the action of violet or ultra-violet
light than when it is in the dark ; since ultra-violet light has
a strong tendency to decompose the molecules of a gas
through which it is passing, this is very strong evidence in
favour of the view that the discharge is caused by the
splitting up of the molecules of the gas.
In the case of the electric discharge through gases the
192
2Q2 DYNAMICS.
insulation seems to be perfect until the electromotive force
reaches a definite value, when a spark passes. Thus the field
can apparently not be discharged by a rearrangement of the
molecules unaccompanied by decomposition. There is evi-
dence however that when the molecules are split up into
constituents a state of molecular structure is produced in
which the discharge may be produced by rearrangement
without further decomposition. Thus Dr Schuster has shown
(Proc. Roy. Soc. XLII. p. 371) that when a strong electric dis-
charge passes through a gas, a very small electromotive force
is sufficient to produce a current in a region of the gas
screened off from the electrical influence of the primary dis-
charge. Again Hittorf found that a gas was weakened for
discharges in the horizontal direction by passing a vertical
discharge through it. The diminution in the electric strength
of a gas after the passage of a spark can be accounted for in
the same way. Again in Mr Varley's experiments on the
electric discharge through gases (Proc. Roy. Soc. xix. 236)
the quantity of electricity which passed through a tube filled
with gas was proportional to E - Q where E is the difference
between the potentials of the electrodes and a constant
electromotive force, in other words the quantity of electricity
which flowed through the tube was proportional to the excess
of the electromotive force above that which broke the dielec-
tric down ; this seems to indicate that the electromotive
force JS produces a supply of atoms in the nascent condi-
tion and that the rearrangement of these atoms discharges
the field.
In the case of fluid insulators the insulation for low
electromotive forces is not as in the case of gases perfect.
A condenser the plates of which are separated by a liquid
dielectric always leaks however small the difference between
the potentials of the plates may be. Some experiments
ELECTRIC RESISTANCE. 2Q3
recently made by Mr Newall and myself (Proc. Roy. Soc.
XLII. p. 410) showed that for small electromotive forces the
leakage obeyed Ohm's law, that is, was proportional to the
difference of potential between the plates. This indicates
that the leakage is produced by the rearrangement under
the electromotive force of some molecular condition, and
that this condition is not produced by the electric field, for
if it were the leakage would vary as a higher power than the
first of the electromotive force. Quincke, who investigated
the passage of electricity through the same liquids, using
however electromotive forces comparable with those which
would produce sparks through the dielectric, found that under
these circumstances the quantity of electricity passing through
the dielectric varied as a higher power than the first of the
electromotive forces, which is just what we should have ex-
pected if the electric field split up the molecules of the fluid.
There are many liquids which, though they only conduct
electricity with great difficulty when pure, yet when salts or
other substances (which may themselves be non-conductors)
are dissolved in them, conduct readily. This kind of con-
duction is called electrolytic and is accompanied by effects
which are not observed in other cases.
Since the solvent is not a conductor, the discharge of
the electric field which constitutes conduction must in some
way or other be due to the action of the substance dissolved
in it. The consideration of the discharge through gases as
well as the chemical decomposition which always accom-
panies this kind of conduction suggests that in this case the
discharge is caused either by the splitting up of the mole-
cules of the salt by the electric field, or else by the
rearrangement when in a nascent condition of the atoms of
a molecule of the salt or the constituents of a more complex
molecule containing both salt and solvent, the splitting up of
294 DYNAMICS.
the molecule being done independently of the electric field.
The first of these methods is unlikely for the following reasons.
(1) If it were true it would require a finite electro-
motive force to start a current through an electrolyte, just
as to send a spark through a gas, whilst from the evidence
of many experiments it seems clear that the smallest electro-
motive force is sufficient to start a current through an
electrolyte.
(2) The experiments of Prof. Fitzgerald and Mr Trouton
(Report of the British Association Committee on Electrolysis,
1886, p. 312) have shown that Ohm's Law is obeyed with
great exactness by a current flowing through an electrolyte,
whereas if the electromotive force had to break up the
molecules the current would be proportional to a higher
power than the first of the electromotive force.
(3) If the molecules were split up by the current then
the salt will form a greater number of individual systems
when the current is flowing than when it is not. Now the
rise of the solution in an osmometer and the lowering of its
vapour pressure depend upon the number of molecules in
unit volume of the liquid and not upon their kind, so that if
the number of separate systems is increased by the passage of
the current these effects ought to be increased by the passage
of a current through the solution. I have lately made some
experiments on both these effects and have not been able
to detect that the slightest change was made by the current.
For these reasons we conclude that the splitting up of
the molecules which allows the current to pass is not caused
by the electromotive force but takes place quite indepen-
dently of the electric field.
The forces between the atoms in a molecule are usually
too strong to allow of any arrangement under the electric
field, but when the molecule breaks up and these interatomic
ELECTRIC RESISTANCE. 295
forces either vanish or become very small the constituents
of the molecule are free to move under the electro-
motive force, and they will move so as to diminish the
strength of the electric field. In order to form a definite
idea of the way in which the field gets discharged we
may take the usual view that the constituents into which
the molecule splits up are charged with opposite kinds
of electricity, and that when the molecule splits up the
positively charged constituent travels in one direction, the
negatively charged one in the other; in this way we get
two layers of positive and negative electricity formed, the
electric force due to which neutralizes in the region between
the layers the external electric force. The positively charged
molecules soon come into the neighbourhood of some
negatively charged ones travelling in the opposite direction
and they recombine, while the negatively charged ones
do the same with some positive molecules, thus the
force due to the layers vanishes and the external electric
field is re-established to be soon demolished again by the
decomposition and rearrangement of other molecules.
Although we suppose that the current is transmitted by
the molecules of the electrolyte breaking up, this does not
necessarily imply that the electrolyte should when free from
electromotive force be largely dissociated, for all that is
necessary on this view for the passage of a current is that
the molecules of the electrolyte should split up, and there is
nothing to prevent our supposing, if other reasons render it
probable, that they would instantly re-unite if no electromo-
tive force acted upon them. And since the state of dissoci-
ation depends upon the ratio of the time the atoms remain
dissociated to the time during which they are combined, we
may make this as small as we please and yet have continual
splitting up of the molecules.
296 DYNAMICS.
There does not seem any necessity for supposing that
the passage of electricity through metals and alloys is
accomplished in a fundamentally different way from that
through gases and electrolytes. For the chief differences
between conduction through metals and through electrolytes
are (i) that in electrolytic conduction the components of
the electrolyte appear at the electrodes, and we have polar-
ization, and (2) that the conductivities of electrolytes in-
crease while those of metals diminish as the temperature
increases.
Let us begin by considering the first of these differences,
that of polarization. A little consideration will show that we
could hardly expect to detect it in the case of metals or
alloys, for here instead of, as in electrolytes, the property of
splitting up being confined to a few molecules sparsely scat-
tered through a non-conducting solvent, the whole of the
molecules can split up, thus the rate of disappearance of
any abnormal condition would be almost infinitely greater
than in the case of electrolytes, so that if any polarization
were produced it would probably die away before it could
be detected. Let us next consider the appearance of the
constituents of the conductor at the electrodes. The only
case in which we could expect to detect this is that of the
alloys, but even in this case Prof. Roberts-Austen was
unable to detect any change of composition in the alloy
round the electrodes ; we must remember however that an
alloy differs very materially from an electrolyte because
while in the latter we have a few "active" molecules
embedded in a non-conductor, in the former it is as if the
solvent as well as the salt conducted, so that the discharge
is not concentrated on a few molecules of definite com-
position but can travel by an almost infinite variety of
paths.
ELECTRIC RESISTANCE. 2Q/
Then again the statements about the effect of heat on
the conductivity of elements and electrolytes though true in
general are subject to exceptions, thus the conductivities of
selenium, phosphorus and carbon increase as the tempera-
ture increases ; that of bismuth is said to increase at certain
temperatures, and I have lately found that the conductivity
of an amalgam containing about 30 per cent, of zinc and
70 of mercury is greater at 80 C. than at 15 C. We must
remember too that the rate of increase of conductivity with
temperature for electrolytes diminishes as the concentration
increases. No sharp line of demarcation can therefore be
drawn between the two classes of conductors on this
account.
There does not seem any difference between metallic
and electrolytic conduction which could not be attributed
to the vastly greater number of molecules taking part in
metallic conduction, whilst assuming that in all cases the
current consists of a series of intermittent discharges caused
by the rearrangement of the constituents of molecular
systems.
We shall therefore proceed to examine the dynamical
results to which such a conception of the electric current
leads.
Let us consider the case of an electric field where the
electromotive force is everywhere parallel to the axis of x.
Let the electric displacement in this direction be /, then in
the Lagrangian function of unit volume of the medium there
is the term
where K is the specific inductive capacity of the medium.
298 DYNAMICS.
This term gives rise to the force
_
K
parallel to the axis of x. In consequence of the continual
rearrangement of the molecular systems//^ is not uniform
but keeps alternately vanishing and rising to a maximum
value. If these alterations are sufficiently rapid the effect
represented by this term will be the same as that of a steady
force equal to its mean value, that is to
K
Let us suppose that in consequence of the rearrange-
ment of molecular systems /vanishes n times a second, and
that T is the period which elapses between the end of one
period of extinction and the end of the next, then
where i is the maximum value of/, and ft a quantity which
depends upon the ratio of the time the field is destroyed to
that during which it exists.
When the molecular systems rearrange themselves so as
to discharge the electric field molecules charged with I
units of electricity pass through unit area in one direction,
while I units of negative electricity are carried by molecules
moving in the opposite direction.
Thus 2f is the sum of the positive electricity moving
in one direction and of the negative in the opposite passing
through unit area in unit time, it is therefore equal to u
where u is the intensity of the current, and since nr is
equal to unity, the force we are considering equals
nK
ELECTRIC RESISTANCE. 299
so that this continual breaking down of the field produces
the same effect as if the substance possessed the specific
resistance zirfilnK. Thus the greater the number of times
per second the displacement breaks down &c., the better
the conductivity.
Now the breaking down of the displacement is caused
by the rearrangement of the molecules, and the rearrange-
ment of the molecules in a solid will produce much the
same effects as the collisions between the molecules of a
gas, and will tend to equalize the condition of the solid,
thus we might expect the rate of equalization of temperature
to increase with the number of molecular rearrangements.
The electrical conductivity would also increase in the same
way, so that this view fits in with the correspondence which
exists between the orders of the metals when arranged ac-
cording to thermal and to electrical conductivities.
The preceding investigation of the resistance of such a
medium is only valid when the electromotive force is ap-
proximately constant over a time which includes a great
many discharges. If the displacement were to be reversed
during the interval between two successive rearrangements
of the molecules the substance would behave like an insula-
tor and not like a conductor. If <r is the specific resistance
of the substance then
27T/3
*'
where all we know about ft is that it cannot be greater
than unity. To find a superior limit to n let us assume
that (3 has its maximum value, and that K is 7/9 x io 80
which is about the same as for light flint glass, then the
300
DYNAMICS.
number of times the field breaks down a second is given
by the following table :
n
Silver
1-6
x io 3
5
xio 17
Copper
1-6
x io 3
5
x io 17
Gold
2'I
x io 3
4
x io 17
Platinum
9
x io 3
9
x io 16
Lead
2
x io 4
4
x io 16
Mercury
9-6
x io 4
8
x io 15
Water with 8-3 per cent, of
sulphuric acid
3'3
xio 9
2-4
x io 11
Copper sulphate and water
i'9
xio 10
4-2
x io 10
According to the electromagnetic theory of light the
electric displacements which constitute light are reversed
nearly io 15 times per second; comparing this with the
number of times the field is discharged in an electrolyte, we
see that the displacement would be reversed many times
a second before it was discharged and hence that such
substances would behave like insulators to these rapidly
alternating displacements, and so according to the electro-
magnetic theory of light should be transparent, which as a
matter of fact most of them are. Again, we have certainly
overestimated /? and probably underestimated K\ if we take
this into consideration we may conclude that the number
of times the field is discharged is probably even in the
best metallic conductors not much greater than the number
of times the displacements accompanying the propagation
ELECTRIC RESISTANCE. 30 1
of light are reversed, hence we need not be surprised that
metals in thin films possess a transparency almost infinitely
greater than that calculated on the assumption that their
conductivity is the same as that for steady currents.
The number of times the field is discharged at any point
will depend upon the number of molecules which split up
in unit time and the distance which these travel before
combining. If m is the number of times the molecules
in unit volume split up in unit time and if it requires q
molecules per unit area to be split up in order to discharge
the field, then if the molecules after being split up travel
a distance x under the influence of the electromotive force
before again entering into combination, we shall have
m
=-*,
since any q molecules which break up within a distance x/2
on either side will discharge the field. Since both x and q
will be directly proportional to the electromotive force,
n will be independent of it, if the splitting up of the
molecules is accomplished by other means.
Since u zni = 2 xt,
q
and since, if the substance is an electrolyte,
where e is the charge on either of the ions into which the
molecule splits up, we have
U = 2mX.
So that if JV"bQ the number of molecules of the salt in
unit volume
u m
3O2 DYNAMICS.
= x x (number of times each molecule breaks up per second)
= the distance between the two ions at the end of one
second.
But ujzNk is, see Lodge, Report on Electrolysis, British
Association Report, 1885, p. 755, the quantity called by
Kohlrausch the sum of the velocities of the ions, and if
we assume that the ratio of the velocities is given by
experiments on the migration of the ions, this view of the
current would lead to the same expression for the absolute
distance travelled by each ion in unit time as that given by
Kohlrausch.
A full discussion of this would however lead us too far
from our purpose, which is merely to use this conception of a
current to deduce reciprocal relations from the effects of
various physical agencies on resistance.
The specific resistance of a substance according to our
view is
27T/3
nK*
and if this varies when the circumstances are changed it
may be because either /?, n, or K are changed. To take
an example the resistance of a "metal wire seems to be
slightly affected by strain, this may arise either from the
specific inductive capacity being altered by strain, or by the
strain altering the number of times a second the molecules
split up, or finally by an alteration in the time the field
remains discharged. The term
K
in the Lagrangian function corresponds, see 35, to a force
equal to
ELECTRIC RESISTANCE. 303
tending to produce an extension e. Thus unless the altera-
tion in the resistance was due to the alteration of K with
the strain there would be no corresponding elastic force.
If however it does arise from the alteration of K with the
strain the mean value of the elastic force is
and
r/'<lt = af' = a-,
JO %
where a is a number which cannot be greater than unity,
and which like (3 depends upon the time the field remains
discharged.
Thus the mean value of the elastic force
$n de '
For good conductors this term will be exceedingly small
on account of the smallness of <r/, see the table p. 300,
and even for bad conductors it will never get large enough
to make it comparable with the large forces required to
produce an appreciable change. in the extension.
If # is a coordinate of any type this term indicates a
force of type x equal to
or as it may be written
27T/3 2 dX
Now .AT is of the order io~ 21 and vu, the electromotive
304 DYNAMICS.
force, even for a fall of 10 volts per centimetre, is only io 9 ,
so that in this case the force of type x is of the order
a dlogK
27T/? 2 dx '
and so is exceedingly small ; hence we conclude that the
reciprocal effects corresponding to the effects observed on
the resistances are probably much too small to be capable
of detection unless for very bad conductors under the
influence of electromotive forces comparable with those
used in experiments on static electricity.
INDEX.
Absolute temperature, 95
Absorbed air, effect of on vapour pressure, 173
Absorption of gases by liquids, 179
- Henry's Law of, 181
Action, Least, 14
Adie, membranes for osmometers, 186
Affinity, determination of coefficient of, by the observation of the
electromotive force of a galvanic cell, 271
Alloys, electric resistance of, 296
Battery, electromotive force of, 265
- Gas, 275
Baur, effect of temperature on magnetization, 105
Berthelot, Law of Maximum work, 220
Bertrand, vapour pressure, 161
Bevan. Young's modulus for ice, 259
Bidwell, effect of magnetization on the length of an iron bar, 54
Boltzmann, residual torsion, 130
on dissociation, 200
Bosscha, forms of clouds, 203
Bunsen, absorption of gases by liquids, 181
Capillarity, effect of, on electromotive force required to decompose an
electrolyte, 87
effect of, on vapour pressure, 162 163
- effect of, on solubility, 251 255
- effect of, on freezing point, 262
- effect of, on density of salt solutions, 191 192
effect of, on dissociation of gases, 203
- effect of, on chemical equilibrium, 234 237
Cassie, effect of temperature on specific inductive capacity, 65, 102
Chemical equilibrium, 215
- effect of pressure on, 221, 237
- effect of surface tension on, 234 237
- effect of magnetization on, 240
- effect of temperature on, 221
Chrystal, article on electricity quoted, 136
magnetism quoted, 61
Circularly polarized light, magnetic effects produced by, 78
Clausius, 143
expression for force between two moving electrified spheres, 36
formula for an imperfect gas, 197
T. D. 20
306 DYNAMICS.
Coefficient of magnetization, effect of temperature on, 105
Coefficient of self induction, 40
Compressibility of salt solutions, 183
Constitution of bodies, Maxwell on, 133 134
Controllable coordinate defined, 94
Coordinates, definition of, 19
" kinosthenic," 9
" positional," 12
specification of, 20
" speed," 9
- " unconstrainable," 94
Critical value of magnetic force, 56
Currents, induction of, 41
mechanical force between, 40
effect of, on elasticity of wires, 43
Czapski, variation of electromotive force with temperature, 99
Dielectric, effect of moving conductors on stiffness of, 38
strain in due to electrification, 62
Diffusion of salts, 182
Discharge, electric, through gases, 291
through liquids, 293
Discharges, number of, in electric field, 300
Dissociation, 193
of nitrogen tetroxide, 200
of phosphorus pentachloride, 208
Boltzmann on, 200
Willard Gibbs on, 200
effect of electricity on, 207
effect of presence of neutral gas on, 207
effect of surface tension on, 204
effect of pressure on, 221
effect of temperature on, 201
of a solid into two gases, 210
of salts in solution, 212
Dissolved salt, effect on vapour pressure, 175
effect on freezing point, 262
Dupre, on vapour pressure, 161
Dynamical interpretation of temperature, 90
Elasticity of a wire, effect of current on, 43
Electric resistance, 289
Electric discharge through gases, 291
through liquids, 293
Electricity, inertia of, 32
specific heat of, 106
effect of, on vapour pressure, 164 167
effect on dissociation, 207
thermal effects produced by charge of, 1 10 1 1 2
Electrification, strain in a dielectric due to, 62
thermal effects due to change of, 102
INDEX.
Electrolyte, effect of pressure on electromotive force required to decom-
pose, 8385
capillarity on electromotive force required to decom-
pose, 87
Electrolytic conduction, 293
Electromotive force due to variation of the magnetic field, 68
produced by twisting a magnetized wire, 7 1
required to decompose an electrolyte, effect of pres-
sure on, 84, 85, 87
required to decompose an electrolyte, effect of capil-
larity on, 87
produced by inequalities in temperature, 106
of batteries, 265
of gas batteries (table of), 275
Energy, "free, "95
Entropy, 157
Equilibrium, chemical, 215
"ect of strain on, 239
- effect of pressure on, 221, 238
effect of surface tension on, 234 237
- effect of magnetization on, 240
effect of temperature on, 221
v. Ettinghausen and Nernst, effect produced by flow of heat in a
magnetic field, 116
Evaporation, 158
effect of strain on, 168
effect of pressure on, 168 172
effect of surface tension OH, 162
effect of electrification on, 164
Ewing, effect of strain on magnetization, 55
critical value of magnetic force, 56
hysteresis, 104
Faraday, force on soft iron in magnetic field, 47
Fitzgerald, on Hall's phenomenon, 73
on Ohm's law in electrolytic conduction, 294
Force on a body in a magnetic field, 47
- a conductor carrying a current, 69
magnetic, due to currents, 68
electromotive, due to variation in magnetic field, 68
Free energy, 95
Freezing point, effect of pressure on, 257 259
effect of torsion on, 260
effect of surface tension on, 262
effect of dissolved salt on, 262
Gas, Lagrangian function for a perfect gas, 154
Gas batteries, 265
Pierce's determination of electromotive force of, 275
with one gas, 276
Gases, absorption of, by liquids, 1 79
308 DYNAMICS.
Gases, electric discharge through, 291
Gibbs, Willard, on dissociation, 200
on the electromotive force of batteries, 277
Glazebrook, on Hall's phenomenon, 73
Groves gas batteries, 266
Guldberg and Waage, on chemical equilibrium, 225
Hall's phenomenon, 72
Hamilton, principle of varying action, 9
Height to which a salt solution rises in an osmometer, 188
v. Helmholtz, conservation of energy, 2
objection to Weber's law offeree between two electrified
spheres, 37
" freie energie," 95
variation of electromotive force with temperature, 99, 269
strain produced by a magnetic field, 52
Hertz, action of light on the electric discharge, 291
Hittorf, discharge of electricity through gases, 292
Hoff, van t', pressure in dilute solutions, 175
osmotic pressure, 188
effect of walls of vessel on chemical action inside, 206
Hopkinson, J. , residual charge of a Leyden jar, 136
Horstmann, chemical equilibrium, 229
Hysteresis, 104
Induction of electric currents, 41
Inertia of electricity, 32
magnetism, 65
Irreversible effects, 281
Jahn, variation of electromotive force with temperature, 99
Joule, elongation of a bar produced by magnetization, 54, 59
effect of magnetization on the volume of a magnet, 56
Kinosthenic coordinates defined, 9
Kirchhoff, strain produced by a magnetic field, 52, 54
Kohlrausch, electrolytic conduction, 302
Lagrangian function, expressions for, 23, 29
mean value of stationary, 145, 146, 147
expression of a gas, 152
. liquid or solid, 155
Larmor on varying action, 18
Law, second, of Thermodynamics, 99
Least action, 14
Leyden jar, residual charge of, 136
Liebreich, inert space in chemical reactions, 236
Light, magnetic force produced by, 78
Liquefaction, 255
Liquids, discharge of electricity through, 293
Liveing, effect of surface tension on chemical action, 237
INDEX.
Lodge, Report on Electrolysis, 302
Magnetic field, stress produced by, 47
5, 68
force due to currents,
- produced by twisting a wire conveying a current, 72
- arising from the Hall effect, 74
produced by circularly polarized light, 78
inertia, 65
Magnetism, Ewing's researches on, 53, 54
Magnetization, terms in Lagrangian function depending on, 44
effect of torsion on, 61
due to torsion, 61
strains produced by, 50 58
change of length due to, 54
effect of strain on, 50 58
thermal effects due to, 103
effect on chemical action, 240
Mass, effects due to, in chemical equilibrium, 223
Maximum work, Berthelot's Law of, 220
Maxwell, Electricity and Magnetism, 41, 44, 45, 47, 49, 67, 289
Theory of Heat, 81, 90, 98, 164
strains produced by an electric field, 64
a magnetic field, 52
on the constitution of bodies, 133, 134
Mechanical force between circuits conveying currents, 40
Mercury vapour, electric discharge through, 291
Meyer, Lothar, Modernen Theorien der Chemie, 225
Meyer, Victor, effects of surface of vessel on dissociation, 207
Monckmann, effect of surface tension on the density of salt solutions,
192
Muir's Principles of Chemistry, 212, 225
Natanson, E. and L., dissociation of nitrogen tetroxide, 200
Neesen on residual torsion, 1 30
Nernst and v. Ettinghausen, electromotive forces due to flow of heat
in a magnetic field, 116
Neutral gas, effect of, on the dissociation of a solid into two gases, 212
Newall, discharge of electricity through liquids, 293
Ohm's Law for electrolytes, 294
Osmometer, 186
Osmosis, 1 86
Peltier effect, 115
Pfeffer on osmotic pressure, 175
Pfeffer's Osnwtische Untersuchtingen, 186
Phosphorus pentachloride, dissociation of, 208
Pierce, electromotive force of gas batteries, 275
Polarization, rotation of plane of, by magnetic field, 78
" Positional" coordinates, 12
Potential energy, 14
310 DYNAMICS.
Pressure, effect of, on chemical equilibrium, 221, 237
solubility, 245 251 ^
the electromotive force required to decompose an
electrolyte, 8485
of gas batteries, 273
the freezing point of liquids, 257
evaporation, 168 172
osmotic, 175, 188
Quincke, discharge of electricity through liquids, 293
strains in a dielectric due to electrification, 64
Rain drops, effect of pressure on the formation of, 172
Raoult, effect of dissolved salt on vapour pressure, 178
the freezing point of solutions, 175, 263
Rayleigh, Lord, coefficient of magnetization for small magnetic forces,
46
reciprocal relations, 81
Theory of Sound, 81
Reciprocal relations, 81
Residual charge of a Leyden jar, 136
Residual effects, 128
Resistance, electrical, 289
effect of strain on, 302
Reversible thermal effects due to a current of electricity, 109, 269
Riemann, law offeree between two moving electrified spheres, 37
Roberts- Austen, conduction through alloys, 296
Rontgen and Schneider on the compressibility of salt solutions, 83, 183
surface tension of salt solutions, 87, 254
Routh, Rigid Dynamics, 9
Stability of Motion, 10
Rowland, rotation of plane of polarization of light in a magnetic field, 79
Salt solutions, compressibility of, 83, 183
surface tension of, 87, 254
Salts, diffusion of, 182
dissociation of, in solutions, 212
effect of, on freezing point, 262
effect of pressure on the solubility of, 245 251
Schneider, see Rontgen and Schneider.
Schumann, compressibility of salt solutions, 183
Schuster, electric discharge through mercury vapour, 291
gases, 292
Second Law of Thermodynamics, 99
Self induction, coefficient of, 40
Solubility, effect of pressure on, 245 251
surface tension on, 251 255
of liquids in fine drops, 252
pores, 255
Solutions, compressibility of, 183
surface tension of, 190
INDEX. 311
Solutions, dissociation of salts in, 212
Sorby, effect of pressure on solubility, 247
Specific heat of electricity, 106
Specific inductive capacity, effect of temperature on, 65, 102
strain on, 64
Spheres, force between two moving electrified spheres, 35
Steady state, 80
Strain in a dielectric due to electrification, 62
- effect on vapour pressure, 168
chemical equilibrium, 239
- freezing point, 259
solubility, 251
electric resistance, 302
change of temperature due to, 101
thermoelectric effects of, 113
Streintz, effect of current on elasticity of wire, 43
Stresses produced by magnetic field, 47
Surface tension of solutions, 190
Surface tension, effect of on the electromotive force required to decom-
pose an electrolyte, 87
- vapour pressure, 162, 163
solubility, 251255
dissociation of gases, 203
density of solutions, 191
freezing point of solutions, 262
chemical equilibrium, 234 237
Tait, Thomson and, Natural Philosophy, 18
Temperature, absolute, 95
effect on specific inductive capacity, 65, 102
electromotive force of batteries, 99
coefficient of magnetization, 105
chemical equilibrium, 221
dissociation, 201
change of, produced by strain, 101
Thermal effects due to change of electrification, 102
magnetization, 103
- change of strain, 101
reversible, due to a current of electricity, 109
produced by communicating a charge of electricity to a
body, no Ti2
Thermodynamics, Second Law of, 99
Thermoelectric effects of strain, 113
Thermoelectricity, 1 1 o 127
Thomsen, Thermochemische Untersuchungen, 227
Thomson, James, effect of pressure on the freezing point of water, 257
Thomson, Sir William, effect of strain on magnetization, 48, 52
- change of temperature due to strain, 101
- magnetization, 103
effect of surface tension on vapour pressure, 163
on the decay of torsional vibrations of wires, 139
312 DYNAMICS.
Thomson, Sir William, effect of pressure on the freezing point of water,
257
thermoelectric effect due to inequalities of tem-
perature in a conductor, 106
Thomson and Tait, Natural Philosophy, 18
Tomlinson, effect of a current on the elasticity of wire, 43
Torsion, in a magnetized wire due to current, 70
produced by magnetization, 61
effect on magnetization, 61
effect on freezing point, 260
Torsional viscosity, 139, 287
Trouton, Ohm's Law in electrolytes, 294
Unconstrainable coordinate, defined, 94
Vapour pressure, 161
effect of surface tension on, 163, 164
effect of electrification on, 165, 166
effect of strain on, 168
effect of pressure on, 168 172
effect of absorbed air on, 173
effect of dissolved salt on, 175
Varley, discharge of electricity through gases, 292
Villari, effect of strain on magnetization, 48 52
Viscosity, torsional, 287
Waage and Guldberg, on chemical equilibrium, p. 225
Waals, van der, 90
formula for an imperfect gas, 197
surface tension of gases, 204
Weber's expression for the force between two electrified spheres in
motion, 36
v. Helmholtz's objection to, 37
Wiedemann, G. , connexion between torsion and magnetization, 61
torsion produced by a current flowing along a mag-
netized wire, 70, 72
Work, Berthelot's Law of Maximum, 220
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