UC-NRLF
QC
NOTES
ON THE
PROPERTIES OF MATTER AND HEAT
BY
PERCIVAL LEWIS, Ph. D.
*
Associate Professor of Physics in the University of California
57
( UNIVERSITY )
BERKELEY, CAL.
1903
Copyright, 1903, by PERCIVAL LEWIS
SAN FRANCISCO, CAL.
PRESS OF THE HICKS-JUDD CO,
1903
PREFACE.
The following Lecture Notes have been privately printed for the use
of students taking sophomore Physics in the University of California.
Students cannot be too often reminded that they cannot really learn
the principles of Physics unless they become acquainted with them at first
hand from daily experience or from individual or lecture experiments;
text-books are merely guides to the use of such material. They are
very necessary guides, however, for without them it would be impossible
for any one person to properly classify, coordinate, and interpret his
knowledge, to fill the gaps in his experience, or to get a proper perspective
view of the whole subject. In reading, however, do not in any case try
to recall the statements as given in the book. See, if you can, the
thing or the phenomenon to which your attention is called by it, and try
to remember what your own eyes have seen; if that is impossible, try to
form a mental picture of the thing or the phenomenon, not of the words
of the book. A proper use of the imagination in this way will make the
study a much easier one.
Some attention paid to the historical development of Physics and to
the biographies of the great physicists will do much to humanize the sub-
ject and thus arouse interest and stimulate the imagination and memory; it
will also add greatly to the culture value of Physics.
In studying Physics one learns many interesting facts, some of which
may be of practical use. Its principal value, however, should be not to
impart information, but to exercise the mind in drawing logical conclu-
sions, and to form the habit of seeking truth for its own sake in fields
where traditional authority and personal prejudice have no weight, this
making it easier for us to seek truth in fields where these have weight.
Berkeley, August, 1903.
114431
Below are the titles of some good available works of reference:
General Physics. Text-books of Deschanel, Daniell, Hastings and
Beach, and Watson.
Properties of Matter. Tait, Properties of Matter; Mach, Science of
Mechanics; Poynting and Thompson, Properties of Matter; Risteen,
Molecules; Boys, Soap Bubbles; Perry, Spinning Tops. The following
little books of the Scientific Memoir Series give the original papers, with
bibliography; Mackenzie, The Laws of Gravitation ; Bams, The Laws of
Gases; Ames, The Free Expansion of Gases; Jones, The Modern Theory
of Solution.
Heat. Text-books by Maxwell, Balfour Stewart, Tait, Madan, and
Preston, and Tyndall's Heat as a Mode of Motion; Stewart, The Conser-
vation of Energy.
Historical. Whewell, The History of the Inductive Sciences;
Routledge, Popular History of Science; Cajori, History of Physics;
Clerke, History of Astronomy; Grant, History of Physical Astronomy;
Williams, The Story of Nineteenth Century Science.
Biographical. Garnett, Heroes of Science (for Boyle, Cavendish,
Rumford, and Maxwell); Lodge, Pioneers of Science (for Newton and
Kepler); biographical articles in Encyclopedia Brittanica and other
encyclopedias.
Such scientific journals as Science, Nature, and the Popular Science
Monthly should be frequently consulted.
Special references will also be given in the text.
CONTENTS.
Page.
PROPERTIES OF MATTER.
Introductory 1
FUNDAMENTAL MAGNITUDES 4
Physical Quantities 4
Time 5
Length 5
Mass 5
GENERAL PROPERTIES OF MATTER 5
Extension 6
Inertia 6
MOTION 6
Speed 6
Angular Speed 7
Velocity 7
Acceleration 7
FORCE.
MEASUREMENT OF MASS 8
Density 8
FORCE AND MOTION 9
Composition of Forces 10
Resolution of Forces 10
Moment of Force 11
Parallel Forces 11
Momentum 11
GRAVITATION 12
Acceleration of. 12
Center of Gravity 12
The Balance 13
Kepler's Laws 14
Universal Gravitation 15
Centripetal Acceleration 15
Newton's Proof. 16
Direct Verification 17
Variations of. 18
Tides 19
Determination of g 21
Falling Bodies 21
Inclined Fall 22
Nature of Gravitation 22
CONSERVATION OF MASS 23
WORK AND ENERGY 23
SPECIAL PROPERTIES OF MATTER 25
Elements 25
Discontinuity 26
Divisibility 26
Molecular Forces ^6
States of Matter ,. 26
Page.
FLUIDS 27
Hydrostatics 28
Transmission of Pressure 28
Pressure Due to Weight 28
Pascal's Principle 28
Archimedes' Principle 28
Specific Gravity 29
GASES 30
Weight of. 30
Boyle's Law 31
Charles' Law 31
Deviations from Laws 32
Amagat's Results 32
VanderWaals' Equation 33
Dalton's Law 33
THE ATMOSPHERE 33
Pressure of. 33
Barometers 34
Measurement of Altitudes 35
Cyclones 36
Manometers 36
Air Pumps 37
Mercury Pumps 37
GASES IN MOTION 38
Efflux of 38
Transpiration 39
Pressure in 39
Diffusion, Free 40
Diffusion, Partitions 41
ABSORPTION OF GASES 41
OCCLUSION 41
ADSORPTION 42
VISCOSITY OF GASES 43
KINETIC THEORY OF MATTER 43
PROPERTIES OF LIQUIDS 47
Hydrostatics 47
Hydrodynamics 47
Jets 48
Flow in Pipes 48
Vortices 48
MOLECULAR FORCES IN LIQUIDS 49
Compressibility 49
Cohesion and Adhesion 50
Viscosity 50
Surface Tension 50
CONTENTS Continued.
Page.
SOLUTIONS 54
Surface Tension of. 54
Solubility 54
Diffusion 55
Osmosis 55
Theory of. 56
SOLIDS -..- 57
Flow of 57
Diffusion of 57
Solution of. 58
Elasticity 58
Hardness 59
Tenacity 60
Friction 60
Crystals 61
MOLECULAR THEORIES.
SIZE OF MOLECULES
HEAT.
Production of
Effects of
Nature of.
Historical
Quantity of Heat
Gl
(52
TEMPERATURE 65
Scales of 65
Thermometers 65
CALORIMETRY 67
Quantity of Heat 67
Specific Heat 67
Methods of Measurement 68
Specific Heat of Gases 69
Change of Specific Heat ; 70
Latent Heat .... ,. 70
MECHANICAL EQUIVALENT OF HEAT
71
Page.
EXPANSION 72
Expansion of Solids 72
Anomalous Expansion 73
Expansion of Liquids 74
Expansion of Gases , 75
Absolute Zero 76
CHANGE OF STATE 77
Internal Work 77
Fusion 77
SOLUTION AND CRYSTALLIZATION 79
VAPORIZATION AND CONDENSATION 80
Spheroidal State 82
Pressure of Saturated Vapors 83
Fogs, Cloud, Rain, Dew 84
Vapor Density 85
ISOTHERMAL CURVES 85
CONTINUITY OF LIQUID AND GASEOUS STATE 86
LIQUEFACTION OF GASES 87
OTHER EFFECTS 88
Molecular Constraints 88
Chemical Action 89
MEASUREMENT OF HIGH TEMPERATURE
AND SMALL DIFFERENCES OF 89
TRANSFER OF HEAT 90
Radiation 90
Convection 90
Conduction 90
THERMODYNAMICS 92
Carnot Reversible Cycle 93
Absolute Temperature 93
Change of Freezing Point 93
Adiabatic Expansion 94
ORIGIN AND MAINTENANCE OF SUN'S HEAT 95
DISSIPATION AND DEGRADATION OF ENERGY 95
Vwi*ivt.rv<2ti
PROPERTIES OF MATTER.
INTRODUCTORY.
1 . All the natural phenomena which appeal to our senses are part of
the subject-matter of Physics. This is true not only of the simpler
processes of inorganic life, but, to a greater or less extent, of all the
phenomena of organic life, such as the rise of sap, the circulation of the
blood, and muscular movements. Physics is, therefore, the foundation
of all the sciences.
The most striking discoveries in Physics were made in the last two or
three centuries, and as a systematic science it is scarcely a century old ;
yet it had its unconscious beginnings before any other science, when man
began to apply his mind to the utilization of natural phenomena. In
supplying their daily wants our distant ancestors must have mastered and
applied many rudimentary principles of Physics, just as people to-day who
have never studied Physics are constantly making use of its principles
under the name of common sense.
2. This knowledge cannot properly be called scientific knowledge,
however, for it does not involve deliberate analysis into their simplest terms
of the phenomena of nature, and comparison of the results with a view to
finding general elementary principles which will give us a clearer insight
into their nature and relations, and guide us to further knowledge. Those
who seek knowledge for immediate practical ends usually have neither
the time nor the disposition to put it on a scientific basis, and the present
state of our knowledge and civilization owes little to such persons. The
progress of the sciences, and incidentally of the practical arts based upon
their principles, has always been due most largely to the craving for
knowledge for its own sake, regardless of the material advantages which
it may secure, or even with a deliberate sacrifice of advantage or pleasure ;
Galileo's reward for truth-seeking was imprisonment, and Faraday resigned
opportunities of making a fortune by applied science in order that he
might give all his time to research. The justification of such a course,
from a material standpoint, is that practically all the applications of
electricity in use to-day are based upon Faraday's discoveries and would
be impossible without them.
Because science has its origin in the love of truth and not in money-
getting, we find its first beginnings in those directions where absolutely no
material reward could be expected in the study of the heavenly bodies.
Some of the ancients were content to observe and even worship the sun
and other heavenly bodies without asking what their nature might be, or
the cause of their orderly motions ; but there were some who sought to
secure at least enough information about these bodies to correctly describe
their paths and predict their future positions. The first physical laws to
be established were those which describe the motions of the members of
the solar system.
3. Although we learn every science through our preceptions, the
most important results are usually not immediately perceptible to our
senses. We do not see directly the real motions of the planets with
2 PROPERTIES OF MATTER.
reference to the sun, because those motions are compounded with that of
the earth, and so it took many centuries of patient observation to pass
from the epicycles of Ptolemy to the ellipses of Kepler. We cannot
see the materials in the sun and the fixed stars, but the spectroscope
enables us to determine some at least of their constituents with as much
certainty as though they could be analyzed in a chemical laboratory. It
is not the substance itself, but the quality of light which it emits, which
appeals directly to our senses ; yet the proof is as valid as the recognition
of a friend, which is made possible by the light he reflects.
Direct or absolute knowledge of things is inconceivable. Our knowl-
edge is all through our senses, or from inferences based on sense
perceptions. We learn the form and color of certain things ; we feel that
they are heavy or light, hard or soft, and that they move in such and such
a way. Most people have seen round things and square things, red things
and green things ; . they have felt hard things and soft things, smelt things
that were pleasant or unpleasant, have heard loud or faint sounds If we
attempt to describe any new object to them, k is by the use of such terms
which suggest known comparisons. It would evidently be impossible to
describe a red apple to one who had never seen the color red or a sphere.
It may be concluded from such considerations that all our knowledge is
relative, or based upon comparison with other objects familiar to our
senses.
4. Sometimes we find that many bodies differing widely in many
respects have a common attribute ; for example, all bodies known to us
will fall to the earth if unsupported, or have weight ; all kinds of matter
have the property of inertia, or the inability to alter their own motion in
any way. Such facts are physical principles, the beginnings of a science.
We find, further, that certain phenomena tend to occur in a definite way;
that, for example, the planets move" in approximately elliptical orbits about
the sun. The statement of this fact is a physical law. This fact was
"explained" later by Newton, who showed that this type of motion
resulted from the more general law that every two bodies in the universe
attract each other with a force varying inversely as the square of the
distance between them. All so-called "explanations" of physical facts
are of this character ; the explanation consists in showing more general
relations or a law of wider scope. A law itself is simply a description
reduced to its simplest and most general terms. The reason why two
bodies attract each other is unknown to us and must always remain
unknown. We may, however, at some future time analyze this attrac-
tion and show that it depends in some way on a medium between the two
bodies. We may then say that we have explained the attraction ; but the
explanation will simply reduce the attraction of distant bodies to a case of
push or pull between two contiguous bodies and who can explain the
why of the push or the pull ?
The student of Physics cannot, then, hope to find out why things
happen; he can only expect to find out how they happen and to formulate
this experience in general descriptions or laws. Such laws greatly
economize mental effort by systematizing our knowledge. The term law
used in this way has none of the legal sense of obligation or necessity.
5. Nature does not always afford us the best opportunities for
observing her phenomena. They may occur so rarely or so fitfully,
or on such a small scale, that we cannot study them; or they may be so
complex as to make it very difficult to resolve them into their simplest
INTRODUCTORY. 3
elements. We therefore resort to experimentation, in which we may
arbitrarily bring about the necessary conditions and exclude those which
might be confusing. If, for example, we wish to find out how changes
of temperature affect a gas, we separate all foreign substances from it and
arrange the conditions so that the temperature alone changes, all other
physical conditions remaining constant.
6. The knowledge which we gain by observation and experiment is
of use to us chiefly because we believe that it will continue to hold good
in the future. It is not merely a mass of historical material describing
what has been; it enables us to describe what is going on to-day, and to
predict what will take place to-morrow with a feeling of great certainty.
Experience invariably teaches the lesson that the processes of nature are
uniform, or that under the same conditions the same routine of
phenomena will continue. Because the sun has risen and set at regular
intervals for all recorded time, we believe that it will continue to do so
unless some catastrophe occurs which alters the conditions. Most
physical laws are based upon the observation of a large number of cases,
but sometimes a surprisingly small number will give us a firm conviction
of the generality of a law. We are content to believe after one experi-
ment that all red-hot objects will burn us. Having once observed that a
pure metal melts at a certain temperature, we believe that all specimens of
that metal will do the same, under the same conditions. It is very
important, however, to be sure that all conditions are the same. In the
last case, for example, the melting point would not be the same if the
pressure to which the metal is subjected were changed. This process of
basing a general law on a limited number of observed cases is called
scientific induction. In no case can it be said to give us a certainty, but
it does give a high degree of probability.
7. Sometimes the progress of science is greatly aided by the
judicious use of the imagination, constantly checked by a comparison
with facts. The closest observation will not show us the paths described
about the sun by the planets, but we may, as Kepler did, venture the
hypothesis that the shape of an orbit is that of an ellipse, or parabola,
or some higher curve; and then we can check our guess by comparison
with given positions of the planet and prove or disprove the hypothesis.
Hypotheses which cannot be tested and verified have no claims to
consideration.
In the same way a theory, due wholly to the imagination, may be of
the highest use. In no way can an atom or molecule, if such actually
exist, appeal directly to our senses, yet the molecular theory of matter
has been an exceedingly fruitful one. We try to picture to ourselves
such a constitution of matter that its various physical properties, such as
elasticity and changes of state with temperature and such phenomena as
diffusion of gases, the pressure of gases, and chemical phenomena would
be natural consequences. So long as the molecular or any other theory
is consistent with all known facts, and even serves to predict new facts, it
matters little whether it is literally true or not; it is just as good as if it
were true, and is probably as near the truth as we can ever get. If a
theory is a sound one, it must always be in accord with the facts; hence
it is absurd to speak of theory as being opposed to practice. If it is
theoretically true that a given gas will diffuse through an opening of a
given size at a certain rate, we shall find it to be practically true if the
conditions are in accord with those demanded by the theory.
4 PROPERTIES OF MATTER.
8. The student must bear constantly in mind that the matter in a text-
book is not Physics. The direct object of his study is nature as we find
it in everyday life, supplemented by the laboratory. Every one of us
began his study of Physics before he could read. The text-book is a
guide to the use of this material, just as a catalogue is a guide through an
art gallery; each is almost indispensable on account of its classification of
details, its historical information, and its discussion and criticisms; but
neither can take the place of the original subject matter.
References. Mach, Popular Scientific Lectures The Economical Nature of
Physical Inquiry The Principle of Comparison in Physics. Clifford, Lectures,
Vol. I Aims and Instruments of Scientific Thought. Tyndall, Fragments of
Science Scientific Use of the Imagination.
FUNDAMENTAL MAGNITUDES AND UNITS.
9. The impressions which we receive- from the external world
through our physical senses can all be reduced to descriptions in terms of
what we call time, space, and matter. Physics does not undertake to
discuss the ultimate nature of these fundamental physical magnitudes, but
leaves such questions, as well as that of the objective reality of matter and
phenomena, to be settled, if possible, by philosophy. The ideas conveyed
by the words space and time are so rudimentary that they admit of no
further description or definition than that furnished by ordinary experi-
ence. The same may be said of matter; but the varied properties of
the latter, together with the time and space relations involved in its
varied activities, furnish an inexhaustible storehouse of material for all the
physical sciences, such as Chemistry, Astronomy, Geology, and the
applied sciences. All of these lie in part at least within the domain of
Physics; but in order to limit the scope of the science to manageable
dimensions it is usually understood to deal onlv with the more general and
fundamental properties of matter and the phenomena which are common
to all these sciences.
There are various physical phenomena, such as those described under
the headings of Light and Electricity and Magnetism, which, although
manifested in connection with matter, do not directly involve its properties.
It will be found as we study these phenomena that they may be consistently
described if we assume that they take place in an hypothetical medium
called the ether, which possesses some of the properties of matter, or may
even be the basis of all matter. It is usually assumed that the ether does
not directly affect our senses, but it may at least be questioned whether the
action of a dazzling beam of sunlight upon the eye is not as direct an effect
as the blow of a club upon the head. In studying the phenomena of light
and electricity we shall find that the reasons for assuming the existence of
the ether are about as convincing as those for the existence of matter.
10. Physical Quantities. Any description of the phenomena of
nature which undertakes to be precise must be founded on accurate meas-
urements, requiring the use of certain standards of comparison, called
units. In the absence of absolute knowledge regarding what we call
matter, force, energy, heat, electricity, etc., we can measure them only in
terms of some of their invariable properties or effects. The term quan-
tity has, therefore, a somewhat arbitrary meaning, and it might easily
happen that quantity measured in one way might not be consistent with
A
PHYSICAL QUANTITIES. O
that measured in another. Such, for example, is the case in measuring
temperature by thermometers of different materials. The most funda-
mental physical quantities are those of time, length, and mass.
Time. The natural unit of time is the solar day, or the interval
between two successive transits of the sun over the same meridian; this,
however, varies in length according to the position of the earth in its orbit.
The average length of the day during a whole year, or the mean solar
day, is adopted as a standard. A more convenient unit for ordinary pur-
. poses is the second, of which there are 86,400 in a mean solar day.
The tidal waves on opposite sides of the earth are held in a nearly
fixed position relative to the moon, while the earth revolves between them
as a car wheel revolves against the shoe of its brake; undoubtedly, there-
fore, the speed of rotation of the earth is diminishing on account of the
tidal friction, but the effect is so small that during historical time it has
not reached a measurable amount, as shown by the comparison of the
times of the recorded with the calculated eclipses of the moon in ancient
times.
Length. Various arbitrary standards of length have been adopted in
different countries. We need concern ourselves only with the English
foot and yard, and the French meter, standards of which are kept by those
governments. The meter was intended to be the one ten-millionth part of
the distance from the earth's equator to its poles, but actually differs
slightly from this.
1 meter = 10 decimeters = 100 centimeters = 100& millimeters.
1 inch = 2 53995 cm.
The meter was defined by a law of France in 1795 as the distance between
the ends of a platinum rod made by Borda, at C.
Mass is sometimes defined as quantity of matter ; more exactly it is
proportional to the inertia or to the weight of matter, these two proper-
ties varying together at any given place. The English unit of mass is the
pound, the French is the kilogram, which is defined as the mass of a cer-
tain piece of platinum made by Borda in 1795. The kilogram is intended
to be equal to the mass of a liter or 1000 cubic cm. of water at 4 C. The
usual scientific unit of mass is the gram, or the mass of a cubic cm. of
water at 4 C.
1 kilogram 1000 grams = 2.2046 pounds.
1 gram = 10 decigrams = 100 centigrams = 1000 milligrams.
Usually the centimeter, gram, and second are used as scientific units.
This is called the C. G. S. system.
The student should, by reference to some standard text-book or labor-
atory manual, familiarize himself with the principles underlying some of
the ordinary instruments for precise measurement, such as the following:
Time Clock, chronograph. Length Vernier, comparator or dividing
engine, cathetometer, micrometer screw, spherometer. Mass Chemical
balance.
GENERAL PROPERTIES OF MATTER.
11. No definition of matter can add anything to the knowledge of it
which we have all gained by experience. All kinds of matter have been
found to possess the universal properties of extension and inertia, as well
as the relative property of weight, so that matter has been defined as that
which possesses these properties.
6 PROPERTIES OF MATTER.
12. Extension. We cannot think of any material body except as
occupying a definite region of space, from which all other bodies are ex-
cluded, and we assume that no two particles of matter, however small, can
simultaneously occupy the same space. This conclusion is of course
purely hypothetical. So far as ordinary sensible masses are concerned,
any apparent violations of the principle of impenetrability can always be
explained as a result of the porosity or non-continuity of matter. Examples
of such apparent penetrability are found in the case of solution, as of sugar
in water. Carbon-monoxide passes readily through red-hot iron; Francis t
Bacon forced water through the walls of a leaden globe; an alloy of tin
and copper is smaller by 7 or 8 per cent than the sum of the volumes of
its original constituents; the same is true of a solution of alcohol in water.
Faraday found that potassium oxide has a smaller volume than that of the
original potassium before oxidization. Glass and other vitreous bodies
alone show no evidence of porosity; such Substances are not leaky to
anything known to us.
13. Inertia. When a car stops suddenly its occupants are pitched
forward. A ball thrown vertically upward from the deck of a moving
vessel will return to the hands of the thrower. A pendulum vibrating
without constraint in a given plane will continue to vibrate in that plane,
while the earth rotates beneath it; to the observer, who is at rest relatively
to the earth, the plane of the pendulum seems to shift. This is known as
the Foucault pendulum. A ball rolling on a horizontal plane will go
further and further *in a straight line before it comes to rest as the friction
'and other constraints are diminished. It is a legitimate conclusion that if
all resistance were removed, the ball would continue to move forever in a
straight line. This is the case of ideal limits, such as often occurs in
mathematical problems. A stone whirled in a sling cannot move in a
straight line, because the constraint of the cord holds it at a fixed distance
from the center of rotation; but it will fly off in a straight line if the string
breaks. So long as it is held by the string, it moves as nearly as possible
in a straight line that is, in a plane unless it is violently displaced. For
the same reason a heavy wheel (gyroscope) offers a strong resistance to
any attempt to change its plane of rotation. A bullet set in rotation by a
rifled barrel will go straighter to its mark than one from a smooth-bore
gun.^ 1 ) We learn from such facts that constancy of motion is charac-
teristic of all matter if left to itself; it is said to be inert, or to possess
inertia, meaning thereby that it has no power to alter its own motion,
either in speed or direction. To bring about such changes, some external
agency is necessary, and while the change is taking place, the external
agency is said to exert a force upon the body.
14. Motion is purely a relative term. All bodies known to us are
apparently in motion. In order to determine the displacement of any
object we must consider some point say a point on the earth's surface
to be at rest, and measure all displacements from that point as/an origin.
Jw4^L^AilMMM, w^^V^
Speed is rate of change of o2aUe*i measured along the Jibe of displace-
ment. If uniform, it is equal to the displacement per unit time; if it
varies uniformly, it is equal to an infinitesimal displacement divided by
the infinitesimal time required for it. If times and displacements be
plotted on coordinate axes, it will be seen that in all cases the speed is
(1) See Perry, Spinning Tops.
MOTION FORCE.
proportional to the tangent of the angle with the X axis of the element of
the curve corresponding to the instant at which the speed is required.
In general, if d is displacement along a path, straight or curved,
where the interval t 2 t^ must be taken infinitesimally small if the speed
is changing.
Angular Speed is the rate at which a line rotating about a point
(radius vector) is changing its direction. The simplest way of reckoning
angles is in radians. The radian is an arc equal in length to the radius
of the circle described by any point on the radius vector, and in a circum-
ference there are 2?r radians. If w be the angular speed,
-
the latter term being used if the speed is uniform. T is then the time of a
complete rotation.
Velocity is speed in a given direction; the term implies direction as
well as rate of displacement. A speed in a given direction may be
resolved into velocities along the X and the Y axes, or in any other
direction.
Acceleration is rate of change of velocity, and like velocity, implies
direction as well as change of speed. If the acceleration a is uniform,
If variable, the interval / 2 ^ must be made very small. The accel-
eration is proportional to the tangent of the angle with the X axis of an
element of the curve formed by plotting successive values of time and
velocity along coordinate axes.
Angular Acceleration is defined in a similar way.
15. Force. When the motion of a body is accelerated, that is,
when its motion is changed either in speed or direction, or both, experi-
ence shows that it is due to the action of some external agency, and to
this action we apply the name force. It is to be noted that force is that
which is either actually producing an acceleration, or which is only
prevented from doing so by the action of an equal and oppositely-directed
force. Force does not have the objective reality of matter and of energy,
but may exist at one instant and be destroyed the next. Press your
hand on a table and a force exists which would move the table were it free
to move; take your hand away, and the force ceases to exist.
It is possible in several ways to make quantitative measurements of
force. For example, we know that it requires a certain force to elongate
a spiral spring by a given amount; the free end of the spring will move
until the elastic force of restitution is just equal to the applied force. We
infer logically that it will require twice the force to elongate two parallel
springs to the same extent. Experiment shows that the force which will
elongate two similar springs simultaneously by a given amount will
elongate one spring twice the amount. By such experiments we arrive
at the law, which experiment shows to be very nearly true, that the
8 PROPERTIES OF MATTER.
elongation or compression of a spring is proportional to the force acting
upon it, and we can use such a spring in the comparison of forces. This
is an example of the statical method of comparison, in which equilibrium
is secured by balancing two equal forces against each other, so that there
is no resulting motion.
If, however, we consider the simpler case of a single unbalanced
force, we must measure it in terms of the acceleration produced by it.
The force is defined as being proportional to the acceleration. It is found,
however, that the acceleration does not depend solely upon the force,
but also upon the nature and size of the body acted upon. That function
of the nature and the size of the body which determines the acceleration
produced by a given force, or in other words which determines the
inertia of the body, we call mass. Usually, but rather incorrectly, mass
is defined as quantity of matter. It is doubtful, however, whether the
term quantity of matter conveys any clear klea when we are comparing
different kinds of matter.
16. The Measurement of Mass. If mass be defined as the quantity
of matter, it is easy to see that two cubic centimeters of water must have
twice the mass of one cubic centimeter; it is not at all evident that a
cubic centimeter of alcohol has the same mass as one of water indeed it
is manifestly untrue if quantity of matter has any relation to either inertia
or to weight, or any other than a geometrical significance.
The following experiments indicate a .logical method of estimating
mass :
1. Suppose that two cubes of the same material, of exactly the same
size, are successively set in motion on a smooth horizontal table by a pull
exerted through a spring balance. It will be found that the spring will
be stretched by the same amount in each case if equal accelerations are
imparted to the two masses. With a given cube it is also found that the
elongation of the spring is proportional to the acceleration, and therefore,
by the definition of force, also proportional to the force. If another cube
of twice the volume be used, the spring will be stretched twice as much
in imparting the same acceleration. In this case one cube has twice the
mass of the other, in whatever way we may estimate mass. Were all
matter alike we might measure its mass in cubic inches.
2. If we take cubes of the same size, but of different substances, the
force required to produce the same acceleration will be found to be
different in each case, showing that inertia is not proportional to volume
if we compare different substances. From (1) we may assume that if the
cubes in (2) are of such relative sizes that the same applied 'force, as
measured by the stretching of the spring will produce the same accelera-
tion, that is, if the inertias are equal, the masses are equal.
This method of measuring mass, which defines it in terms of inertia,
is called the dynamical method. It is difficult to apply practically. The
statical method of weighing, in which we compare masses in terms of the
balanced attraction between them and the earth, is more convenient.
This method will be discussed in detail later.
Density is the mass of unit volume, while specific gravity is the ratio
between the density of a given substance and that of water taken as a
standard. The numerical value of the density of a substance depends
upon the system of units adopted; for example, in the English foot-
pound system the density of water would be 62.321. Specific gravity is
a numerical ratio, and independent of the system of units.
V
DYNAMICS.
FORCE AND MOTION (DYNAMICS).
17. The most familiar of all forces is that of gravitation, and it is
not strange that it was the first to be investigated. It is surprising, how-
ever, to consider how many centuries elapsed after this subject became
an object of attention before any one was able to accurately describe the
way in which a body falls. In their philosophic zeal to find why bodies
fall, the ancient Greeks overlooked the simpler problem of how they fall.
They explained the fall of bodies by saying that every body seeks its
proper place, and that the place of heavy bodies is below that of light
bodies. It was to them a logical inference that a very heavy body would
seek its place faster than one less heavy. It seems to have never occurred
to them, nor to any one until many centuries later, to try the simple
experiment of watching bodies fall, and thus test this assumption. There
can be no better illustration of the danger of a priori reasoning.
Galileo, a professor in the University of Pisa, in the early part of the
seventeenth century, seems to have been the first to abandon the vague
methods of the older thinkers. He undertook to formulate an accurate
description of how bodies fall, and in so doing laid the foundations of
Dynamics, or that branch of Mechanics which treats of matter in motion.
He found by dropping bodies from the Leaning Tower at Pisa that the
speed of fall is proportional to the time of fall, regardless of the nature or
size of the object, provided that the weight is sufficient to make the
resistance of the air negligible. He thus introduced the idea of accelera-
tion, or change of velocity per unit time. He also perceived a very
fundamental principle of mechanics that a body may be subjected to
two or more simultaneous motions, which go on absolutely independently.
Before his time it seems to have been tacitly held that one effect must
cease before another begins that a projectile must cease to move hori-
zontally before it can drop vertically. Galileo was the first to show that
the rate of vertical fall is absolutely independent of any horizontal motion
which the body may possess, as any one may see in the case of a stone
projected horizontally.
These ideas were further elaborated and more definitely expressed by
Sir Isaac Newton in his Principia (London, 1687). He recognized the
property of mass as inherent in bodies, and considered the action upon
matter not only of gravitation, but of all forces which may cause motion.
He also distinctly formulated the principle of the parallelogram of forces,
which had been more or less vaguely perceived in special cases by others
before him.
The conclusions of Newton may be summarized substantially as
follows :
1. Every body tends to move uniformly in a straight line unless
acted upon by an impressed unbalanced force.
2. Change in motion (acceleration) is in the direction of the
impressed force and proportional to its intensity. This statement con-
tains within it the definition of force.
3. Action and reaction are equal and oppositely directed.
These laws are not the results of a priori reasoning, but are simply a
generalized description based on our daily experience. Their application
to cases outside of our experience is a typical example of scientific induc-
tion. As a matter of fact, we have no knowledge of any body which has
ever moved uniformly in a straight line; but we see that all bodies -tend
10 PROPERTIES OF MATTER.
to do so more and more as we remove opposing forces, and the conclu-
sion of the first law of Newton seems to be a logical necessity. (1)
We hear much of the impossibility of perpetual motion. As a matter
of fact, perpetual motion is the rule of nature. It is impossible only in
the limited sense that a body cannot overcome resisting forces, thus doing
work, without being supplied with energy from some external source, or
in other words having its motion replenished at the expense of other
moving bodies.
18. Experiments such as those described in section 16 show that the
acceleration imparted to a given mass varies directly as the force, and
that the acceleration due to a given force varies inversely as the mass
acted upon. This proportionality is expressed as follows :
(4) F= Kma,
K being a numerical constant depending onthe system of units adopted.
This is the algebraic expression of Newton's second law.
The C. G. S. unit of force, called the dyne, is defined as the force
required to produce in one second a change of velocity of one centimeter
per second in a mass of one gram. Under these conditions, K = 1.
19. Composition and Resolution of Motions, Velocities,
Accelerations, and Forces. If a body be moved from A to B,
thence to C, its final position is the same as though it had been moved
directly from A to C. Geometrically, in dealing with directed or
* ' vector ' ' quantities, we may write,
(5) AB + BC = AC.
If AB is proportional to and in the direction of a given velocity v t
(motion per unit time), and BC in the same way represents another
velocity z/ 2 , then the line AC will represent the resultant velocity v. In
other words if, by the simultaneous application of two forces, the two
velocities v l and z/ 2 proportional to AB and BC are simultaneously
produced, the body will actually move along the line AC with a resultant
velocity v proportional to the length of that Tine.
The same reasoning applies to any number of velocities. If lines
representing them be drawn end to end, the closing side of the polygon
thus formed will be the resultant velocity.
Since acceleration is change of velocity in unit time, the same rules
apply to the composition of accelerations; and also to forces, since
accelerations are proportional to forces.
Suppose that v t is the velocity of a body at the time tf r , v 2 the velocity
(differently directed) at the time 4. Draw from a point two lines AB,
AC, proportional to the two velocities and similarly directed. Then BC
represents the change of velocity, both in amount and direction, or
(6) ^-z^aa-O-BC
This is always geometrically true of the first equation, but is arith-
metically true only when the velocities are in the same direction; it is
always true of the second equation.
20. Resolution of Vector Quantities. By a converse method,
any one motion, velocity, acceleration, or force may be resolved into any
number of components in any desired direction.
(1) For discussion of work of Galileo and Newton see Mach, Science of Mechanics, Ch. II; also
articles Galileo and Newton, Ency. Brit.
r
FORCE MOMENTUM. 11
In general, we wish to resolve these quantities into two components at
right angles to each other, along the X and the Y coordinate axes. If a
force F have a direction at an angle a with the X axis, and A' and Y be its
components,
X = F COS a
Y=
All these results are necessary consequences of the definitions of
velocity, acceleration, and force, but they may be established as well by
direct experiment.
21. The moment of a force is the tendency of the force to
produce rotation about an axis, and is measured by the product of the
force into the perpendicular distance from the axis to the line of action of
the force.
22. There are two necessary conditions of equilibrium for any rigid
body :
1. The algebraic sum of the forces acting at any point of the body
must be zero for both the X and the Y direction, or in any direction,
otherwise there will be motion in that direction,
, 7 , sA r =S^cosa =
2 Y=z Fs'm a = 0.
2. The algebraic sum of the moments about any point whatever
must be zero, or there will be rotation about that point.
2^ =
S Yx = 0.
Rotations to the left (anticlockwise) are taken as positive, those to the
right as negative, because angular measurements are usually regarded as
increasing in the anticlockwise direction.
23. Parallel Forces. If R be the resultant of a number of
parallel forces acting on a body,
(9) R = F l + F 2
the proper algebraic sign being given to each force.
The moment of the resultant is the same as the resultant of the
moments, or
(10) Rx = F, x, + F 2 x z -f . . . etc.,
x, x^ x 2 , etc., being measured in the same direction from any common
origin.
24. Momentum. By the definition of force,
F= ma = 1
or
(11)
Experiment shows that the effect of a blow given by a moving mass is
proportional to the mass and to its velocity. The product mv is called
momentum. The above equation shows that change in momentum is
proportional to the force acting on the body and to the time the force acts.
12 PROPERTIES OF MATTER.
QUESTIONS AND PROBLEMS.
1. Formulate from your own experience some physical law not explicitly
stated in the text-books.
2. Rotate a cup containing water or coffee, and note whether the liquid moves.
Explain the result.
3. A wagon moves with the uniform speed s. In what path does a point on
the circumference move relative to the axle ? relative to the earth ? What is the
speed of the top point of the wheel relative to the earth? f the bottom point?
4. If w be the angular velocity of a point with radius vector r, prove that the
linear speed ^ = rw.
5. The wheels of the wagon make 10 revolutions per minute. What is their
angular speed per second with reference to the axle ? If the radius is 50 cm. what
is the linear speed of the top point with reference to the lowest point ? its angular
speed ?
6. What are the angular speeds of the minute and the hour hands of a clock?
7. The wagon referred to above starts from rest with a uniform acceleration,
and at the end of 30 seconds is traveling at a spee*d of 4 meters per second. What
is its linear acceleration ? the angular acceleration of the top point with reference
to the axle ? with reference to the low r est point ?
8. What is the resultant velocity (speed and direction) of a boat rowed at a
rate of 4 miles per hour eastward and drifting at a rate of 3 miles per hour
southward ?
9. A mass of 100 grams is at one instant moving eastward with a speed of
50 cm. per second. Five seconds later it is moving in a direction 60 north of east
with the same speed. What is the direction and amount of its acceleration?
10. In the above case what force in dynes acts on the body?
11. A force of 50 pounds weight acts vertically downward on a bicycle when
its crank-arm is at an angle of 45 with the horizon. What is the component force
which tends to cause rotation ? What does the other component do ?
12. In the above case, if the crank-arm is six inches long, what is the moment
of the force about the axle expressed in the foot-pound system ?
13. A force of 1000 dynes acts for 10 seconds on a mass of 20 grams. What
is its change in velocity ?
14. A horizontal meter rod is acted upon in an upward direction by the forces
200, 500, 300, and 700 dynes, at distances of 20, 25, 35, and 60 cm. from one end
respectively, and by a downward force of 800 dynes at a distance of 40 cm. from
that end. What is the resultant force, its point of application, and the resultant
moment?
15. Why is a heavy fly-wheel often used in running machinery?
16. Why is a balance weight used on the driving-wheel of a locomotive
opposite the crank-pin ?
GRAVITATION.
25. Acceleration. Experience shows that every unsupported body
falls to the earth with a uniform acceleration. In all cases where bodies
are seen to rise for instance, balloons or where the acceleration is not
constant and the same for all substances, the effect can be traced to the
buoyancy of the surrounding medium, or to some resisting force. The
force acting between any body and the earth is called the weight of the
body, and like any other force it is defined by the relation
(12) W=mg, ,..
yft+iu&
where g is the acceleration due to rnnnlMjjfiii (about 980 cm. or 32 feet
per second gain of velocity per*s>econd).
* jtf^LifjLjFr
26. Center of QimvWp. Each element of a given mass is acted
upon independently by gravity. (See section 23.) The total weight
(resultant force) is the sum of the elementary weights, or
(13) W=W^+W 9 + . . . . ZV
THE BALANCE. 13
The moment of the entire weight about any given point must be the
same as the sum of the elementary moments. If the elements are at the
horizontal distances x^ x z . . etc., from the point considered,
(14) Wx w^Xi + WtX^ . . . etc.
The point of application of the resultant weight must lie in a vertical
line at a horizontal distance from the origin chosen, where
V * ffftt \ /*/ I
This point is called the center of gravity, or of weight. The body
acts with reference to external forces as though its entire mass were con-
centrated at that point. If the body be suspended by a string, the center
of gravity will lie in the prolongation of the string, and as low as its con-
straints will permit. If balanced on a knife edge, the center of gravity
will be vertically above, at or below the knife edge. When the center of
gravity is above the point of support, equilibrium is unstable; when coinci-
dent with it, neutral; when below it, stable. The conditions are respec-
tively similar to those of a body on a hill top, one on a plain, and one in
a valley.
27. The Balance. From the relation W= mg we may determine
the mass of bodies by a statical method, by balancing their weights from
the ends of an equal-armed lever. In general it is impossible to secure
lever arms of exactly equal lengths. There are several conditions upon
which the sensitiveness and accuracy of a balance depend, some of them
inconsistent with each other.
Let the weight of the beam be w, and its center of gravity at a dis-
tance h below the point of support. P^ and P 2 are the weights of the
scale pans and their appendages. The system should balance with no
weights on the pans. If the arms of the balance are of lengths a t and a 2
the condition of equilibrium is
(16) P t a, = P 2 a 2
If the weights W t and W 2 be added, the condition of equilibrium
becomes
(17) (W, + P t }a^(W,+P,}a,
By subtraction of (16) from (17)
(18) W 1 a 1 =lV 2 a 2
If a t = a 2 , then W l = W 2 . If this is not the case, and if W t is the
weight of the standard required to balance the unknown weight W in
one pan, and W 2 the required weight when the object is placed in the
other pan, we have
W, a t = W 2 a 2
(19) IV a,= W 2 a 2
w=-
This is Gauss's method of double weighing.
The conditions for a good balance are that it shall be true; that is to
say, its arms horizontal and of equal length; it shall be stable; it shall be
14 PROPERTIES OF MATTER.
sensitive; it shall have a small period of oscillation, so that readings may
be quickly taken.
The sensitiveness of a balance is proportional to the deflection of the
beam produced by a small overweight placed in one pan; usually this
overweight is one milligram.
Let the angle between the axes of the arms (deflection downward) be
2a. If the overweight x deflects the beam or a pointer attached to it
through the angle 8, the condition of equilibrium is
(20) ( W+ x~} a . cos (a+0) = Wa . cos (a 0) + wh . sin e
Expanding and collecting coefficients,
[ (2 W-\- x) a . sin a + w/i] sin e = ax . cos a . cos 6
ax . cos a
tan0 = - , --_ , -r
wk + (2 W+vc) a sin a
In above expressions, 2 W includes the weight of the pans.
If the beam be straight (by which is meant that the fulcrum and the*
two knife edges supporting the pans are in the same straight line),
(22) tan* =4=^
/ wh
From this expression we see that the sensitiveness is greater the
greater the length of the arms, the lighter the beam, and the nearer the
center of gravity of the beam is to the fulcrum. It is entirely independent
of the load' W if the beam is straight. To preserve this condition under
all loads the beam must be rigid.
Stability and quickness of vibration are directly proportional to h\
consequently it is impossible to secure these advantages simultaneously
with great sensitiveness. A compromise is necessary.
Balances are made for which a sensitiveness sufficient to show differ-
ences of weight of one-thousandth milligram is claimed ; probably the
limit of accuracy is about .01 mg.
For accurate determinations of mass it is necessary to correct for the
buoyancy of the atmosphere. A delicate balance is very sensitive to
temperature changes and to air currents, and must be protected from
them when readings are made.
PROBLEMS.
17. A meter rod balances on a knife-edge at division 40 if a 100 gram weight
hangs from it at division 32. What is the weight of the rod ?
18. From a uniform circular disc of 20 cm. radius a smaller disc of 10 cm.
radius is cut tangentially. Where is the center of gravity of the remaining portion ?
28. Kepler's Laws. The German astronomer Kepler, about
1609, after many vain attempts to formulate simple laws descriptive of the
motions of the planets, on the basis of Tycho Brahe's astronomical
observations, succeeded in establishing three such laws, which have
become the basis of mathematical astronomy. .They are as follows :
1. All the planets move in elliptic orbits, with the sun at one focus.
2. The motion of each planet is such that the radius vector drawn
from it to the sun sweeps over equal areas in equal times.
3. The squares of the periods of rotation of the planets about the
sun are proportional to the cubes of their mean distances from it.
K
UNIVERSAL GRAVITATION CENTRIPETAL ACCELERATION. 15
29. Universal Gravitation. The ancients observed that bodies
fall, and speculated as to why they fall. Galileo was the first to observe
how they fall. About the middle of the seventeenth century, several
persons suggested the possibility that the moon and the planets were held
in their orbits by a force similar to that of weight, but Newton, about
1666, was the first to clearly recognize this law of attraction as possibly
a general principle, and to state it as a function of the mass and the
distance apart of the two bodies. The form which he gave to the law
was this
(23) f=J&2*.
F being the attractive force, m t and m 2 the masses of the two bodies,
d the distance between them, and K a numerical constant depending on
the system of units adopted. If ra T be the mass of the earth, m 2 that of
another body at the earth's surface, at a distance r from the earth's center
of gravity, and JFthe weight of the second body,
(24)
This equation was simply the statement of a hypothesis until it was
verified indirectly by astronomical observations and directly by laboratory
experiments.
30. Centripetal Acceleration. The orbits of the moon and the
planets are ellipses differing very slightly from circles. To make a body
follow a circular path by constantly drawing it from the tangential path
which it tends to follow on account of its inertia, the constant action of
some force is required; if the body moves at a uniform rate around its
orbit, the deflecting force must always act at right angles to the path, so
that there is no component of acceleration in that path. To fulfill this
condition for a circular or nearly circular path, the force must be always
directed toward the center of the circle, and the acceleration produced by
it is the centripetal acceleration. To calculate this acceleration, consider
an element of the orbit so short that it may be considered a straight line.
In a very short time / suppose that the body deviates from a tangent
drawn to the circle at its original position by a distance d, owing to the
central attraction. If a is the acceleration produced by this force, the
average speed in a radial direction during the time / is
The arc traversed is
b = vt,
Also, from similar triangles,
d : b = b : 2r.
Therefore,
(26) =.
This may be tested experimentally as follows : Two spheres of masses
m^ and m 2 slide on a smooth rod attached to a whirling table at right
16 PROPERTIES OF MATTER.
angles to its axis of rotation. If these two masses be connected by a
string and the table rotated, it will be found in general that the connected
masses fly off to one end or the other of the rod; but if placed in such a
position that the centers of the spheres are at distances r^ and r z from the
axis of rotation, such that r t : r 2 = m 2 : m lt it will be found that the
bodies will be in equilibrium whatever the speed of rotation. If (26) is
true, - ^
or if ^ be the angular velocity,
v, = r,
Substituting above,
(27) m
in accordance with the results of the experiment.
31. Newton's Proof. If the moon moves in a nearly circular orbit
about the earth as a center, with a nearly uniform velocity, a central
attracting force is necessary. Newton assumed that this force is due to
the attraction of the earth, and that it varies, just as the intensity of light
for example, inversely as the square of the distance. If R be the radius
of the moon's orbit, r the radius of the earth, g the acceleration of
gravity at the earth's surface, and g t the acceleration of gravity at the
distance of the moon,
By the law of centripetal acceleration we also have
Newton tested these relations by substituting values of v and R from
the imperfect data then obtainable; the equality
S
did not appear to be verified. Many years later the distance of the moon
was recomputed by Picard; Newton substituted the new values in his
equation, and found it completely verified.
Newton likewise found that Kepler's laws were in almost exact accord
with the hypothesis of a central attractive force, varying inversely as the
square of the distance. Any apparent deviations from Kepler's laws are
explained by the perturbing influence of other attracting bodies. The
position of the planet Neptune was almost exactly computed from its
perturbations on Uranus, by Adams and by Leverrier, before it had ever
been seen, although these perturbations had the effect of displacing
Uranus by only 2' of arc. Galle, of Berlin, pointed his telescope in the
direction assigned by Leverrier, and within half an hour discovered the
planet. The accurate fulfillment of astronomical predictions is a constant
verification of Newton's law of universal gravitation.
. DIRECT VERIFICATION.
32. Direct Verification. The resultant attraction of the ear
would be exactly directed toward its center if it were of uniform density,
or if its density varied uniformly between its center and circumference.
As a matter of fact, local irregularities may cause measurable deflections
of a plumb line. In 1774 Maskelyne, the astronomer royal of England,
determined the deviation of a plumb line produced by Mount Schehallien,
in Scotland. The mountain is of very regular shape, and a rough calcu-
lation of its density could be made by a study of its geological structure.
From these data its mass and the position of its center of gravity could
be roughly determined It was found by comparison with the directions
of the stars that the deflection of the plumb bob toward the mountain was
about 6" from two opposite stations on the north and the south. If the
acceleration toward the mountain be g^
(29) 1-=* = t ana
o
Also, if m t be the mass of the earth, m 2 that of the mountain,
(30)
m n d z tan a
We can thus calculate the earth's mass as compared with that of the
mountain. Thfc s number is inconveniently large, so the results are usually
expressed in terms of the earth's density, from the relation
(31) m t
This method, which is evidently liable to great inaccuracies, gave
(32) P = 4.71
Henry Cavendish, in 1798, secured more trustworthy results by the
use of the torsion balance. Two lead spheres, s t and s 2 , about two inches
in diameter, were attached to the ends of a light wooden rod several feet
long. This was suspended horizontally from a fine wire, and the entire
system enclosed in a glass case. Two large spheres, S t and S 2 , were
attached to a frame so that they could be brought to any required position
with respect to s t and s 2 . In one position the torsion arm would be
twisted in one direction; in another position, in the opposite direction.
The deflections were read with a telescope. The magnitude of the deflect-
ing force could be expressed in terms of the known force required to twist
the wire through 1, and the actual angular deflection produced by the
attraction of the large spheres. The density of the earth as found by
Cavendish was
(33) P =
18 PROPERTIES OF MATTER.
Professor Vernon Boys, of England, in 1894 devised a much more
sensitive and accurate torsion balance. The small spheres were of gold,
about 5 mm. in diameter, and were hung by fine quartz fibers from a short
torsion beam 2.3 cm. long, which was itself suspended by a quartz fiber,
within a closed cylinder. Within a concentric cylinder which could be
rotated about the first were hung two lead spheres about 10 cm. in diam-
eter. The pairs of attracting spheres were in different horizontal planes,
in order to eliminate as far as possible the effect of each large sphere on
the more distant small sphere. The quartz fibers were used because of
their small torsional rigidity and their constancy of zero position. Boys
was able to show the attraction between two small shot, the actual magni-
tude of this force being about 1/200,000 dyne. He found that
P = 5.527
Richarz and Krigar-Menzel (1898) used a balance method. A large
cube of lead, weighing about 100,000 kilograms, was placed under a sensi-
tive balance which had four pans, two above the block and two below it,
suspended by fine wire passing through holes in the block. The process
of double weighing was repeatedly applied to two approximately equal
masses, m^ and m 2 , first one above, the other below, then interchanged.
The apparent weight of the upper mass would be increased, that of the
lower lessened by the attraction of the lead block. The attraction between
the block and a kilogram mass was about 1.4 milligram weight, which
indicates how much care was necessary in taking the readings. They
found
A^=6.685X 10- 8
P-5.505
t
The discrepancy between the results of Boys and of Krigar-Menzel
and Richarz maybe due in part to local differences of density of the earth.
33. Variations of the Force of Gravitation. Mass is an
invariable property of any body; its weight depends on the mass and dis-
tance of another body. At some point between the earth and the moon,
for example, an object might have no resultant weight ; but its mass
would remain unchanged, and could be determined by a dynamical
method.
The weight of a body varies at different points of the earth's surface,
for several reasons:
1. There are local and irregular variations of g due to mountains,
masses of rock beneath the surface, etc., which cannot be computed.
2. The earth is not a perfect sphere, but is flattened at the poles, so
that its cross-section is approximately an ellipse, the minor axis coinciding
with the axis of rotation. If a body be carried from the equator to the
pole, it is constantly approaching the center of the earth. In the case of
a spheroid such as the earth there is no definite center of gravity, toward
which the plumb line would point wherever it might be, so that the
increase of g cannot be simply calculated in terms of the change of the
earth's radius. If the latitude is X, and g & the value of g at the equator,
it is found that .
(36) *' (!+-
VARIATIONS OF THE FORCE OF GRAVITATION THE TIDES. 19
3. At a given latitude there are changes of g with the altitude above
sea level. We have
g. (r+*r
or
(37) g^ = ( 1-y-) ^-. = (1-0.000000314^)^-0
This applies strictly only to values observed on a high tower or in a
balloon. The value of g on a mountain or plateau is influenced by local
masses.
4. In order to hold bodies in their places on the surface of the earth
a centripetal acceleration is necessary. At the latitude X the necessary
acceleration, in a plane at right angles to the polar axis, is (assuming the
earth to be spherical)
(38) = r x <* = ^rcos\
The necessary acceleration is furnished by gravity, but of course not
in the line of action of gravity. Let G represent the direction and
intensity of gravitation if the earth were at rest, a the centripetal acceler-
ation with the earth in motion, and g the actual observed acceleration
after a component of G has been taken to furnish the necessary centripetal
acceleration. From the above equation it is easy to calculate the relation
(39) g = G (1 7^ cos 2 X)
Combining above equations and substituting for the value of g at the
equator that at latitude 45, (980.61) the value of g at any latitude X and
height h above sea level may be expressed by the equation
(40) g = 980.61 (1 - 0.00259 cos 2 x - 0.0000003/z)
It is the residual component g which determines the weight of bodies.
After the direct pull of gravitation has lost a component to supply the
necessary centripetal acceleration at a point, the remaining component is
of course no longer directed toward the earth's center except at the
equator and the poles; only in these places does a plumb line point
exactly toward the center of the earth.
The surface of a liquid is always normal to the resultant force acting
on it, and the ocean is, therefore, when not disturbed by wind, always
normal to a plumb line. Were the earth to cease rotating, the water
would rush from the equator to the poles, assuming a spherical surface.
Renewal of rotation would restore the present shape. We can thus realize
how the earth was flattened at the poles when in a semi-molten state.
The residual component of gravitation has a component tangential to the
earth's surface which keeps water and other objects from rolling down the
13-mile hill to the poles.
34. The Tides in the earth's oceans are caused by the attraction of
the moon and the sun. The component due to the former is more than
twice as great as that due to the latter, on account of the nearness of the
moon. At the times of new moon and full moon both effects are in the
same phase and the tides arehjgjiest (spring tide). At half-moon the
RA^
OF THE
1IN1IVFRSITY
20 PROPERTIES OF MATTER.
crest of one tidal wave coincides with the trough of the other, and the
tides are low (neap tide).
There are two tidal waves, one toward the side of the moon (but not
directly under it on account of a lag or lead due to the solar action, fric-
tion, etc.) and one away from it. To explain this we must consider the
nearer and further masses of water and the intermediate solid earth to
behave as three bodies. The nearer is more accelerated toward the moon
than the earth, the further is less accelerated ; and so they move apart,
the nearer body of water being heaped up, and the earth leaving the
further body behind. Both the earth and the moon are rotating about
their common center 'of gravity (situated within the earth). Looking at
the problem in another way, we can consider the tides as arising from the
centrifugal tendencies of the earth and water with respect to the axis
passing through the center of gravity of the two bodies regarded as one
system. If the moon were at rest with respect to the earth the results
would be different (see Problem 26). There is necessarily a loss of
energy by friction between tidal waves and the solid earth, and when the
wave is stopped by impact against the shore; this must be borrowed from
the earth's energy of rotation, hence the days must gradually be growing
longer.
The height of the tides is very different in different places. In the
open Pacific it is only 2 or 3 feet. In the North Sea interference of two
tidal waves, one from north of Scotland, the other through the English
Channel, minimizes the effect to a few inches. In nearly enclosed seas
like the Mediterranean they are only a few inches high. South of England,
at St. Malo and the Isle of Jersey, they are some 35 feet high, and in
narrow bays such as the Bay of Fundy, into which large masses of water
are driven by their momentum, the tide may rise to 70 feet or more.
References. Newton, Kepler, Tides Mach, Science of Mechanics; Lodge,
Pioneers of Science ; Ball, Time and Tide. Verification of law of gravitation
Mackenzie, Laws of Gravitation ; Tait, Properties of Matter ; Burgess, Physical
Review, May, 1902 ; Boys, Nature, August 2, 9, 23, 1894, and Phil. Trans. Royal
Soc., 1895; Poynting and Thomson, Properties of Matter; Poynting, Scientific
American Supplement, May 2, 1903.
QUESTIONS AND PROBLEMS.
19. Derive the expression for centripetal acceleration by applying the method
of section 19, equation (6).
20. Do you know of any case of centrifugal acceleration or force ?
21. A mass of lead weighing 100 kilograms has its center of mass 100 cm.
above the pan of a balance. How much will it diminish the weight of 100 grams
in the pan ?
22. If the earth rotated seventeen times as fast as it does, what would be the
value of g at the equator ?
23. If the earth rotated more than seventeen times as fast as it does, and if
we were unaware of its rotation, how would we be apt to state the law of gravita-
tion ? Would it be correctly stated ?
24. Explain the trade winds and cyclones. Show that a projectile will be
deviated to the right in the northern hemisphere no matter in what direction in a
horizontal plane it may be fired.
25. If a projectile could be weighed while moving east or west would it
weigh the same as if at rest ?
26. If the moon and the earth had no mutual acceleration, how would the
tides be affected ?
27. What shape is assumed by the water surface in a cylindrical vessel when
the latter is rotated rapidly? Explain.
28. Explain the action of the Watt's governor for steam engines.
29. Explain the " centrifugal " cream separator.
DETERMINATION OF G - FALLING BODIES. 21
35. Determination of g. Consider a smooth sphere rolling
down a smooth inclined plane at an angle of a with the horizon. The
resultant acceleration down the plane is the resolved part of g in that
direction, or
(41) a = g sin a.
In this way a may be made so small as to be conveniently measured,
and g computed from the above relation. This method was used by
Galileo, but gives only crude results.
The most accurate method is by means of pendulum observations.
Consider a conical pendulum that is, a simple pendulum which rotates in
a circle of radius a, so that the supporting cord describes a cone in space.
The tension in the cord must be such that it will furnish a vertical com-
ponent to support the weight of the pendulum, and a centripetal force
holding the pendulum in its orbit. We have evidently, if / be the length
of the pendulum, and h the height it is raised,
v 2 j a : g = a : I h.
If the displacement is so small that h may be neglected in comparison
with /, this gives
If T be the period of rotation of the conical pendulum,
(42)
Observation shows that if the same pendulum vibrates in a plane the
period will be exactly the same in fact we shall see later, in discussing
vibratory motion, that the circular rotation is simply the resultant of two
plane vibrations at right angles to each other, and necessarily takes place
in the same time as either vibration. Furthermore, for all small displace-
ments or amplitudes a, such that h may be neglected, the period is
independent of the amplitude a. This fact was discovered by Galileo.
From the above,
47T 2 /
If the pendulum bob is small and heavy, this gives a very accurate
method of determining g, since both T and / can be very exactly
measured. In the case of the compound pendulum (a massive bar), the
problem is not so simple, since each element of the pendulum tends to
vibrate at a different rate. There are relations, however, which make it
possible to determine the length of the equivalent ideal simple pendulum,
so the compound or Kater pendulum is more frequently used.
Some values of g are as follows :
Latitude.. 15 30 45 60 75 90
g ........... 978.1 978.4 979.4 980.6 981.9 982.7 983.1
36. Falling Bodies. Within observable distances from the surface
of the earth the acceleration of gravity is practically constant, or the gain
in velocity per second is the same during each successive second. If v be
22 PROPERTIES OF MATTER.
the initial velocity of a falling body, and it falls from time t to time x
or t seconds,
(44) g =
(45) v,=
Since the velocity changes at a uniform rate, the average velocity is
f Ad\ ^t+^o
(46) v = .
In falling from the height h to h t ,
_ V t + 27 1
(47) *.-A-f- -^-
Multiplying (44) by (47), we get
This relation is perfectly general ; it applies to bodies projected
upward with velocity v , as well as to falling bodies. If, however, g and
distances fallen through be counted as positive, the velocity of upward
projection and the height of ascent must be counted as negative.
37. Inclined Fall. Consider a body rolling down an inclined
plane. The acceleration a is found by the relation,
(49) a f - and also a = -
v~~ * t ' *
The average velocity down the plane is
(50) v = ^+^
/" f "\ \ ^+ I 7^ r*> *
/.-/,= *= -^ *
The speed acquired by falling depends, therefore, solely on the vertical
distance through which the body has fallen ; but the direction of the
velocity is determined by the constraint at right angles to the direction of
motion.
The same considerations apply to the pendulum ; its velocity at any
point is that due to the fall through the height h hi. In passing
through its resting point, however, the actual motion is entirely in a
horizontal direction.
38. Nature of Gravitation. No reasonable hypothesis has ever
been found to account for gravitation. Le Sage explained it as the result
of the bombardment of etherial particles against bodies, driving them
together; but this would imply that a flat body would be heavier when
presented broadside than when presented edgewise toward the earth, and
f~ OF THE \
UNIVERSITY)
CONSERVATION OF MASS. 28
would also involve a continuous expenditure of energy. There is no way
of shielding matter from its effects; were this possible, an inexhaustible
supply of work might be obtained by so shielding half of a vertical wheel,
which would be driven by the pressure of gravitation on the unshielded
half.
39. Conservation of Mass. There is no external resemblance
between metallic sodium, the violet sodium vapor at high temperatures,
and the glowing yellow sodium vapor at still higher temperatures. A
chemical compound, such as sodium chloride, has no resemblance what-
ever to either of its constituents. It is rather doubtful whether we can
say in such cases, as is usually done, that in all these changes the matter
remains unchanged. The total mass, however, does seem unalterable by
any physical change or chemical combination. Some experiments by
Landolt and others on the weight, of substances in a sealed tube before
and after combination seem to indicate the possibility of a very slight loss
in weight; but such measurements are difficult and the results of doubtful
validity when compared with the enormous number of chemical determi-
nations showing the contrary.
QUESTIONS AND PROBLEMS.
30. Draw a curve with the values of latitude and g given in section 35 as
coordinates, and by interpolation find the value of g in Berkeley.
31. Calculate the length of the seconds pendulum in Berkeley.
32. How would a large block of lead under a pendulum affect its period ?
33. If a seconds pendulum is constrained to swing in a plane at an angle of
45 with the vertical, what will be its period ?
34. A body is thrown upward with a velocity of 10 meters per second. How
far will it rise ?
35. How long will it be before the body in preceding problem returns to the
earth ?
36. A sphere rolls from the top edge to the bottom of a hemispherical bowl
of 50 cm. radius. What will be its velocity at the bottom?
WORK AND ENERGY.
40. When we raise a weight or in any other way move an object
against external forces, we say that we do work. Not only is the idea of
force involved, but the idea of motion as well. A weight lying passively
on the earth does no work; when by sinking it drives the wheels of a
clock, it does work. A hod- carrier who carries a load of brick to the
third story of a house evidently does twice as much work as though he
carried it to the second story. Work done iipon a body may be quanti-
tatively defined as being equal to the product of force overcome by the
distance through which the displaced body is moved. The work done by
a given force (the product of the force and the distance through which it
acts) is in general greater than that done upon any one resisting force ;
there are usually unknown frictional forces to be overcome as well, or the
excess of applied force may not do useful work, but produce acceleration
of the body moved. A moving object may transfer its motion in part or
wholly to another body when it strikes it ; so the possession of motion
implies the power of doing work. A body at rest may also under certain
conditions have the power of doing work. A raised weight, for example,
or a coiled spring may set a clock in motion when the escapement is
released. The possibility of work in such cases is a consequence of the
position or the state of a working body. A body which for any reason
24 PROPERTIES OF MATTER.
is able to do work is said to possess energy; this energy is called kinetic
if it depends upon motion; potential if it depends upon position or state.
The total energy of a body may be part kinetic, part potential. There
are other classes of energy; that of heat, for example, which seems on
analysis to be due to molecular kinetic energy; and electrical and chemical
energy.
If a body of mass m is raised to a height k , the required work is
W= mgh
and this represents the potential energy of the body at this height. If
the body falls freely to the height h lf we have from equation (48)
Multiplying by m,
(53) 4 m (y* v mass and position of the molecules, and the forces acting between
them, and the effects of external forces upon molecular conditions.
Besides producing motion of bodies as a whole, external forces may
cause strains in matter, or deformations of volume or shape involving
relative molecular motion, and work in opposition to resistance of inter-
molecular forces.
46. States Of Matter. There are two kinds of strain change of
volume, which depends upon the compressibility of the substance; change
STATES OF MATTER FLUIDS. 27
of shape, which depends upon its stiffness or rigidity. We may roughly
divide matter into three classes or states, depending upon the nature of
the strain produced by an applied force. These are solids, which are
almost incompressible, and almost rigid; liquids, which are almost incom-
pressible, and not at all rigid; and gases, which are compressible and not
at all rigid. Both liquids and gases have great molecular mobility, causing
them to flow or to yield to the action of the smallest applied forces.
They are called flidds in consequence. The distinction between liquids
and gases is that the former have definite volumes and may have free
surfaces, the latter will fill any vessel in which they may be placed, and
usually have no well-marked free surface. ( Some heavy vapors like iodine
vapor may have an indistinct surface boundary.)
In many substances there appear to be intermolecular forces which
oppose any changes in the relative positions of the molecules, and which
restore them to their former positions if the applied forces are not so
great as to cause permanent rupture. Such bodies are called elastic, and
the force of restitution is a measure of the elasticity of the substance. The
elasticity is expressed in terms of the force of restitution per unit area.
When elasticity is perfect this stress is equal to the applied force ; in
recovering its original condition the body does as much work as that done
upon it, and there is no loss in overcoming frictional resistance. A well-
tempered spring nearly fulfills this condition. No solids are perfectly
elastic in any respect, however. All fluids are perfectly elastic as regards
volume changes, but none have elasticity of shape. Some substances,
such as putty or clay, offer considerable resistance to forces tending to
change their shape, but no force of restitution; the work of the force is
entirely spent in overcoming internal friction, or viscosity.
The various coefficients of elasticity are numerically defined as the
ratio of the force of restitution and the strain produced by the applied
force.
(55 )
strain
FLUIDS.
47. Fluids (including both liquids and gases) will all yield in time
to the action of any distorting force, however small it may be. All known
fluids offer some resistance to changes of shape, but it breaks down with
time in some cases almost instantaneously, in others very gradually.
This resistance is of a frictional character, and seems to exist between the
smallest parts, or molecules. In the case of fluids it is called viscosity,
and bodies which yield very slowly to distorting forces are said to be very
viscous. Molasses and tar are instances of this kind. Even the lightest
gases have some viscosity. Air, for example, offers a frictional resistance
to projectiles, which increases very rapidly with increased velocity.
Meteors are so heated by this means, even in the very rare upper atmos-
phere, that they become incandescent and vaporize. Between the least
viscous gas, hydrogen, and solids at the other extreme, there is no sharp
dividing line. Hydrogen, air, carbon dioxide, ether, water, oil, molasses,
shoemaker's wax, sealing wax, are examples of fluids in increasing order
of viscosity. Many metals, such as lead, will flow under the action of
forces sufficiently great, but an infinitesimal force will not cause this effect,
so that lead would not be called a fluid [ A. rod of sealing wax will in
OF THE
UNIVERSITY
28 PROPERTIES OF MATTER.
time bend under the action of very small forces; so in a sense sealing wax
may be classed among fluids, although when acted upon by a strong and
sudden force it may exhibit the properties of a brittle solid, and appear to
be less fluid than lead. Heat in nearly all cases increases molecular
mobility indeed, is held to be synonymous with it and thus changes
solids to liquids, liquids to gases.
4:8. Hydrostatics is the branch of Mechanics which deals with the
equilibrium of fluids at rest.
Fluid Pressure. When a fluid, either gaseous or liquid, is contained
in a vessel, the weight of the fluid, its elastic force of expansion (if a gas)
and forces transmitted from external sources will act upon the walls of the
vessel. Some conclusions, easily verified by experiment, are necessary
consequences of the mobility of fluids. One of these is that the force of
a fluid at rest upon the walls of the vessel^ or upon any immersed surface,
must be everywhere normal to the surface. If there were an uncompen-
sated tangential component, the fluid would move in that direction, and
thus not be at rest. With gravity alone acting, all liquid surfaces are level;
when the wind blows strongly they are normal to the resultant of gravita-
tion and wind. If a vessel containing ether or water is slightly tipped,
readjustment of level takes place very quickly; if it contains molasses the
raised portion of the liquid flows very slowly down hill, but in time the
surface will become horizontal.
It is convenient to consider the force acting normally on each unit
area rather than the total force. Pressure is defined as normal force per
unit area. When we speak of pressure at a point we mean the ratio
between the normal force acting on an infinitesimal element of surface and
the area of that element.
49. Transmission of Pressure. Imagine a fluid compressed by
a piston in a cylinder. If the reaction pressure in any one region of the
boundary were less than the applied pressure, the fluid would move in that
direction, displacing the boundary. It follows that the applied pressure
must be transmitted uniformly in all directions, and must be the same at
every point of the surface.
The applied pressure upon any imaginary element of the fluid within
its volume must for the same reason be the same at every point and act
uniformly in all directions.
50. Pressure due to Weight of the Fluid. The pressure due
to an applied force is the same throughout a fluid. This is not true of the
pressure due to the weight of the fluid. Consider any imaginary horizon-
tal surface within the fluid. Each unit area supports the weight of the
column of fluid resting on that area. This downward pressure is balanced
by an equal upward pressure. At every point in the horizontal plane the
pressure is the same, but in going downward it increases with the depth.
These principles were first clearly recognized and stated by Pascal.
In case of a fluid of uniform density p the pressure on a plane at a
depth h is
(56) P
51. Buoyancy. Any interior portion of a fluid in equilibrium must
be buoyed up by a force exactly equal to its own weight, due to the
upward pressure of the fluid beneath it. If we substitute for this portion an
equal volume of any other material of the same density as the fluid, we
should expect that to be also in equilibium ; and experiment verifies this
BUOYANCY SPECIFIC GRAVITY. 29
supposition. In other words, any submerged body is buoyed up by a
force equal to the weight of the water it displaces. If the body is denser
than water, it will sink, with an apparent weight equal to the difference
between its own weight and that of the displaced water. If it is less dense,
it will float, but still displace by its submerged portion its own weight of
water. This is known as the principle of Archimedes, and is one of the
oldest laws of Physics.
Archimedes' principle is a consequence of Pascal's principle. Consider
an imaginary cube of a fluid of sides a, in equilibrium with the same surround-
ing fluid. If the upper surface be at a depth h, below the surface, the
pressure per square centimeter on this surface is Pg/i, the total downward
pressure being a*pgh. The lateral pressures on opposite sides balance each
other. The pressure on a horizontal plane at a depth h-\-a below the sur-
face is pg{h\a~). Since this acts uniformly in all directions, the upward
pfessure on the bottom of the cube is a 2 pg(/i j ra). The difference between
this and the downward pressure gives as the total buoyant force, a?pg
but this is equal to the weight of the cube. If this cube be replaced
by one of a different density, the upward force from without will be
unchanged and the resultant weight will a 3 (p t p)g.
In the case of two liquids in communicating vertical tubes (U-tubes)
the heights in each limb when in equilibrium depend only upon the density
of the two liquids, not upon the cross section or the shape of the tube.
(57) P.g/1, = P 2 gh 2
This applies to gases as well as to liquids. The barometer is an
example, in which a certain column of mercury in a small tube is counterbal-
anced by a column of air in an unlimited space. If a U-tube with unequal
arms be held in a cone of light passing from a lens to a screen, ether
vapor poured into the long arm will, by projection on the screen, be seen
to flow out of the short arm; if it be poured into the short arm, it will
simply flow over when that arm is full, not rising to the level of the top
of the long arm.
52. Specific gravity is the ratio between the weight of a given
substance and that of an equal volume of a standard 'substance, usually
water. It is usually found by an application of Archimedes' principle.
A body weighs in air W^ in water IV 2 . By this principle the weight of
an equal volume of water is W^ W z and
(58) s= T **;..
In the C. G. S. system the specific gravity of a substance is numerically
equal to its density.
The C. G. S. densities of some substances are as follows :
Copper ................................... 8.9
Water at 4 C 1
Camphor 1
Pine wood 0.5
Potassium 87
Aluminum.., 2.56
Mercury ................................... 13.6
Gold ....................................... 19.3
Platinum ................................. 21.5
Density varies with temperature, and in case of solids with the treat-
ment to which subjected (hammering, etc.).
For historical discussion of laws of hydrostatics see Mach, Science of
Mechanics, p. 86.
80 PROPERTIES OF MATTER.
QUESTIONS AND PROBLEMS.
41. Show that the total fluid pressure on the base of a vessel depends solely
on its area and the height of the free surface above it, not at all on the shape of
the vessel.
42. Explain the hydraulic press.
43. A cubical block of wood sinks to 0.6 of its height in alcohol (s. g. 0.82).
What is its specific gravity?
44. Given, a meter rod balanced on a knife-edge, a beaker of water, and a
stone. Show how to measure the specific gravity of a body.
GASES.
53. The capacity of indefinite expansion possessed by gases indicates
that they possess much more molecular freedom than liquids, and for this
reason many of their properties are simpler than those of liquids; conse-
quently they will be first considered.
Since gases are fluids, Archimedes' and Pascal's principles apply to
them without modification.
Weight of Gases. Before the time of Galileo the use of syringes
and pumps and many of the phenomena of suction were known, but their
mode of operation was not understood. The only explanation advanced
was that nature abhors a vacuum. Galileo first began to consider this
question when he was told that a certain pump with a long suction pipe
was unable to raise water to a height of more than thirty-three feet. He
had previously shown that air has weight, by expelling some air from a
large bottle by heating it and noting the loss of weight, and he seems to
have suspected some connection between the weight of the air and the
height to which water could be pumped.
Torricelli verified this conclusion. One of his students, Viviani, in
1643, filled a long glass tube, sealed at one end, with mercury, and
inverted it in a vessel of mercury. As Torricelli had expected, the mer-
cury column did not entirely drop, but remained at a height of about
one-fourteenth that of the column of water that a pump can raise. Thus
it was shown that the unknown cause could raise equal weights of different
liquids, and Torricelli inferred that this cause must be the counterbal-
ancing weight of the atmosphere.
In 1646 the news of Torricelli' s discovery came to Pascal, in France.
Pascal likewise explained the result as being due to the weight of the
atmosphere, and proved this conclusion in 1648 by sending a barometer
to the top of the Puy de Dome. As he expected, the height of the mer-
cury was less, on account of the reduced weight of the atmosphere above
the open surface of mercury.
In 1755 Black discovered carbon dioxide, and eleven years later
Cavendish discovered hydrogen. Since then many other gaseous sub-
stances have become known. Their relative weights or densities may be
determined by a refinement of Galileo's method. Large globes may be
weighed when exhausted by an air pump, and again when filled by the
gas in question. The movements of one gas in another may be shown
visually by passing a divergent cone of light through the region occupied
by them; on a screen beyond the phenomena will be projected, being
made visible by the different refractive powers of the different gases.
Ether vapor will be seen to sink in air, hydrogen to rise. (This is known
as Topler's schlieren method.)
BOYLE'S LAW CHARLES' LAW.
31
The density of a gas depends of course upon the pressure and the
temperature to which it is subjected. The standard conditions are taken
to be an atmospheric pressure of 76 cm. of mercury and C. tempera-
ture. Under these conditions the densities of some gases are :
Air .................................... 0.001293
Argon ................................ 0.00170
Carbon dioxide .................. 0.00195
Chlorine ................... . ......... 0.00317
Helium 0.00021
Hydrogen 0.0000898
Nitrogen 0.001254
Oxygen 0.001429
54. Boyle's Law. About 1662 Robert Boyle became interested
in the experiments of Pascal, which showed that ' ' the greater the weight
is that leans upon the air, the more forcible is its endeavor of dilatation. ' '
Boyle undertook a careful investigation of the relation between volume
and pressure by compressing air in the short closed branch of a U-tube
by a column of mercury in the long open branch. His investigations
went from pressure of about one inch of mercury to four atmospheres, and
he found that the volume varied very nearly inversely as the pressure, or
(59)
pv = constant.
Boyle also noted that the volume and pressure were increased by a
rise of temperature, but he did not study this point closely.
In 1676 Mariotte, of France, published an account of similar experi-
ments, but credit belongs by priority to Boyle.
Boyle's law is a consequence of the perfect elasticity of gases. Elas-
ticity is numerically measured by the ratio of stress per unit area to the
strain per unit volume; if it is perfect, this ratio should be equal to the
applied force per unit area. If a gas is compressed by an increase of
applied force from/> to/ x> and the volume thereby reduced from v to v lt
we have
(60)
stress / p v , r
= = /_, the applied force.
offoi-r* *?t ^ TI ** x A L
strain
which gives
(61)
i v i = P v ~ constant
55. Charles' Law. Toward the end of the eighteenth century it
was discovered independently by Charles and by Gay-Lussac that when
the temperature of a gas changes it increases in volume or in pressure or
both conjointly by the same fraction of its volume at C. for each degree
change of temperature above zero C. The value of a, the coefficient of
expansion or of compressibility, is J/ 73 or 0.00367. The change is an
arithmetical, not a geometrical progression, so that at a definite negative
temperature of 273 the product pv would vanish, provided the gas
retained its gaseous properties ; as a matter of fact, all gases become
liquids before that point is reached. We cannot conceive that under any
circumstances the volume could become zero ; but the pressure may.
We may write
(62)
pv =p.v. (1 j- af) = ap v (273 ff)=
where T is the temperature measured from the absolute zero of 273 C. ,
and R is the so-called gas constant, which is different for different gases.
32 PROPERTIES OF MATTER.
If p and v are the pressure and the volume of a unit mass of the gas
considered, the value of the product will be different if a different mass m
is employed. In general,
(63) pv = mRT
where R is the constant deduced from observations on unit mass.
56. Deviations. It is found, as in the case of many physical laws,
that Boyle's and Charles' laws are only approximately true, and that wide
deviations may occur at high pressures and temperatures. Regnault;
Natterer, and many others have investigated these deviations, but the latest
and most reliable measurements are due to Amagat, now of the Ecole
Polytechnique of Paris. His first experiments were made by an extension
of Boyle's method, the long branch of his U-tube being a steel tube 330
meters long passing down the shaft of a coal mine at St. Etienne. Later
he used an improved method by which he could attain pressures of about
3000 atmospheres. The total hydrostatic pressure due to a column of
mercury acting on a large piston A was transmitted through the piston
rod to a smaller piston a which compressed the gas. This scheme is easily
seen to be an inversion of the hydraulic press. Instead of multiplying
the pressure exerted on a small area by applying it to a larger area by
means of a liquid piston, and thus securing a larger total pressure, the
total pressure secured in such a manner is by means of a solid piston con-
centrated on a small area. If A be the area of the large piston, a that of
the small piston, h the height of the mercury column which produces the
pressure, and p the density of mercury,
(64) P =
is the pressure applied to the gas in the small cylinder.
Amagat lubricated the pistons with very viscous liquids, such as castor
oil and molasses, which effectively prevented any leakage of mercury or
gas around the pistons.
Some of his results for different gases are summarized below. The
product pv at atmospheric pressure and C. is taken as unity; if Boyle's
law were exactly obeyed, this product should remain constant with con-
stant temperature. The curves drawn with the product pv as ordinates
and/ as abscissae, show deviations from the law. Each curve should be
a horizontal line if the law were correct.
Value ofpv (p v =1).
Pressur- in Atmos. Oxygen. Hydrogen. Nitrogen. Carbon Dioxide.
1.0000
0.1050
0.2020
0.3850
0.5595
8905
1.6560
1
1 0000
1 0000
1 0000
50
100
9055
1 0690
9910
200
9140
1 1380
1 0390
300
9625
1 2090
1 1360
500
1 1570
1 QfSfir;
i QUOO
1000
1 7360
1 79^0
9 0700
2000
2 8160
2 38QO
Q 3270
2800...
3 6176
2 8686
4 2700
In the case of hydrogen, the value of pv uniformly increases as the
pressure rises, or the gas is not so compressible as required by Boyle's
law. In the case of oxygen, nitrogen, and the heavier gases, such as
carbon dioxide and ethylene, pv diminishes to a minimum and then rapidly
increases with increasing pressure, being at first more then less com-
DEVIATIONS D ALTON'S LAW. 33
pressible than the law would require. This suggests that for all the gases
except hydrogen there is up to a certain point a slight attraction between
the molecules which helps condensation; then a crowding together which
resists it. As the crowding increases until the point of liquefaction is
approached, the deviation from Boyle's law becomes very great. The
easily-liquefied gases, such as carbon and sulphur dioxide, chlorine,
ethylene, etc., show marked deviations. In the case of hydrogen there is
always the internal resistance to compression with little or no molecular
attraction. In a gas which strictly obeys Boyle's law there could be no
molecular attraction or repulsion; the pressure would depend solely on
the number of molecules within a given volume, without regard to any
forces between them.
It is seen that Boyle's law is more nearly true at high temperatures;
it is believed to hold very closely for low pressures, but it has been found
hard to verify this conclusion because of the difficulties of measuring
small pressures accurately.
It would seem natural, as indicated above, to explain deviations from
Boyle's law as being due to two causes, (1) intermolecular forces of
attraction, helping the applied pressure to condense the gas; (2) the
actual volume occupied by the molecules, or the volume they would fill
were there no free space between them. Attempts have been made to
generalize Boyle's law by taking account of these factors. One of the
most successful is that of van der Waals, who in 1873 proposed the
formula
(65) ^ + ^) (v-b} = mRT,
which very closely conforms to observation for all gases and vapors.
The term a / v 2 takes account of the effect of molecular attractions; the
term b expresses the molecular volume, or the part of space actually
occupied by the molecules. For air a = 0.0037; = 0.0026. For
carbon dioxide a = 0.01 15; *=-0.003.^U ^* /**, W. * M> *f I *+}
57. Dalton's Law. In a mixture of several gases having no chem-
ical action on each other the total pressure is equal to the sum of the
partial pressures produced by each separately.
(66)
(For original papers of Boyle and Amagat see Barus, the Laws of Gases; also,
for results, Tait, Properties of Matter.)
THE ATMOSPHERE.
58. The pressure at any point in the earth's atmosphere is due mainly
to the weight per unit area of the air above that point, although the
pressure may be somewhat modified by atmospheric currents. The
density diminishes very rapidly, therefore, in going upward.
The composition of the atmosphere at sea level is approximately as
follows: Nitrogen, .78; oxygen, .21; argon, .01; varying traces of car-
bon dioxide, ammonia, and water vapor; very small traces of the gases
neon and krypton. These properties are subject to slight local variations.
Since oxygen is somewhat heavier than nitrogen, there is a slight
tendency for it to accumulate more at the bottom than at the top of the
34 PROPERTIES OF MATTER.
atmosphere. In ascending, therefore, the proportion of oxygen becomes
slightly less.
At the sea level the atmospheric pressure is on the average equal to
the weight of a column of mercury 76 cm. high and of 1 square cm. cross
section. This is equivalent to a pressure of about 1,013,000 dynes per
square cm., or 14.7 pounds weight per square inch.
The law of diminution of pressure in ascending above sea level is
readily calculated by -the application of Boyle's law. The calculated
percentages, at several different heights, of the principal constituents, and
the pressure of the mixture, are given below:
Height in Relative Percentage.
Meters. Oxygen. Nitrogen. Total Pressure.
21 78.96 760 mm.
1,000 20.71 79.25 670.7
10,000 18.35 81.63 218
20,000 15.92 84.07 62.8
40,000 11.54 88.46 5.2
60,000 8.89 91.11 0.4
The quantity of oxygen becomes not only absolutely but relatively
less in ascending.
Theoretically there is no limit to the height of the atmosphere ; but at
an infinite distance the density would be infinitely small. At a height of
several hundred miles the density is sufficient to raise meteors to incan-
descence by the frictional resistance offered to their very rapid motion.
The greatest altitude was probably reached by Glaisher in a balloon ascent
in 1862. He went to a height of nearly seven miles, his barometer falling
to a height of less than nine inches.
The usual method of indicating and measuring the atmospheric pres-
sure is by means of the barometer, which was devised by Torricelli in
1643. Shortly afterward Otto von Guericke invented a mechanical air
pump and showed the atmospheric pressure by the classical experiment of
the Magdeburg hemispheres. These hollow metallic hemispheres, fitting
closely at their edges, were held together with great force when the air
between them was exhausted.
The distinction between the weight and the elastic pressure of a gas
must be carefully borne in mind. Any portion of air at sea level is com-
pressed by the weight of the atmosphere above and exerts an equal and
opposite elastic force of one atmosphere, whether the actual mass of the
portion be great or small. If we introduce in a barometer tube enough
air to depress the mercury to one-half its former height, it by no mea'ns
follows that we have introduced seven and a third pounds of air.
59. Barometers. The simplest type of barometer is the siphon, in
which the pressure of the atmosphere is equal to the weight of the column
of mercury between the levels h^ and h 2 .
The Fortin type is ordinarily used in scientific work. The bottom of
the mercury reservoir is of leather, and may be raised or lowered by a
screw, in order to keep the surface at a constant level.
In the aneroid barometer, changes in pressure are registered by the
motion of an index connected with the elastic corrugated metal cover of a
partially exhausted box.
Various forms of self-registering barometers are used in meteorological
work.
OF THE
X^ r* OF
BAROMETERS. 35
60. In order that observations made at different times and places may
be comparable, they must be reduced to standard conditions, and expressed
in dynes. The standard condition of the barometer is a height of 76 cm.
of mercury at a temperature of C., at sea level, in latitude 45. On
account of the variations of g, the same height will not indicate the same
pressure in dynes in any other latitude.
Four corrections have to be applied to a barometric reading, in order
to determine the absolute pressure in dynes, on account of
(#) Capillarity. On account of surface tension, the mercury is de-
pressed in a glass tube. This depression may be quite large in small
tubes, but for a tube of 1 cm. diameter it is only about 0.002, or neg-
ligibly small.
(^) Change of the density of mercury with the temperature, and
(c] Change of the length of scale with the temperature. These two
corrections may be considered together. If r is the scale reading, k Q
the reduced height, and c the cubical coefficient of expansion of mercury,
If the scale has a linear coefficient of expansion .
An infinite number of strokes would thus be necessary to secure a perfect
vacuum. Practically a pressure of a few millimeters of mercury is the
lowest usually attainable, as the valves cease to act when the pressure
becomes too small to move them.
65. Mercury Pumps are much more efficient. There are two
typical forms, the Geissler and the Sprengel. The Topler-Hagen is an
improved form of the Geissler pump, working without any stop-cocks.
When the reservoir R is raised, the mercury rises toward the bulb B,
cuts off communication with the space ,S which is to be exhausted, and
drives the imprisoned air out through the capillary fall tube T. On lower-
ing the reservoir a Torricellean vacuum is left behind the mercury, air
rushes in from S, and the operation is repeated, one bulb full of air being
expelled at each stroke. There is no definite limit to the possible
exhaustion, except the almost unavoidable presence of small traces of
water-vapor, air, and other gases which become condensed on the glass
walls of the pump, and gradually escape into the vacuum.
38 PROPERTIES OF MATTER.
The Sprengel pump acts on an entirely different principle. If a jet of
mercury moves past the mouth of a tube communicating- with S, and if
the point P be at a height greater than that of the barometric column above
the mercury surface at the bottom of the fall tube, it is evident that the
hydrostatic pressure in the mercury at P will be zero. The weight of the
atmosphere is unable to sustain the weight of the mercury column below
P, and there is a tendency for the column to break, as the lower parts are
moving faster than the upper parts, which have just begun to fall. Air is
consequently sucked in laterally and carried down with the mercury. The
same effects would be produced if the fall tube were much shorter, and
the mercury driven through the nozzle under the action of a pressure
greater than that of the atmosphere. This is the case with the Bunsen jet
or filter pump, which is short, but operated by water under high pressure.
The action of such pumps may seem at first sight paradoxical; but we
must remember that while the jet is produced by high pressures, this
pressure does not exist in the freely falling jet itself. As a matter of fact,
each part of the jet, in addition to the momentum which enables it to move
against atmospheric pressure, is subject to the acceleration of gravity, so
that the lower parts tend to separate from the upper parts, thus creating a
tension, or negative pressure. This is a special example of the general
law of reduction of pressure in a moving fluid, to be considered later.
The Sprengel pump is even more efficient than the Geissler pump and its
modifications.
GASES IN MOTION.
66. Unbalanced forces produce motion in matter ; conversely, from
motion of matter we may infer the existence of unbalanced forces. When-
ever we see fluids in motion Pascal' s principle cannot hold ; the pressures
on all sides of a moving element cannot be the same.
67- Efflux of Gases. Consider the flow of a fluid through a small
opening a in a very thin wall, the pressures on the two sides being / x and
p^ and the work done in driving through this opening a small cylinder
of the fluid of cross section a and length /. This work is evidently
(70) w=Fl=(p-p^al=(
where v is the volume of the cylinder.
If all of this work is expended in producing kinetic energy,
(PP*}V = \ms* or p-p 2 = &S*
(71) s= |2(A-A)
y '
In the case of liquids this expression (known as Torricelli's law) is a very
close approximation, as there is only a slight loss of energy due to friction,
if the wall is thin. In the case of gases, there is not only their frictional
resistance, but also loss of energy in expanding against the exter-
nal pressure. This changes the numerical value of the expression but
still leaves the speed of efflux of different gases inversely proportional
to the square root of the density of the gas. This principle has been used
by Bunsen in comparing the densities of gases.
To the extent that the above formula is true we see that since p varies
as / the velocity of efflux of a gas is the same at all pressures. For
hydrogen escaping into a vacuum under a pressure of one atmosphere,
TRANSPIRATION CHANGES OF PRESSURE DUE TO MOTION. 39
,$=0.000089 and /=!, 013,000 dynes; consequently s is about 1500 meters
per second; the speed of air is a little more than one fourth of this.
68. Transpiration is the name applied to the passage of gases
under a difference of pressure through long capillary tubes. The gas un-
usually condenses upon or ' ' wets ' ' the walls of the tube, hence the flow
is impeded by viscosity, or the friction of the gas upon itself. The rate
of flow no longer obeys any simple law. Below are given the relative
rates of transpiration of some gases.
CO 2 1,376
O 1
N 1,150
NH 3 2
H ............................................ 2,26
In general the lighter gases transpire more rapidly, but there are excep-
tions, as seen by comparing CO 2 and NH y
69. Changes of Pressure due to Motion. If there is in any
fluid a positive acceleration in a given direction, there must also be a tend-
ency for the parts going in that direction to separate from the parts
behind them, and to lag behind the parts in front, thus diminishing the
pressure between them; and conversely a retardation involves a crowding
together and consequent increase of pressure. This point will be made
clear by considering an analogous case. If a weight rests on a board, it
exerts a certain pressure on the latter; if the two are allowed to fall
freely while in contact there is no pressure whatever between them; the
weight simply moves continuously into the space formerly occupied by the
board, without pushing it; if the board is not allowed to fall freely, but is
attached to a cord moving with friction over a pulley, thus reducing its
acceleration to # t , there will be more or less pressure depending on the
relative acceleration, but it will not be equal to the total weight. Again,
imagine a heavy cylinder with a tightly-fitting piston moving in it without
friction. If the cylinder be held vertically so that the piston starts to fall out
of it, and the cylinder be allowed to fall itself an instant later, the piston
having started first, will at each instant be falling faster than the cylinder,
and will thus increase the volume and diminish the pressure of the air
contained in the latter.
In the case of incompressible fluids it is possible to calculate a fairly
exact relation between velocity and pressure, neglecting the effects of
friction, which are small except in the case of flow through small pipes,
such as capillary tubes, in which viscosity plays a large part. Consider
any small parallelepipedon of fluid of cross sections and length b with pres-
sure /, and p 2 acting on its two ends. If the element moves, these pres-
sures must be different, and work is done on the element by a force per unit
area equal to the difference of pressure. If we neglect losses by friction
all this work is transferred into kinetic energy (or conversely if the
momentum of the element is carrying it against a higher pressure); hence
(72)
From this equation we see that
(73) /,+ \ps r 2 =p
This equation shows that wherever the speed of a fluid is greatest the
hydrostatic pressure is least, or conversely; and the sum of the pressure at
a point and the kinetic energy of the unit volume embracing that point is
a constant. This law, known as Bernouillis law, is only qualitatively
true in the case of compressible fluids like gases.
40 PROPERTIES OF MATTER.
If a jet of fluid issues from a reservoir under atmospheric pressure
(=1,014,000 dynes)-
(74) A =/ - fas = when J= J^.
\ P
For this speed, the pressure within the jet is zero; for greater speeds, it is
negative or the parts will tend to fly asunder, causing suction laterally on
any surrounding fluid at rest.
There are many examples of this principle, both in liquids and gases.
The Sprengel pump is one illustration. If a jet of liquid or air be driven
from a tube through a funnel-shaped expansion, light objects may be
sucked up into the funnel. A light sphere tends to rest against an up-
ward jet of water. Air blown through a horizontal tube will suffer a re-
duction of pressure on expanding into a larger tube, and the reduction of
pressure will raise a liquid in a connected vertical tube T. If a jet of air be
blown against a metal plate near a flame the flame will be sucked toward
the plate. A downward jet of air will pick up a card. Winds in general
reduce barometric pressure. A condition for success in all such experi-
ments is that the speed of the moving fluid must be sufficiently great to
maintain itself against any obstructions, such as opposing atmospheric pres-
sure, and flow in steady stream lines. A jet of air or water fulfills these
conditions, the momentum of the fluid keeping up a steady, state of motion.
Other examples will be given in discussing the motion of liquids.
70. Diffusion. The cases of fluid motion so far considered are mass
motions, depending on differences of pressure on the two sides of any ele-
mentary mass considered; if any very small element could be frozen solid
the resulting motion would be the same. In the case of all fluids which
are capable of mixture, however, (excluding such cases as oil and water in
contact), we observe that there are relative motions not dependent upon
differences of pressure, and which appears to be molecular, not mass,
effects.
Dalton showed that if a vessel filled with one gas be placed in commu-
nication with another vessel filled with a different gas, the two would in
time be found to be uniformly mixed, no matter how small the channel of
communication; this is true even if the lighter gas be placed above
the heavier. Such experiments illustrate the tendency of every gas to uni-
formly fill all the space open to it, without regard to whatever other gas
may be present. This process is called diffusion. The presence of
another gas will retard the rate at which a gas diffuses, but it in no way
interferes with the final equilibrium of the gas with itself. In a sense any gas
is a vacuum to any other gas. The attraction of gravitation is some check
on indefinite diffusion, as shown by the atmosphere. Heavier gases col-
lect below the lighter, as in the case of "fire damp" in mines and wells.
It follows that Pascal's principle must apply to each gas in a mixture sepa-
rately, and it can only be in equilibrium when its pressure is the same in
all directions, or its density the same at all points in a horizontal plane. If
we imagine the state of a rarefied gas, however, as composed of isolated
molecules with large spaces between them, it is difficult to picture to our-
selves anything like a uniform hydrostatic pressure in a gas such a pres-
sure as that between two surfaces everywhere in contact. The phenomena
of diffusion suggest very strongly that motion has something to do with
what we call the pressure of a gas, and we shall presently find that all the
phenomena of gases can be explained very simply and very satisfactorily as
DIFFUSION - ABSORPTION OCCLUSION. 41
a result of motion alone, without any sort of contact or pressure between
neighboring molecules except that resulting from momentary impacts.
This is called the kinetic theory of gases.
71. Diffusion Through Porous Walls Atmolysis. If two
vessels containing different gases be separated by a porous partition, say of
unglazed earthenware, it will be found after a time, even if both gases be
originally at the same pressure, that they will become uniformly mixed.
During the process of diffusion the equilibrium of pressure will be for a
time disturbed, the lighter gas moving more rapidly than the heavier.
This may be shown as follows: A glass tube is sealed into a porous cup
and the whole inverted so that the end of the glass tube is submerged in
water. Surround the cup by an atmosphere of hydrogen or coal gas, and
the pressure within the cup will rise, as shown by the escape of bubbles.
After a time renew the original surrounding atmosphere. The pressure
will now diminish, as shown by the rise of water in the tube. The coal
gas gets in at first more rapidly than the air can get out; afterward it gets
out more rapidly than the air can get in. With carbon dioxide the con-
verse is true. Graham showed that the rates of interdiffusion of two gases
are inversely as the square root of their densities. We may infer from
such experiments that any gas is always diffusing within itself, or that the
molecules are always in motion. This method enables us to sift one gas
from another (atmolysis).
We have here another illustration of Dalton's law. Each gas strives
to secure equilibrium for itself in all the available space, without regard to
any other gas that may be present
72. Absorption of Gases by Solids and Liquids. Boyle's law
and the phenomenon of diffusion indicate that the attractive forces between
the molecules of a gas or of different gases are exceedingly small. This
is apparently not the case with the attractive forces between gases and
solids or liquids. Water will absorb or dissolve more or less of any gas
in contact with it. Some gases are absorbed but slightly; for example,
oxygen and nitrogen; others are absorbed in large quantities; for example,
ammonia gas and hydrochloric acid gas. It is found that the quantity of
a gas absorbed by water or other fluids is directly proportional to the
pressure of the gas above the liquid surface. The statement of this fact is
known as Henry's law. It does not hold in the cases of very soluble
gases, such as ammonia. In the following table are given the volumes
of various gases absorbed under a pressure of one atmosphere by unit
volume of water at different temperatures:
t H N O C0 2
.0193 .012035 .04114 1.7967 1050
10 " 0.1607 .03250 1.1847 813
20 " 0.1403 .02838 0.9014 586
As we may infer from the above figures, dissolved gases may be driven
off by heat.
73. Occlusion. Some metals absorb some gases in large quanti-
ties. This phenomenon is sometimes called occlusion. Platinum will
absorb hydrogen so violently that if the platinum is already warm it may
be raised to incandescence; a spiral of platinum wire, for example, if
heated in a Bunsen flame may be allowed to cool below red heat when the
flame is extinguished, and if the gas be again turned on the platinum will
42 PROPERTIES OF MATTER.
absorb hydrogen from the gas and become so hot as to ignite the latter.
Palladium may absorb more than a thousand times its own volume of
hydrogen, which must in consequence be reduced to a density comparable
with that of water. The attractive force required to condense hydrogen
to this extent must be enormous. It is possible that the hydrogen, which
has many metalic properties, may form something like an alloy with these
metals. Iron will absorb carbon monoxide in considerable quantities.
The gases absorbed by both liquids and solids may be driven off by
heating or by placing them in a vacuum. On heating water, for example,
air bubbles are seen to form and rise throughout its entire mass; the same
effect will be observed under the receiver of an air pump. Platinum,
palladium, and iron will give up their absorbed gases at high temperatures,
or will transmit them if they are present on one side only. If a stream of
impure hydrogen be passed through a platinum or palladium tube at a high
temperature, pure hydrogen will pass through the walls of the tube. In
this way the purest hydrogen may be secured. Carbon monoxide will
readily pass through hot iron, and may thus contaminate the atmosphere
of a room by passing through the walls of a stove in which the combus-
tion is imperfect, resulting in the formation of this gas.
74:. Adsorption. Glass is apparently impervious to any known gas
or liquid; yet it is easily shown that when a piece of glass is heated, con-
siderable quantities of air, water vapor, and carbon dioxide are given off.
These gases seem to be strongly condensed on the surface of the glass;
they may be said to ' ' wet ' ' it, and the molecules cling together in much
the same way that liquid water clings to the glass that it wet's. This phe-
nomenon, which is called adsorption, shows itself in filling a barometer
tube; the tube must be repeatedly heated after being filled with mercury,
partly to drive off the air absorbed by the mercury, partly to remove the
condensed gases and vapors on the glass. In the vacuum tubes designed
to show the electric discharge through gases at low pressures it is almost
impossible to get rid of the water vapor adhering to the \valls of the tube.
Prolonged heating in the presence of metallic sodium or potassium seems
to be the only effective method of removing it. For the same reason
glass insulating stands for electrostatic experiments usually fail to insulate
until they are well dried by warming. It is difficult to test Boyle' s law at
very low pressures on account of the tendency of the gas to condense in
the containing vessel or for already condensed gas to pass off. This con-
densation is clearly a case of molecular attraction, for different solids differ
very widely in the extent to which they condense gases on their surface.
Finely divided platinum sponge will absorb much more hydrogen than
the same quantity of solid platinum. The amount of available surface
seems important, so that in such cases both absorption and adsorption
seem to be active. The same is true of other solids, such as charcoal,
which absorbs several hundred times its own volume of ammonia gas and
large quantities of all other gases; hence its value as a deodorizer. The
more porous or finely divided the charcoal is that is, the more surface it
exposes the more effective it is in this respect.
If a letter or design be firmly traced on a glass plate with a soft stick,
and if the glass be afterward breathed upon, the water vapor in the breath
will condense more freely on the untouched portions, bringing out the
design plainly. The scraping away of the condensed gases on the glass
seems to affect its power of condensing water vapor. If a coin be laid for
a time on a clean glass or metallic surface, and then removed, its image
VISCOSITY. 43
may be brought out in the same way. These images are called Moser* s
breath figures.
75. Viscosity. There is more or less molecular friction in gases,
or between gases and solids, which is called viscosity when it concerns the
friction between parts of the same substance. The viscosity of gases is
strikingly shown in the transpiration of the gas through a fine capillary
tube, on the walls of which the gas becomes adsorbed. Maxwell compared
the viscosities of different gases by observing the vibrations of a thin disc,
suspended as a torsion pendulum, over a similar disc placed under it.
The viscosity of the gas between the two discs acted as a friction brake
to bring the pendulum to rest ; the less viscous the gas, the longer the
vibrations persisted. Hydrogen is the least viscous of gases; air has about
twice as great a viscosity. It is to this property that the suspension in the
air of fine dust and smoke particles, etc., is due. The weight of these
particles diminishes as their volume, or as the cube of their linear dimen-
sions, while their surfaces diminish as the square of their linear dimensions,
or less rapidly. When the particles are very fine, therefore, friction against
the air becomes large as compared with their weight, and they fall very
slowly. The friction increases very rapidly with velocity; so that meteors
in the upper atmosphere, traveling thirty or forty miles a second, may
become incandescent, even in greatly rarefied air.
The resistance to the motion of a projectile through the air is due to
several causes; partly to the friction, partly to the mass of air dragged
along with the projectile, thus increasing its inertia, and partly to the
elastic resistance of the compressed air before it. On account of the dif-
ferences in the density of the air, and consequent differences in its refrac-
tive power, instantaneous photographs of projectiles may be made, which
show the compression wave in front and the eddies and wake behind, their
general appearance being strikingly similar to the water waves, eddies, and
wake around a rapidly moving ferry boat. Friction plays a considerable
part in such phenomena, causing a mass of the fluid to be dragged along
with the moving body.(^
The viscosity of liquids is diminished with rising temperatures ; mo-
lasses becomes very thin when it is heated. With gases the contrary is
true ; the viscosity increases with the temperature. The reason for this
we shall find suggested in the next section.
KINETIC THEORY OF MATTER.
76. It was once supposed that the pressure of gases is due to a
repulsion between their molecules. If such were the case, or even if it were
due to an elastic reaction against actual molecular contact (which sup-
position seems to be excluded by the unlimited expansiveness of gases), a
compressed gas would possess a large amount of potential energy which
would become kinetic if the gas were allowed to expand without doing
external work. Molecular kinetic energy, however, is heat; consequently,
a gas expanding into free space should be heated. As shown by the
experiments of Thomson and Joule, which will be discussed later, this is not
the case ; the temperature changes of a gas expanding into a vacuum are
vanishingly small.
Hooke, in 1678, Daniel Bernouilli, in 1738, and others at various
times, suggested that the pressure of a gas might be due to the impact of
its molecules. Clausius was the first to state the theory in precise terms.
(1857).
(1) See Boys, Nature, 1892.
44 PROPERTIES OF MATTER.
According to this theory all molecules are supposed to be in constant
motion. In gases and liquids they are free to travel in any direction, im-
peded only by frictional forces ; in solids they have only a small degree of
vibratory freedom about a mean position. The diffusion phenomena ob-
served in fluids are strong evidences in favor of this view ; and if the
theory that heat is molecular kinetic energy is correct, the molecules of
all kinds of matter above a temperature of absolute zero must be in motion
of some sort. Thus direct observation and agreement with another well-
supported theory both strengthen the validity of the kinetic theory of
matter.
In the case of gases there are probably frequent impacts between the
molecules as they move at random, the number diminishing as the
density diminishes. We have no reason to believe that all the molecules
of a given gas move with the same velocity ; it seems, on the contrary,
exceedingly probable that they are moving in all directions and with all
speeds between zero and a maximum depending on the temperature.
Furthermore, these speeds are probably constantly subjected to sudden
changes in amount and direction of motion by impacts against each other
and against the walls of the containing vessel. Let us consider whether
the pressure exerted on the walls can be explained as the effect of motion.
In a given gas all the molecules have the same mass m. If these
molecules behave like elastic spheres, and if one having a component
velocity u^ in the X direction strikes the wall of a cubical enclosure viith
sides of length /, it will rebound with the same velocity u t . The impul-
sive pressure on the wall during the time of contact / is/,, and by the
principle of change of momentum we know that
(75) . m \U T -(-,)] = 2mu T =/, L
If the molecule meets no other molecule, but rebounds freely back and
forth between opposite walls, collisions with the wall will occur at intervals
equal to 2/ / u y and the number of impacts per second will be n = u t / 2/.
A rapid succession of impacts is equivalent in its effects to a steady
pressure, if exerted on bodies having inertia, such as the walls of a vessel.
We might, for example, imagine a ballistic pendulum permanently and
steadily deflected by the rain of bullets from a rapid-fire gun. The
equivalent pressure is the time average of the impulsive forces, or
and if T is taken as one second,
mu.
(77)
I
The partial pressures given by the molecules moving with other velocities
can be expressed in the same way, and the total pressure p is
m
There is an ideal velocity u, called the velocity of mean square, which
is defined by the relation,
(79*) , #V = */ + V + * 3 a .'
Perhaps no molecule may have a velocity of this exact value, but if all the
KINETIC THEORY OF MATTER. 45
molecules did have this velocity the total pressure effect would be the
same as in the actual case.
This relation now gives
(80) ^mu*=p.
Since we are dealing with a column of unit cross section and of length
N
I, N j I is the total number of molecules per cubic cm., and-y- m = p, the
density of the gas, hence
(81) -^mu* = PU 2 =p.
Furthermore, the law of averages applied to large numbers would lead
us to infer that the velocities of mean square v and w in the Kand Z
directions, are each equal to u, and that the relation of these components
to the actual speed in the line of motion must be
(82) w a + z/ a + o/ a = 3 a = F 2 ,
from which
Multiplying by v, the volume of the gas,
(83) pv = i MV* = constant.
This evidently corresponds to Boyle's law.
The mechanical theory of heat suggests the probability that the
temperature of a mass of gas or other substance is proportional to the
mean kinetic energy of its molecules. F 2 would then be proportional to
the absolute temperature, and the product pv, being also proportional to
F 2 , would likewise be proportional to the absolute temperature, or
(84) PV = ^MV* = MRT.
The kinetic theory is thus a satisfactory explanation of Charles' law.
If the average molecular kinetic energy of two different gases at the
same temperature were not the same, we should expect their temperature
to change if they were mixed, as a result of the equalizing effect of
impacts. Such a change is not observed, hence
(85) \m T V^ = \m 2 F 2 *.
If we consider the same two gases separately at the same pressure
(86) P = \N, m ^ V* = $N 2 m 2 V;.
Comparing the last two equations,
(87) N^ = N^
or the number of molecules per unit volume of all gases at the same
pressure and temperature is the same. This is Avogadro' s law, first deduced
by an entirely different method from chemical considerations.
A gas when rapidly compressed becomes heated, because work is
expended against the impacts of its molecules, increasing their kinetic
46 PROPERTIES OF MATTER.
energy; when they do work against pressure in expanding they lose
kinetic energy and are cooled. If they expand against a vacuum, and if
there are no intermolecular forces, no work is done and there will be no
change of temperature. It will be found that the gases investigated by
Thomson and Joule only approximately fulfilled these conditions, indicat-
ing that there are slight molecular forces, and that there is n.o' perfect gas,
or one which perfectly fulfills the gaseous laws.
Dalton's law is evidently in accord with the kinetic theory.
We may deduce the velocity of mean square for different gases from
the relation
(88)
From this we may calculate the velocity of mean square for hydrogen at
C., and find it to be about 1,860 meters per second; that of oxygen is
465. The mean-square velocity of different gases varies inversely as the
square root of the density, as in the cases of efflux and diffusion, but in
each case the numerical value of the velocity is different. In the case of
efflux there is a single definite mass velocity; in the other case the
molecular velocities may range from zero to high values; but we can at
least say that on the average the velocity is inversely proportional to the
square root of the density of the gas.
Many phenomena connected with liquids and solids make it seem
probable that in them likewise there is constant molecular motion, but on
account of the proximity of the molecules the molecular forces and
constraints involved make the problem too difficult to be treated mathe-
matically.
It is probable that the kinetic theory of gases can never be directly
verified. The fundamental assumption that molecules are spherical and
perfectly elastic bodies is almost certainly false. The hypothetical velocity
of mean square is one which perhaps not a single molecule has. Neverthe-
less, the kinetic theory fulfills all the purposes of a useful theory. It
accounts for gaseous pressure; for diffusion of gases and liquids; for the
evaporation of liquids, which is the result of isolated molecules breaking
through the surface on account of their momentum ; and for the fact that
all these effects are increased by rise of temperature. It also throws light on
the fact that the viscosity of gases increases with rise of temperature, while
that of liquids diminishes. In gases the apparent frictional effect cannot
be the result of actual contacts. We must assume that the molecules in
one stratum are constantly passing into adjacent strata and conversely. If
there is relative motion of the strata, the result will be that the molecules
passing from a stratum at rest into one in motion will slightly retard the
motion of the latter, as the moving molecules must impart some momentum
to those relatively at rest; those passing from the stratum which is in motion
will in a similar manner set the other stratum in motion. The consequence
is a dragging effect, just as in the imaginary case of two boats moving with
different velocities parallel to each other. If heavy objects be constantly
thrown backward and forward between them the exchange of/ momentum
would finally equalize their velocities. In liquids, on the contrary, the
friction is due to actual contacts and attractions; the increased velocities
due to rise in temperature tend to break down molecular constraints and
increase freedom of motion.
All these facts give a high degree of probability to the kinetic theory,
HYDROSTATICS HYDRODYNAMICS. 47
although its exact details can be understood but vaguely. We must never
lose sight of the fact, however, that it is only a theory, and that it may in
time be replaced by a more satisfactory one; that is to say, one which will
closely associate and explain a greater number of phenomena than those
which have led us to the kinetic theory as a comprehensive description.
SPECIAL PROPERTIES OF LIQUIDS.
77. Those fluids which have a nearly constant volume, very slightly
affected by pressure or heat, and which may in consequence have a free
surface, are called liquids.
The mechanics of liquids may be divided into two branches,
Hydrostatics, which deals with liquids at rest; Hydrodynamics, which
deals with liquids in motion.
78. Hydrostatics. The fundamental laws of hydrostatics are those
of Pascal and Archimedes, which apply to all fluids, whether gaseous or
liquid. These have already been discussed.
Equilibrium. An immersed body is in equilibrium when the center of
gravity of the body lies vertically below the center of buoyancy (the
center of gravity of the displaced liquid). Equilibrium may also be
stable in some cases when the center of gravity is vertically above
the center of buoyancy, as in a floating board. If on slightly displacing
the body so that a vertical line through the center of buoyancy cuts the
vertical through the original center of buoyancy and the center of
gravity, the point of intersection lies above the center of gravity, equi-
librium is stable; if they coincide, neutral; if the point of intersection is
below the center of gravity, unstable. The point of intersection is called
the metacenter, and in ships it is important that it should be as high as
possible, to secure stability.
The surface of the liquid at rest must always be normal to the result-
ant of the forces acting on it, or level when gravity alone acts. Inequali-
ties of atmospheric pressure may produce perceptible differences of level
between different parts of seas and lakes. High water or low water may
be thus maintained by winds. If a cylindrical vessel containing a liquid be
rotated at a uniform rate about its vertical axis, the surface of the liquid
will assume the shape of a paraboloid of revolution, owing to centrifugal
action.
79. Hydrodynamics. Since a liquid is practically incompressible,
the quantity flowing through every cross-section of a stream must be the
same, and the speed of a liquid at a given point must be inversely as the
cross-section of the stream at that point.
If a liquid flows from an aperture, the pressure producing flow, if
gravity alone acts, is p = pgh, h being the height of liquid above the orifice,
or the "head."
80. Torriceltf s Law. As shown in discussing efflux of gases, if
the work done in driving a mass of fluid through an orifice is entirely
converted into kinetic energy,
(89) s* = 2pjt> = Zgh.
As this expression does not involve density, it follows that all liquids,
from ether to mercury, flow with the same speed under the same head.
This is not the case with gases.
48 PROPERTIES OF MATTER.
81. Jets. If we neglect the effect of friction and the resistance of
the air, a jet of liquid, like a projectile, will fall in a parabolic path.
Owing to lateral flow, a jet does not have a cylindrical form, but con-
tracts to a minimum cross-section just above the orifice. This is called
the vena contracta. The volume of liquid flowing out is equal to the
area of vena contracta multiplied by the speed. Experiment shows that
for an orifice of area a in a thin wall V= .62as per second. The volume
of flow is altered by mouth-pieces, or ajutages. If these project inward,
flow is diminished; if outward, it may be increased. If cylindrical and wet
by the liquid, there is a suction, which tends to enlarge the vena contracta,
causing a flow of about . Sas. With a conical ajutage the flow may be . 9as.
82. Flow in Pipes. If a liquid flows in a horizontal pipe under a
given head, the velocity in the pipe will not correspond to that indicated
by the head, owing to the fact that some energy is expanded in overcom-
ing friction. The pressure indicated by vertical manometers called piezo-
meters, communicating with the pipe, gradually diminishes in going
toward the end of the pipe, vanishing at the point of exit. The actual
speed of the fluid is that due to a certain head h less than H. The pres-
sure at any point may be ascribed to an imaginary head h'. At the point
where the liquid enters the pipe h + h' = H. The pressure head at any
point is modified by the resistance of the entire pipe beyond that point.
If the pipe is of variable cross-section, the velocity at any point is inversely
as the cross-section, and piezometers show that the pressure head is great-
est where the velocity head is least i. e. , as shown in the case of gases,
the pressure is greatest when the velocity is least. As expressed by
Bernouilli's law.
/ + \ P s 2 = constant,
or as it may also be stated, the sum of the potential and the kinetic
energies of a unit volume remains constant.
The speed of a liquid is impeded by viscosity. A stream flows most
slowly near the sides and bottom of its channel.
Flow through capillary tubes is greatly affected by viscosity. Poiseuille
found that the rate of flow is proportional directly to the fourth power of
the radius and inversely to the length of the tube.
83. Practical Application. A moving stream by reason of its
momentum can if checked produce an instantaneous pressure greatly in
excess of the actual hydrostatic pressure. This is utilized in the case of
the water ram, by which water may be raised higher than its source by
the momentum of a larger mass of water. The pressure of the stream
R closes a valve v, and the momentum opens w and compresses air in L.
This closes w; v again opens when the flow ceases while the compressed
air raises the water above N. The kinetic energy of moving water is
transformed into work by means of various forms of water wheels, such
as the overshot, undershot, and turbine (illustrated by Barker's mill).
The more nearly the water loses its entire velocity, the more efficient the
wheel.
84. Vortices. In flowing over obstructions or around corners,
water acquires a rotational motion, producing vortices or whirlpools.
Similar phenomena in gases are illustrated by smoke rings, and also, on a
larger scale, by cyclones. It is by friction that such vortices are
produced and brought to rest. Helmholtz has shown that in a "perfect"
COMPRESSIBILITY. 49
fluid that is to say, one having no viscosity it would be impossible to
originate vortices by any means known to us, or, assuming such vortices
to exist, it would be impossible for us to destroy them. These properties
suggested to Lord Kelvin the vortex-atom theory of matter.
MOLECULAR FORCES IN LIQUIDS.
85. Compressibility. Liquids are all slightly compressible, by an
amount which is difficult to measure. Lord Bacon tried to determine the
compressibility of water by compressing it in a closed hollow leaden
sphere; the Florentine academicians (1692) repeated the experiment with
a gilded silver sphere; in each case the water escaped through invisible
pores of the metal. Canton, about 1762, was able to prove that water is
compressible. Oersted, about 1822, devised an improved form of
piezometer, or pressure measurer, which gave more reliable results. The
thermometer-shaped vessel V containing the water was immersed in a
larger vessel also containing water, to which the pressure was applied.
The glass walls of Fwere thus subjected to equal pressures without and
within, while the change of volume of its contents could be determined
by the motion of a pellet of mercury in the capillary neck. If the
volume of the bulb is V, that of each division of the stem v, and the
water is compressed from the mih to the nth division by a pressure of P
atmospheres, the reduction of unit volume by one atmosphere pressure,
or the compressibility, is
(m-n)v
~ P(V+mvy
The volume of V^ is, however, on the whole diminished by the pressure
which thins its walls (as may readily be seen by imagining the vessel a
part of a solid block of glass submitted to uniform external pressure).
This diminishes the apparent contraction of the water, so the contraction
c' of the vessel per unit volume for each atmosphere must be added to c.
Where c' c there would be no apparent change of volume in the water.
The compressibility of most liquids increases with the temperature.
That of water, however, has a minimum about 61. Below are some
values of c.
Water Ether
0000503 00011 Mercury 0000029
16 450 14 16
61 389 100 56 Liquid CO 2 00590
77.4 398
99.2 409
Salts dissolved in water usually diminish its compressibility, as shown
below for sodium chloride solution :
Per Cent. NaCl c
5 .0000455
10 397
20 306
25 258
Water is about 25 times as compressible as copper, 40 times more than
iron, 80 times more than nickel. Compressibility diminishes as pressure
increases, indicating that there is a definite limit to the process.
Liquids, from which all air has been removed by boiling, can support
a tension, stretching without rupture. (Berthelot). Pure water will resist
50 PROPERTIES OF MATTER.
a tension of 50 atmospheres weight ; sugar solution, nearly 100 atmos-
pheres. The sugar seems to increase cohesiveness.
86. Cohesion and Adhesion are the names applied to the forces
which bind together the molecules of liquids and solids. They are
sensible only through very small distances. The adhesive properties
of glue, gilding, silvering, etc. , depend on intimate contact. Similar
forces exist between liquids and solids and within liquids. Clean glass
is wet .by water ; that is, the water clings to it with such force that it
cannot be directly removed, this adhesion being of course much greater
than the cohesion of water. In mercury the cohesion is greater, but it
still requires considerable force to detach a glass plate from mercury,
which does not wet it ; much more force to detach a copper plate, which
the mercury does wet. (Amalgamation).
87. Yiscosity. Although all liquids are mobile, the rates of flow are
very different, owing to internal forces, frictional or otherwise. If we
imagine two parallel surfaces in a liquid moving with relative velocity v,
the viscous resistance R offered to the relative motion will be found to be-
proportional to the velocity v, the area of the surfaces s, and the coefficient
of vicosity /, and inversely proportional to the distance d, between the
(9.) *or/
The viscous resistance offered by parallel layers of liquid to relative
motion may be compared with the resistance offered by a book to a tan-
gential force tending to slide its leaves over each other.
For glycerine, /= . 00238; for olive oil, .001 ; for water, .0000137.
Salt solutions are usually more viscous than water.
A large part of the resistance to the motion of a ship is due to the
viscous resistance between the stationary water and the water film on her
hull. The efficiency of lubricants depends on the existence of films of the
lubricant between and wetting the moving solid parts of machinery ;
the viscosity between the films of oil is less than the friction between the
dry solids. Viscosity brings winds and waves to rest. Rise of tempera-
ture decreases viscosity, that is, increases molecular mobility.
Small particles will remain suspended in liquids for some time, owing
to viscous resistance to fall. Enormous quantities of soil are thus trans-
ported into the ocean. The Mississippi deposits in its delta every year
a mass of silt about one mile square and 240 feet deep, besides a much
larger quantity carried by flotation or pushing.
88. Surface Tension. The unbalanced attractions acting on the
molecules on the surface of a mass of liquid pull them toward the center
and toward each other. The effect is as though the liquid were contained
in a skin or elastic membrane, which takes the shape having the smallest
surface a sphere. This is illustrated in water drops, mercury drops,
melted glass and sealing-wax, and the manufacture of shot. The perfect
sphericity of raindrops is proved by the circular shape of rainbows. The
action of gravity ordinarily masks this result, but Plateau overcame this
difficulty by suspending drops of oil in a mixture of water and alcohol
having the same density as the oil, which then assumed a spherical shape.
The existence of surface films may be shown by sprinkling lycopodium
powder over clean water. On dipping a slightly greasy glass rod in the
SURFACE TENSION. 51
water, the film is weakened and retreats with the powder, leaving a clear
space around the rod. Heat will cause the same result. Alcohol and
ether likewise weaken the tension, causing the water to retreat, like a
stretched sheet of rubber weakened at a given point. Pieces of camphor
will dart along the surface of pure water in all directions, owing to local
variations of tension due to the dissolved camphor.
Pure water has a stronger surface tension than contaminated water,
but separate films and bubbles cannot be formed from it. A vertical film
must necessarily have a varying tension at different points, while that of
water is constant; consequently equilibrium is impossible. In order to
form films and bubbles heterogeneous mixtures, such as solutions of soap
and glycerine in water, must be used. In these the tension can adjust
itself to the necessary conditions. Such a solution has a great superficial
tenacity as distinguished from tension, and with it soap bubbles may be
blown and films formed on frames of various shapes, the films always
taking the minimum possible surface under the conditions. If a loop of
silk is placed on such a film, which is then broken inside the loop, the
latter will at once become an exact circle. These films will sustain con-
siderable weight.
Surface tension is defined as the force acting across unit length in a
surface, or by one-half this quantity in a film, since it has two surfaces.
The contraction of liquid films is another example of potential energy
tending. to a minimum.
The equilibrium position of three surfaces in contact is determined by
the relative surface tensions between each pair of fluids. Consider a
drop of liquid on the surface of another liquid in air. Three forces,
x T 2 , 2 Ty 3 T It act on any element on the common boundary line. These
forces will be in equilibrium when the angles between them are such that
the resultant of any pair is equal and opposite to the other force. If one
force exceeds the sum of the other two, equilibrium is impossible.
Suppose a T w > w T H- T & as is the case when a drop of oil floats on water.
The drop of oil will be drawn out in a thin film covering the water.
Likewise in the case of glass, water, and air,
(92) a 7;> g 7; + w 7;,
and the water surface spreads over the glass.
This effect is also observed in a vacuum, but air has probably some
small influence on the result which cannot be readily determined, since
even in a vacuum there is generally a film of air or other gas condensed
on the solid.
Oil on water prevents waves, partly by reducing the friction whbh
enables wind to heap up waves, partly by reducing the surface tension
which holds a heap together, as a bag might hold sand.
The tension of a bubble compresses the air within it, so that a puncture
is followed by a collapse. A candle may be blown out in this way.
89. Capillarity deals with various phenomena due to surface
tension," such as the ascent or depression of liquids in small tubes. If the
solid is wet by the liquid, the latter spreads over the surface, dragging
with it the surface film of the liquid until the weight of the column of
liquid balances the force due to tension. In general, the surface assumes
a definite contact angle with the solid. If 6 is this angle, h the height of
52 PROPERTIES OF MATTER.
the column, and p the density of the liquid, when equilibrium is estab-
lished
2 irrTcos e = irr 2 hpg.
Therefore,
(93) T = r hpg I 2 cos e, or h = 2 Tcos / rs>g,
or the height varies inversely as the radius of the tube at the point to
which the liquid rises. T may be measured in this way. If h is the
height to which a liquid will rise in a given tube, the column which it will
hold when freely suspended is h-\- h^ h^ being the height sustained by
pressure due to curvature of lower drop. If the length of column is /?, the
bottom is plane; if less than h, concave. Between two flat plates,
similarly, the height may be proved to be inversely as the distance between
the plates. If they are inclined to each other, this leads to the relation
0ti = constant, or the liquid rises between them in the form of an equi-
lateral hyperbola. Liquids which do not wet the solid are depressed
according to the same law e. g., mercury in glass.
The mechanical structure and size of soil particles play an important
part in soil physics, as the distribution of moisture by capillary action is
thereby determined.
The contact angle in the case of water and glass is zero; for mercury
and glass, about 135. In a globular vessel of glass there is a certain
point, for which the tangent plane has a slope of about 55 with the
vertical, at which the curvature of the mercury surface disappears.
Above this point it becomes concave. The contact angle may be found
in any case by dipping a plate of the solid in the liquid and inclining it
until the curvature disappears on one side. The inclination to the vertical
is the contact angle.
90. To find the pressure inside a bubble, consider the energy
change due to a slight expansion. The work done by the pressure must
be equal to the gain in potential energy due to increased surface, or
when the increase of radius is infinitesimal. (2 T is used because there are
two surfaces).
The formula may be deduced in another way. If / is the pressure on
any diametral plane, the total pressure must just balance the total tension
around the circumference, that is,
(94) rr^p = ^r T, p = 2
or twice this value for a bubble.
The same expressions are used to determine the tension required in
steam-boilers.
When a soap film is exposed to equal pressures on both sides, the
condition, 1 /> + 1 / r' = must be satisfied.
In the case of melted solids T may be approximately determined by
melting the end of a rod of known radius and weighing the drop which
falls. The tension around the circumference just before falling may be
put equal to the weight of the drop :
(95) 2
Water-petroleum oo
These values vary greatly with the purity of the liquids.
The ripples on liquids are due to surface tension, the wavelets running
along the surface in the same way that waves are propagated along
stretched elastic cords. The above results were determined in terms of
density of the liquid and the wave-length of these disturbances.
92. A cylinder of liquid becomes unstable when its length is greater
than its circumference, for it may then have a smaller surface by breaking
up into drops. A jet thus breaks up into drops, which, under the action
of surface tension, vibrate between the forms of an oblate and a prolate
spheroid. A jet ordinarily appears to have an undulating contour. A
view by the instantaneous light of the electric spark shows that this shape
is due to the formation of contractions preceding rupture and to the chain
of separate drops of the shapes referred to which follow rupture.
93. Effect of Temperature. Heat diminishes all kinds of molecu-
lar cohesiveness, and consequently surface tension diminishes with rise of
temperature. If T G be the value at ,
T= T (\-cf)
where c has the value : water, .0019 ; alcohol, .0024 ; ether, .0047. For
54 PROPERTIES OF MATTER.
each liquid it is evident that there is a certain temperature at which the
surface tension would vanish. The significance of this will be shown in
discussing critical temperatures.
94. Surface tension of solutions. Impurities (mechanical mix-
tures) in most cases reduce surface tension ; but in definite solutions of
salts, such as that of sodium chloride in water, the surface tension is
usually increased in proportion to the concentration. This is one of a
number of cases which indicate that salts in solutions tend to increase the
cohesiveness of the solvent. They generally diminish the volume, or pull
the molecules together ; they raise the boiling point ; they increase vis-
cosity of liquids ; they increase surface tension.
References. Tait, Properties of Matter ; Boys, Soap Bubbles ; Mach, Popu-
lar Scientific Lectures The Forms of Water; Wm. Thomson, Popular Lectures,
Vol. I Capillary Attraction.
QUESTIONS AND PROBLEMS.
48. Why are shot and raindrops not flattened by their weight while falling ?
49. Explain the action of benzene or a hot iron in removing grease spots.
50. A drop of water is placed in a small conical horizontal glass tube. What
will it do ? What will mercury do under the same circumstances ?
51. A little water lies in a uniform horizontal glass tube, and one end of the
column is warmed. What will it do ?
52. Blow on warm soup or chocolate and observe effect. Explain.
53. Hold a camel's hair brush in water, then take it out. Observe and
explain difference of appearance in the two cases.
54. How far will water rise and mercury fall in a glass tube of .01 cm.
diameter ?
55. What is the object of the slit in a steel pen ?
56. Why will ink spread in ordinary printing paper and not spread in writing
paper ?
57. What is (approximately) the density of sea water 10,000 feet below the
surface, at ?
58. Explain the washing process of separating gold from gravel ; the blast
process of winnowing grain. Is the same principle used in each case ?
SOLUTIONS.
95. The term solution has a wide range of meaning, embracing phe-
nomena attended by violent chemical and thermal action (solution ol
zinc in H^SO^ of H^SO^ in water), and also cases where the components
become incorporated together with little or none of these effects (sugar
in water). There is a clear distinction, however, between solutions and
mechanical mixtures. The latter can be separated by mechanical means;
the former cannot. Very finely divided particles ca*n remain suspended
indefinitely in liquids, as in the case of emiilsions, without really being in
solution (cream in milk, chocolate in water). In such cases the com-
ponents may be separated by centrifugal action, for example, as in
separating cream. This would be impossible with a solution.
96. Solubility. The relations between liquids and solids which
render solution possible in some cases and not in others, are not clearly
understood ; but it must in general depend upon relative molecular
attractions the same sort of thing which determines whether a liquid wets
or does not wet a solid the predominance of adhesion or of cohesion. In
general, work is either done by or upon the molecules of the substance
dissolved, resulting in either cooling or heating. A liquid at a given tem-
perature can generally dissolve only a limited portion of a substance. The
SOLUBILITY DIFFUSION. 55
quantity usually increases with the temperature, but there are exceptions
sulphate of soda diminishes in solubility from 33 to 120 and calcium sul-
phate above 40. Common salt has practically the same solubility at all
temperatures. Some substances have no definite point of saturation, but
dissolve in all proportions, as glue in water. Solutions of liquids in liquids
also occur. Water mixes with alcohol in all proportions. The conduct
of sulphuric ether is peculiar; Water will dissolve about 3 per cent of
ether, or ether about 3 per cent of water. Beyond this they will not mix,
but remain in separate layers.
97. The volume of a solution is not always equal to the sum of the
volumes of its constituents. Generally there is contraction, as in the case
of alcohol and water, salt and water. In the case of a few substances, such
as ammonium salts, there is no contraction.
98. Diffusion of solids and liquids into liquids was first carefully
studied by Graham about 1850. Any substance which is soluble in a liquid,
will, if left to itself, gradually diffuse until it is of uniform density throughout
just as in the case of gases, but far more slowly. The rate of diffusion
between two regions of a liquid is directly proportional to the difference
of concentration between them, and inversely to their distance, and is
accelerated by rise of temperature, (another instance of increased mole
cular mobility accompanying rise of temperature). Stirring promotes dif-
fusion and uniform mixture by bringing the masses of different concentra-
tion nearer together. The effect of gravitation seems vanishingly small in
retarding diffusion of heavy molecules (e. g., CuSO 4 in water becomes
of uniform concentration throughout) . Heavier molecules as a rule diffuse
more slowly. The differences of concentration in a liquid gives rise to
diffusion convection currents similar to those caused by heat.
99. Osmosis. This name is applied to the diffusion of a substance
through a membrane or through porous walls. Since the rate varies with
different substances, phenomena similar to those of gaseous diffusion are
observed. A rubber membrane between alcohol and water allows the first
to pass, but not the other. In an animal membrane the reverse is true.
If a porous vessel, such as an earthenware cup, is filled with a concentrated
solution of any salt and the vessel immersed in water, the latter will pass
through more readily than the molecules of salt ; consequently there is an
increase of the contents of the vessel, which will produce a rise of the solu-
tion in a vertical tube. The force causing this is called osmotic pressure.
Pfeffer, ofLeipsic, in 1877, found that the osmotic pressure for small con-
centrations increases directly as the concentration and the absolute tem-
perature /. e., the relations of density, pressure, and temperature are
those expressed by Boyle's and Charles' laws. In order to isolate the effect
due to the salt from that due to the solvent, Pfeffer used what are called
"semi-permeable membranes," made by precipitating a layer of ferrocy-
anide of copper in the walls of porous cups. This substance allows water
to pass freely, but is impermeable to such substances as metallic salts,
sugar, etc. By connecting the vertical tube with a mercury manometer
Pfeffer measured the osmotic pressures due to a number of substances.
Another important fact discovered by him was that solutions with concen-
trations proportional to the molecular weights of the substances used
(equimolecular solutions) exerted the same osmotic pressures, showing
that this pressure depends solely on the number of molecules present, not
their nature. This is analogous to Avogadros' law. De Vries found that
66 PROPERTIES OF MATTER. .
certain plant cells contain protoplasm surrounded by a semi-permeable
membrane. If placed in a solution of osmotic pressure less than or equal to
that of the contents of the cell, there is no change. If the solution is more
concentrated, water is withdrawn from the cell, the membranous walls of
which contract. Bonders and Hamburger showed that the same is true of
blood corpuscles, so that these organic cells have been found useful in
experimentation, especially in reducing solutions to equal osmotic pressure.
100. Dialysis. Some substances, such as glue and jelly, scarcely
diffuse at all through porous walls. These are called colloids. Crystal-
loids, or substances having a crystalline structure, diffuse readily through
most porous membranes. Mixtures of two such substances may then be
separated by diffusion, which process is called dialysis.
101. Theory of Solution. The resemblance between some of the
properties of substances in solution and those of gases suggests a physical
analogy between the two states. It is an experimental fact that (with the
exceptions noted in section 103) a substance in solution exerts an osmotic
pressure equal to the pressure that the same mass would exert as a gas if
it occupied the same volume at the same temperature. Van't Hoff, now
a professor at Berlin, in 1887 suggested the hypothesis that the molecules
of a substance in solution act like those of a gas in space. This hypoth-
esis has much to recommend it, and has been generally accepted. If the
analogy were perfect, the only function of the solvent would be to serve
as the medium in which the molecules may have free motion. Since salts
are not equally soluble in all liquids, it seems evident, however, that the
solvent takes a more active part, modifying the phenomenon in a manner
not yet understood, so that we must be cautious in applying the laws of
gases to solutions, except in a purely formal way.
102. We may imagine an experiment with gases analogous to Pfeffer's
experiments with semi-permeable membranes, and making the results some-
what clearer. A closed palladium or platinum cylinder filled with nitro-
gen is placed in an atmosphere of hydrogen at the same pressure. The
hydrogen freely enters the cylinder until its own partial pressure is equal
to that outside (Dalton's law). The nitrogen cannot escape, and so the
pressure rises on the inside.
103. There are many cases of abnormal osmotic pressures which
may be explained as the result of molecular dissociation, each part of a
dissociated molecule being as effective in producing pressure as an entire
molecule. This explanation is borne out by similar abnormal conduct in
the raising of the boiling-point and lowering of the freezing-point of solu-
tions, and in electrical conductivity. All of these phenomena indicate
that metallic acids, bases, and salts are in part dissociated in solution (all
electrolytic conductors), while organic solutions (non-conductors) contain
no dissociated elements. Many salts appear to be completely dissociated
in very dilute solutions.
Osmotic diffusion probably plays a large part in plant and animal
circulation and assimilation. Poynting suggests that osmotic pressure
may be due to the loading of water molecules by combination with those
of a salt, making it impossible for those complex physical molecules to
escape, while the ordinary water molecules can enter.
References. Jones, the Modern Theory of Solution, and Physical Chemistry;
Ostwald, Solutions; Whetham, Solution and Electrolysis.
FLOW DIFFUSION. 57
QUESTIONS AND PROBLEMS.
59. Five grams of sugar are dissolved in one liter of water. What is the
osmotic pressure ?
60. Ten grams of NaCl are dissolved in water, and one-half becomes disso-
ciated. What is the osmotic pressure ?
61. In what case would the centrifugal machine fail to separate the compo-
nents of a mechanical mixture ?
SOLIDS.
104. A solid may be defined as a substance that is more or less
rigid i. e. , its molecules occupy definite equilibrium positions. No solid
is perfectly rigid; all yield more or less to a deforming force. We assume
that the permanent shape of a solid is due to cohesive forces between very
complex molecules. If two very smooth and clean plane surfaces of glass
or metal be brought in close contact, these forces will hold the surfaces
together even with a thin air film between them (as shown by the "New-
ton's rings" between two glass surfaces). It is possible that adhesion
between air and glass may play some part in this result. Still more
familiar are examples of adhesion, as shown by pencil marks on paper,
gilding, cements, etc. Spring has shown that powdered bismuth under
6,000 atmospheres pressure becomes a crystalline solid; graphite solders
in a solid mass at 5,500 atmospheres ; copper filings mixed with powdered
sulphur under great pressure form solid and homogeneous cuprous
sulphide.
105. Flow. As Tresca has shown, many solids, such as the metals,
may be made to spread laterally or to flow through openings by applying
sufficient pressure. This phenomenon occurs in wire-drawing, in the
manufacture of metal tubes, and in coinage.
106. Diffusion. There are instances of the diffusion of one solid
in another. Carbon diffuses through iron in the cementation process of
making steel. When metals are welded at high temperatures there is
some interdiffusion at their common boundary. Mixtures of metals in the
form of powder have been known to form alloys when subjected to pres-
sures of several thousand atmospheres.
Joseph Henry showed that a leaden U-rod with one limb in a vessel of
mercury acted as a siphon. The mercury diffused through the rod and
dropped from the lower end of the outer limb until the mercury in the
vessel was exhausted. He also showed that on heating silver-plated copper
the silver will disappear beneath the surface, but may be again exposed
on dissolving the surface copper in acid.
Roberts- Austen^ showed that if a lead cylinder 7 cm. long was placed
on a piece of gold, with close contact, the gold diffused rapidly into the
lead at high temperatures (but below the melting point of lead). At a
temperature of 251 C. the gold penetrated to the top of the cylinder in
three days. Traces of such diffusion were observed even at ordinary
temperatures. The lead was left in contact with the gold for four years,
when traces of gold were found in four parallel slices, each .75 mm. thick,
cut from the bottom of the cylinder. Such results indicate the existence
of translatory molecular motions even in solids, and give us reason to
suspect that there may be constant diffusion of gold within itself. Roberts-
Austen also found some diffusion of gold into silver and copper at higher
temperatures.
(1) Nature, Vol. 54, p. 55, 1896; Proc. Royal Soc., 67, 1900.
58 PROPERTIES OF MATTER.
107. Solution. We might say in the above case that the gold was
dissolved by the lead. Selective solubility is shown in many cases by
melted metals. Mercury will dissolve copper, but not iron. Melted lead
will dissolve tin but not zinc. Alloys of the metals in many cases resembles
solutions in their properties. In some cases these solid solutions may be
saturated, that is, one component will not alloy with more than a certain
proportion of another. In all these cases we see how molecular mobility
is increased by heat.
MOLECULAR FORCES AND STRUCTURE.
The properties of solids depend on the nature of the molecular con-
straint, which determine their rigidity, hardness, ductility, etc. The
molecular force which tends to make molecules resume their equilibrium
positions after being displaced is called elasticity.
108. Elasticity. In order that a body may be elastic it must have
the properties of (1) resistance to a distorting force; (2) a force of restitu-
tion (resilience), which will restore it to its original position on removing
the disturbing force. In perfectly elastic bodies, such as fluids, the force
of restitution is equal to the distorting force.
In bodies having a crystalline structure, elasticity and other properties
are different along different axes. Bodies with properties uniform in all
directions are called isotropic. The elastic relations of crystalline
(aelotropic) bodies are very complicated and cannot be discussed here, but
in the case of isotropic bodies they depend upon two elements only; the
resistance to compression and the resistance to change of form.
The measiire of elasticity is the ratio stress / strain. The coefficient
of volume elasticity or of compressibility is
i stress per unit area pv o
K = - - - - -
- ; - ; - r- = - r
strain per unit vol. (& v )
The resistance offered to change of shape is called^ rigidity. This is
the force called out by the torsion of a wire. In cases where there is no
change of volume a simple shear is produced, similar to the effect produced
by applying a tangential pressure to the top of a pack of cards. The strain
is measured by the angular shear, which for small deformations is b / a,
the tangent of the angle of twist between two planes 1 cm. apart. The
coefficient of rigidity is defined as
(99) n =pa / b,
p being the tangential force per unit area.
As a rule, the compressibility of solids is less than their rigidity. Cork
is an exception.
In cases of longitudinal compression or elongation the coefficient of
elasticity used is Young's modulus:
nnm Jl/r stress per unit area PL
( A.\J\J ) J.YL -- " . ~. . 7~ ~~77~ -- 1 *
strain per unit length al
P being the applied force, a the cross-section of the body, L its length,
and / the elongation or compression. M may be thus defined as the
force per unit area required to double the length of a given substance,
assuming the law of stretching to hold indefinitely.
ELASTICITY HARDNESS. 59
Elongation in general involves lateral contraction. It may be seen
that an imaginary cube in the substance having the diagonals respectively
parallel and perpendicular to the direction of elongation will have one
diagonal lengthened and the other shortened, equivalent to a shear and a
rotation Consequently Young's modulus is a function of both the com-
pressibility and the rigidity of the substance.
The three elastic coefficients k, n, and J/are usually very large quanti-
ties. Their values for some substances are:
k n M
Steel 18X10" 8x10" 20x10"
Brass 10x10" 3.5x10" 9.5X10"
Glass 3.5X10" 2.2x10" 6x10"
The three coefficients may be respectively defined as the force in dynes
required to reduce a unit volume to zero, to shear a unit cube 45, and to
double the length of a unit cube.
In the bending of rods, Young's modulus is the coefficient involved;
in the torsion of rods or wires, it is the coefficient of rigidity.
109. Limit of Elasticity. If bodies are strained bevond their elastic
limit, there is a permanent set produced. When a solid is strained for the
first time, there is a small permanent set even for the smallest distorsions;
for future strains not exceeding that producing the set, Hooke's law is
almost perfectly obeyed. Many metals have a crystalline structure, and
the limit seems to be reached when the crystals begin to slip.
110. Hooke' s Law. In most cases it is found that the deformations of
elastic bodies are proportional to the forces producing them. Exact
measurements show that this is not strictly true for large deformations,
but the law is nearly enough true in most cases to be made the basis of
numerical calculations.
Increased temperature usually diminishes elasticity another example
of the effect of heat in breaking down molecular constraints.
111. Elastic Fatigue. When a body has been subjected to rapidly
alternating strains for some time, elastic fatigue occurs; for example, a
tuning-fork which has been maintained in continuous vibration seems to
lose a considerable portion of its elasticity. Kelvin found that a torsion
pendulum which diminished its amplitude of vibration by one-half in 100
vibrations would, after fatigue set in, drop to one-half its original amplitude
in 45 vibrations. If the wire of such a pendulum be kept twisted to the
right for some time, and then to the left for the same time, on releasing it
the last twist will be first undone, and then the first (after-affecf).
A perfectly elastic body set in vibration would continue to vibrate
indefinitely. All the work spent in producing the displacement would be
converted into potential energy of the molecules. In all actual cases
some of the work is expended in overcoming molecular friction or viscosity
at each vibration ; so that there is constant expenditure of energy, or rather
its transformation to heat, and the vibration ceases.
Such properties as hardness, tenacity, ductility, etc., depend on the
nature of the molecular constraints.
112. Hardness. Below is Mohs' scale of hardness, each substance
being able to scratch all placed before it on the scale : 1, talc ; 2, gyp-
sum ; 3, calcite ; 4, fluorite ; 5, apatite ; 6, feldspar : 7, quartz ; 8,
topaz ; 9, sapphire ; 10, diamond.
60 PROPERTIES OF MATTER.
V-^.
A rapidly moving body is in some cases able to cut bodies harder than
itself. A soft-iron wheel revolving at a great speed can cut steel or
even quartz. In such cases there is mutual wear, but the action is con-
centrated on the hard object and distributed over the other. Similar
effects are seen in the action of a sand blast, and the erosion of rocks by
water. This erosion is due to suspended particles of sand. In Sicily a
deposit of lava made by Mt. Etna in 1603 had a gorge 30 feet deep and
several hundred feet wide cut through it by a stream in little more than
two centuries.
113. Tempering and Annealing. Some metals, such as steel,
when heated and cooled suddenly by immersion in a liquid are rendered
much harder. In case of steel the resulting density is less than before.
In one case the density was 7.817 before tempering and 7.743 afterwards.
On reheating it returned to nearly its original density. In brass and
alloys containing tin there is an increase of density. Bronze is tempered
in a way the reverse of that applied to steel ; it is softened by rapid
cooling.
Annealing is the name applied to the tempering of such substances as
glass. Glass is made much tougher by slow and uniform cooling. If the
surface is cooled rapidly, the subsequent contraction of the interior pro-
duces an interior tension causing an unstable state, as shown in Prince
Rupert drops, made by dropping melted glass in water. Such bodies
will stand a severe blow or pressure, but burst into powder when the stem
is broken off. Barus has shown that the unstable condition is relieved by
dissolving the external layers in hydrofluoric acid.
114. Tenacity is the resistance to fracture due to elongation.
Fracture implies a lack of homogeneity, since a perfectly uniform body
subjected to stress would rupture at all points simultaneously /. ^., fall
to powder. Steel is the most tenacious and lead the least tenacious of the
metals. Toughness implies tenacity rather than hardness. The tenacity
of wires is increased by the process of drawing, so that a very small wire
will support proportionally twice as much as a large one.
115. Friction is the resistance offered to the motion of one body
over another, due to the interlocking irregularities of their surfaces. It
is not a force in the most general sense, since it can only destroy, never
originate, motion. Experiment shows that the resistance due to friction
varies with the substances tested and with the state of their surfaces, as
first shown by Coulomb. It is approximately proportional to the normal
pressure between them and approximately independent of the area of con-
tact and velocity of sliding. The coefficient of friction between two sub-
stances is the ratio of the tangential force of friction and the normal
pressure.
(101) * =
If, for example it requires a force of two pounds weight to pull a
weight of 10 pounds along a level surface, /* = .2. Some values of the
coefficient of friction are: Oak on oak, .41; wagon on ordinary road,
.04; iron on iron rails, .004. Rolling friction (last two cases) is much
less than sliding friction. The fact that the total force of friction depends
upon the pressure of the surfaces in contact is illustrated by the great
firmness of grip of a rope wrapped only once or twice around a tree or
CRYSTALS MOLECULAR THEORIES. 61
post. As shown in discussing surface tension, the normal pressure of a
curved surface is inversely proportional to the radius of curvature hence
the great resistance to slipping when the tree or post has a small radius of
curvature ; even when it is smooth.
116. Crystals. Nearly all inorganic substances have in their natu-
ral state a crystalline structure. Crystals are bounded by plane faces
symmetrically arranged with reference to three or four diametral lines
called axes, and in a given species the angles of inclination between the
faces is constant.
Differences in length of axes are accompanied by differences of
physical properties, so that, except in the case of crystals belonging to
the isometric system, in which all the axes are equal (such as common
salt), there are differences along different axes in hardness, elasticity,
thermal expansion and conductivity, and optical effects.
Crystalization is favored by changes of state, such as passage from
gaseous to solid state (phosphorus volatilized in a vacuum at low tem-
perature, sublimation of arsenic at dull red heat), solidification from
fusion (bismuth), evaporation of solution of salt.
In all supersaturated solutions of salt, crystallization is started by
the addition of a small crystal of the same form, which serves as a
nucleus. A broken crystal placed in a solution of the substance of
which it is composed will repair the flaw before general growth takes
place.
It is difficult to explain the conduct of crystals except by assuming a
very regular geometrical arrangement of their molecules, tending to pro-
duce stability under the action of molecular forces.
Many organic compounds have the same chemical constitutions (are
isomeric), but very different physical properties. This can be explained
only as a result of different arrangement of atoms in a simple molecule, or
different molecular aggregations.
117. Molecular Theories. There have been various hypotheses
as to the nature of molecules, chiefly the following : Lucretius, the Roman
poet, believed that the atoms of which they are composed are hard,
indivisible bodies, whose properties depend on their shape. For instance,
those of honey and milk are round and smooth, while those of disagree-
able substances are rough and hooked. Boscovich held that atoms are
merely centers of force. Sir William Thomson (Lord Kelvin) advanced
the hypothesis that each atom is a little vortex in a perfect fluid the
ether that transmits the waves which produce the effects of light and heat.
Helmholtz showed that vortices in such a fluid that is, one which is per-
fectly elastic and non-viscous could neither be originated nor destroyed
by any means known to us. A special creative act would be necessary.
Many of the properties of such vortices are illustrated in a crude way by
the conduct of smoke rings. The equilibrium form of such rings is a
circle. If distorted, they oscillate about this form. They are indivisible
because it is impossible to touch them. They rebound from each other
and from other bodies with perfect elasticity. Two rings which approach
each other under certain conditions will not separate, but will move on
together, rotating about each other, suggesting chemical combination.
According to this hypothesis, all matter is fundamentally the same, the
differences of the elements depending on the size or complexity of the
vortex atoms, which may consist, in many cases, of a number of rings
inseparably linked together.
62 PROPERTIES OF MATTER.
These speculations are interesting, but no theory has yet been found
which is self-consistent and satisfactory in all respects.
118. Size of Molecules. There appears to be no possible means
of ascertaining the size of a molecule, but some physical phenomena give
us a rough idea of the maximum distance between their centers or the
diameter of their spheres of activity.
The thickness of the film of a soap bubble may be determined with
considerable accuracy by the colors due to interference of light waves of
known length reflected from its outer and inner surfaces. Reinhold and
Riicker have shown in this manner that a minimum value of the surface
tension is reached when the bubble is .0000012 cm. thick. It is not prob-
able that such a film is many molecules thick.
Quincke, by observing the angle of contact between liquids and glass
coated with wedge-shaped films of silver, observed that the effect of the
glass upon the contact angle began to be exerted through the film when
its thickness (determined by color phenomena) was about .0000050 cm.
Wiener has been able to detect optically the existence of a film of
silver not more than .00000002 cm. thick deposited on mica.
By extending a cubic centimetre of water into a film of .00000001 cm.
thickness it may be shown that the heat necessary to be supplied to main-
tain a constant temperature will be sufficient to vaporize the water, or at
this thickness the molecular forces begin to break down.
These figures give some idea of the maximum limit of range of molec-
ular forces.
119. Free Path. As the rarefaction of a gas increases, the length
of the mean free path, or the distance the molecules can move without
collision, is increased. By applying the kinetic theory of gases to the
phenomena of diffusion and viscosity, this distance may be determined.
In air at ordinary pressures it is about .00001 cm. In a good vacuum
it may be several inches. The effects observed in Crookes' radiometer
are due to increased length of path. A little windmill, with vanes of mica
or aluminum, is mounted on an axis in a very low vacuum. One side of
the vanes is blackened, and on exposing the instrument to a source of
heat the blackened side becomes w r armer than the other, owing to its
greater absorptive power. The molecules which strike on that side have
their kinetic energy increased and rebound with greater velocity than those
on the other side. The unbalanced reaction causes rotation.
References. Maxwell, article "Atom," Ency. Brit.; Wm. Thomson, Pop.
Lectures, Vol. I Size of Atoms; Tait, Properties of Matter.
QUESTIONS.
62. A piece of glass fastened at one end or a dead tamarack limb will fly in
several pieces if broken by pulling on free end. Explain.
63. A stretched horizontal wire will not support as heavy a weight at its
center without breaking as it would if hung vertically. Explain.
64. What kind of knots in ropes hold most firmly without slipping ?
65. Explain the amalgamation process of collecting gold from crushed quartz.
HEAT.
120. When we touch bodies we generally become aware of two
distinct sensations one of material resistance, or force, and one of
hotness or coldness. To the latter sensation we give the name of heat or
cold. We also give the name of heat to the cause of these sensations,
regarding cold not as a positive property, but merely as the absence of
heat. The same confusion exists in the use of the words sound and light,
which are applied both to the sensations and to the agents producing
them. In the study of physics, however, we shall concern ourselves only
with the nature of the agencies producing these sensations.
121. Production of Heat. Experience shows that heat may be
produced in several ways:
By chemical action, as in combustion.
By change of state, as in the condensation or freezing of water.
}$y friction. Savages produce fire by friction between two pieces of
wood. When a copper or lead wire is bent rapidly back and forth heat
is produced by molecular friction.
By compression or percussion, as by suddenly compressing air or
hammering a piece of metal.
By an electric current in overcoming resistance, as in an incandescent
lamp. In all these cases except perhaps the last, we may imagine a clash-
ing of molecules, increasing their kinetic energy.
122. Effects of Heat. When bodies are heated the following
effects may be produced:
Change of hotness, or of temperature.
Change of volume, as in the case of the thermometer.
Change of state, as in melting of ice and evaporation of water.
Change in molecular constraints, shown by change in elasticity and
viscosity, in magnetic strength.
Increased molecular velocities, as in diffusion of gases and liquids and
in evaporation.
Chemical dissociation.
Production of electric current, in thermal couple.
123. Nature of Heat. A Form of Energy. If a hot body is in
contact with a colder one, we find that the colder body becomes hotter
and the hot one colder until both come to the same state. Something
must pass from one to the other to produce this change: There are
apparently only two things in the physical world which appeal to our
senses matter and energy. To which of these classes does heat belong?
is a question which can only be decided by a careful study of the effects
and method of production of heat, and its relation to other physical
phenomena.
Many of the ancient Greeks believed in the existence of self-repellent
fire atoms, which caused expansion by forcing themselves between the
atoms of matter. Modifications of this theory held their ground until the
beginning of the nineteenth century. At that time the generally accepted
view was that heat or caloric is a subtle, elastic, weightless fluid, whose
64 HEAT.
parts are self-repellent, thus causing expansion, conduction, and radiation.
This idea was strengthened by the discovery of the fixed gases, which
appeared to have many properties in common with this hypothetical fluid.
Opposed to this material theory of heat is the dynamical, which assumes
it to be the effect of rapid motion in the particles of matter (and conse-
quently a form of energy). This idea has had some advocates from the
earliest days of Greek speculation. Lord Bacon, after considering the
effects of heat, concluded that it must be due to motion. Newton held a
similar opinion. All this was, however, scarcely more than speculation,
and no definite experiments to test the theory were undertaken until the
end of the eighteenth century. Count Rumford was the pioneer in this
work. His attention was called to the subject in 1778 by observing that
when a blank cartridge is fired from a gun, the barrel is more heated than
when a ball is fired (and more mechanical work done), suggesting some
relation between heat and energy. Later he showed that if caloric is a
material fluid it has no weight. In 1798, while boring cannon for the
king of Bavaria, he observed that heat was evolved without limit as long
as the boring continued. He raised the temperature of the cannon to the
boiling point of water by boring two and a half hours. If, as the calorists
believed, heat is a material fluid, we should expect some limit to its
production. Rumford concluded that ' ' anything which any insulated
body or system of bodies can continue to furnish without limitation cannot
possibly be a material substance, and it appears to me to be extremely
difficult, if not quite impossible, to form any distinct idea of anything
capable of being excited and communicated in the manner in which heat
was excited and communicated in these experiments unless it be motion."
The calorists explained that the capacity for heat of the iron (the
amount of heat required to change its temperature by a definite amount)
was diminished by being reduced to powder or borings, thus causing an
evolution of heat. Rumford showed that the capacity was unchanged,
but his proof was not conclusive, because he did not show that the total
amount of heat in the one case was equal to that in the other, as he might
have done by finding whether the same quantity of heat was required to
melt equal masses of the iron and the borings, or was evolved in dissolving
equal amounts in acid, and thus reducing them both to an identical
physical condition. Nevertheless, the explanation of the calorists fails to
hold when we consider the case when friction produces heat without
altering in any way the physical condition of the substance, as when it is
evolved by churning a liquid.
About the same time Humphrey Davy melted two pieces of ice, at an
original temperature below the freezing point, by rubbing them together
in a receiver exhausted of air and surrounded by ice. In this case the
communication of heat from external sources was prevented, and an
expenditure of energy alone resulted in the production of heat. There
was no question of the capacity of the substance, for that of ice is less than
that of water, not greater, as demanded by the caloric theory.
In considering the methods of production of heat it is found that when-
ever it is evolved there is a disappearance of mechanical, chemical, or
electrical energy. If we impart heat to matter, we find that, in general,
expansion is produced, which may do mechanical work against molecular
attraction or external resistance; or the molecular constraints are partially
or completely overcome, as in melting ice, vaporizing water, or producing
chemical dissociation and changes in elasticity and viscosity; or electrical
SCALES OF TEMPERATURE THERMOMETERS. 65
energy is developed, as in the production of thermal currents; or an
increase of pressure, associated with an increase of temperature, is imparted
to gases, which we explain by the kinetic theory as the result of increased
kinetic energy of the molecules. As shown by Joule, a gas is cooled by
expansion when it does work against external pressure, but not when it
expands into a vacuum. The conclusion seems irresistible that heat is a
form of energy. In the case of gases, at least, it seems as though it must
be in the form of molecular kinetic energy; and the same seems to be
probably true of solids, as indicated by the communication of heat by
contact, and by the demagnetization of a magnet by heat.
The caloric theory was completely overthrown only after the experi-
ments of Mayer (who first distinctly stated the idea of conservation of
energy in 1842) and of Joule, who established a definite numerical relation
between heat and energy, known as the mechanical equivalent of heat.
These experiments will be described later.
References. Stewart, Conservation of Energy; Tait, Heat; Tyndall, Heat a
Mode of Motion; Tyndall, Fragments of Science Count Rumford.
124. Quantity of Heat. Heat being a form of energy, it is
capable of quantitative determination in terms of work done or other
effects produced. This subject will be discussed later.
125. Temperature. Another attribute of heat which more directly
affects our senses than its quantity is the quality of hotness, or its
intensity. To this quality we give the name of temperature, meaning
thereby the degree of hotness, or the condition which determines the
flow of heat from one body to another. It is analogous to pressure in
liquids or electromotive force in electricity, which determine the direc-
tion of flow of the liquid or electric current.
THERMOMETRY.
126. Scales of Temperature. We may say that one body is
hotter than another when heat will flow from the first to the second; or
that two bodies are at the same temperature when in thermal equilibrium.
Further than this there is no absolute method of measuring temperatures,
or saying that one temperature is so many times greater than another, in the
sense that we say that one length is so many times greater than another.
It follows that all scales of temperature are more or less arbitrary. Such
scales may be based on any continuously changing effect of heat, which is
always the same under the same conditions such as the series of colors
assumed by polished steel when tempered; the intensity of a thermo-electric
current; the change of electric resistance of a metal; the change in elasticity
of a substance; the rate of evaporation under given conditions, or the
expansion of a given substance. The latter is the effect most commonly
employed in the construction of thermometers. As a rule, fluids such as
air or mercury are used, owing to their great expansibility. The expansion
of the containing vessel may generally be neglected.
127. Thermometers. One of the earliest forms of thermometer
was made by Galileo, about 1592. It consisted of a glass bulb with a long
neck inverted in and partly filled with water or alcohol, which rose or fell
in the tube with changes in the volume of air. There was no definite
method of calibrating these early thermometers, except by marking on
66 HEAT.
the stem the points corresponding to the greatest heat of summer or cold
of winter, until the time of Boyle, who about 1665 used the temperature
of melting ice as one fixed point; and Huyghens about the same time
suggested the use of the temperature of boiling water for another. This
method was adopted by Newton. The intervening space on the stem was
divided into any number of parts, at the whim of the maker, and the
increments of temperature were assumed proportional to the apparent
increments of volume.
The first thermometers of modern form were made by Fahrenheit, of
Holland, about 1714. He was the first to note that the temperature of
boiling water varies with the pressure, and to adopt a standard barometric
pressure 29.8 inches of mercury. The corresponding point of the scale
he called 212 ; that of melting ice 32, and that of a mixture of salam-
moniac and ice, 0.
Reaumur, of France, about 1730 made a thermometer in which the
point corresponding to the freezing of water was marked 0, and that
corresponding to its boiling, 80.
Celsius, professor of astronomy in the University of Upsala, about
1740 made a mercurial thermometer in which the scale between these
points was divided into one hundred equal parts, and the boiling-point
numbered 0; the freezing-point, 100. The modern centigrade ther-
mometer, almost exclusively used in scientific work, is the same, with the
scale reversed.
Guy Lussac discovered that the temperature of boiling water depends
somewhat on the nature of the containing vessel, and it has also been
found to depend on the amount of foreign substances in solution. The
temperature of steam from water boiling under a given pressure, however,
is found to be invariable, and the temperature of melting ice is practically
independent of pressure.
Construction. The ordinary mercurial thermometer is made by blow-
ing a bulb on a small glass tube, filling the bulb and part of the tube with
mercury, and expelling the air by boiling the mercury until the stem is
filled with mercury vapor alone. The tube is then sealed. The points at
which the mercury stands when the bulb is placed in a mixture of ice and
water and in steam from water boiling under a pressure of 76 centimeters
of mercury are marked, and the intervening space divided into 100 parts
of equal volume, called degrees. Since glass tubes are not usually
uniform, they must be calibrated by means of a thread of mercury, the
length of which is measured in different parts of the tube. It is to be
noted that increments of temperature are proportional to apparent, not
true, changes of volume, since the expansion of the glass is ignored.
Different substances, as a rule, have different laws of expansion, so
that thermometers made of them will differ in their readings, except at
the fixed points. Below some comparisons are made of different thermo-
metric substances:
Mercury in glass 45 50 100 150 200
Sulphuric acid 41 ... 100
Cadmium ... 53.3 100 141. 4 179. 1
Silver ... 51.7 100 145.3 188.2
It is found, however (as expressed by Charles' law), that thermom-
eters made of different gases agree almost exactly through wide ranges of
temperature. "Furthermore, it will be shown later that the increase in
Q
Vn
,
E (__,
M
A Q
o
.,
AT
AQ
,
^c -
llr -
^JL
QUANTITY OF HEAT' SPECIFIC HEAT. 67
volume of a gas under given conditions is almost exactly proportional to
the amount of heat absorbed by it. A gas or air thermometer is there-
fore adopted as the most appropriate for scientific measurements. Such a
thermometer is, however, inconvenient to use. It is found that of all
liquids mercury has the most uniform expansion through ordinary ranges,
as shown by comparison with the air thermometer:
Air 20 40 60 80 100 200
Mercury 19.98 39.67 59.62 79.78 100 202.78
Mercury is also a good conductor of heat and has a low specific heat,
causing it to respond rapidly to changes of temperature. It is readily
obtained pure, does not stick to glass, and remains liquid between 40
and 350 C. It is therefore commonly used for making thermometers.
Ether and alcohol are sometimes used for sensitive thermometers on
account of their great expansibility.
As time goes on it is observed that there is a gradual change in the
zero of a thermometer, caused by contraction of the glass. After heating
there is a depression of the zero, since the glass does not for some time
resume its original volume. A special kind of glass made in Jena is less
subject to this defect than other kinds.
Other methods of measuring very high temperatures or very small
differences of temperature, depending on other effects of heat, will be
explained later.
CALORIMETRY.
128. Quantity of heat may be relatively measured in terms of any
of its effects. Being a form of energy, the most scientific method would
be to measure it in terms of energy in ergs. This would be inconvenient
in practice, however ; hence it is usually measured in terms of physical
effects easily produced. Two methods of calorimetry, or heat measure-
ment, are ordinarily used, based upon
1. Change of State. It is found that a certain amount of heat is
necessary to melt a given mass of ice or other substance, or to vaporize
it, without change of temperature. It is expended in work done against
molecular forces. It may be assumed that to change the state of twice or
n times the mass would require twice or n times as much heat. By this
system quantity of heat would be actually measured in terms of work
done, although not in the ordinary work units.
2. Change of Temperature. If a certain amount of heat is required
to raise the temperature of a given mass of a substance from to 1, it
will evidently require n times as much heat to raise the temperature of n
times the mass through the same interval. It is not in general true, how-
ever, that n times the heat will raise the temperature of the original mass
n degrees, although it is approximately true in most cases.
The practical unit of heat is the calorie, or the amount required to
raise the temperature of one gram of water at its maximum density
from 4 to 5.
The thermal capacity of a body is the amount of heat required to raise
the temperature of a body one degree.
129. The specific heat of a substance is the ratio of the amount
of heat required to raise the temperature of a gram one degree to
that required to raise the temperature of a gram of water one degree,
HEAT.
or, practically, the number of calories required to raise the temperature of
one gram one degree. As indicated above, this quantity is usually
variable with the temperature, although for practical purposes we may
assume that the amount of heat which will raise the temperature of n
grams of water from 4 to 5 will raise the temperature of one gram
n degrees. The variations of the specific heat of a substance are due to
the fact that not all of the heat goes into increased molecular kinetic
energy (temperature), a part being expended in overcoming molecular
and external forces. Only in the case of gases under uniform conditions
of pressure or of volume is the specific heat constant. Black, in 1760,
discovered the fact, previously unsuspected, that the specific heat of
all substances is not the same. For example, if 100 grams of mercury
at 100 are mixed with 100 grams of water at the resulting tempera-
ture is not 50, but about 3. 2. In general, if m be the mass and s the
specific heat of the mercury, M the mass of the water, and t the final
temperature,
(102) ms
There are a number of methods of determining specific heat.
130. Black' s ice calorimeter consists of a block of ice, with an
excavation from which all water has been sponged out. Into this is
placed a mass m of a substance at a temperature of t which is covered
with a slab of ice. In falling to the heat evolved melts a mass M of
ice, which can be removed by a dry sponge and weighed. If L units be
required to melt one gram of ice,
(103) mst = LM=SQM.
131. The method of mixture is most commonly employed. If m
grams of a substance at temperature T be mixed with M grams of
water at / in a calorimeter (thin metal vessel) of mass m t and specific
heat s lt the final temperature being t
(104) ms (T-^ = (M+m I s 2 ~) (t-Q,
from which s may be calculated. The product m^s x is called the water
equivalent of the calorimeter. There is some loss by radiation, which
may be corrected by observing the rate of cooling of the water at its
mean temperature; or observations of temperature may be made at regular
intervals while stirring, and temperatures and times plotted as coordinates.
The resulting curve beyond its maximum ordinate gives the law of cooling,
and if continued backward until it cuts the axis of temperatures it will
give approximately the temperature which would have been attained had
the heat been imparted instantaneously and without loss to the water.
Radiation losses may also be eliminated by making the initial temperature
of the water about as much below the temperature of the room as its
final temperature will be above it. This is easily done if the specific heat
of the substance is approximately known from a previous experiment.
132. The Bunsen calorimeter consists of a bulb of glass with a
laterally communicating capillary tube, and a test tube sealed into it.
The bottom of the bulb and a part of the capillary tube are filled with
mercury, and the rest of the bulb with water from which all air has been
removed by boiling. The water is partly frozen around the test-tube by
A Q - -
-w*^^
. v*f
- C-v '-
SPECIFIC HEAT. V ~ 69
X^t/FOfr
placing the bulb in a freezing mixture. If m grams of water at tempera-
ture t be placed in the test-tube, it will be cooled down to zero, the heat
melting the ice without any appreciable loss by radiation. The volume of
the contents of the bulb will be diminished, and the mercury column in
the tube will recede n divisions. Evidently a change of volume of one
division corresponds to the absorption of mt / n calories, and from the
amount of melting produced by any substance introduced into the test-
tube its specific heat may be determined. This is one of the best methods,
especially when only small quantities of a substance are available.
133. Joly Steam Calorimeter. One pan of a balance is suspended
in a steam-tight chest at temperature t; m grams of the substance are
placed on the pan and counterbalanced; dry steam is admitted, and M
grams of steam are condensed on the pan and the substance, the latent
heat set free raising its temperature to 100; or the substance may be
freely suspended from the balance-arm, no pan being used. If a pan is
used and has the water equivalent m^s^ and if one gram of steam in con-
densing gives out 536 calories
(105) (ms + m,s^ (100 f) = LM= 536 M.
134. Specific Heat of Gases. When a compressed gas expands
against an external pressure (no heat being applied) it does mechanical
work, at the expense of its own potential energy. If we place a thermo-
pile or sensitive thermometer in such an expanding gas we find that it is
cooled, as we might expect for if there is no other source of energy, the
work must be done at the expense of the kinetic energy of the molecules.
If the compressed air in a vessel saturated with water-vapor be allowed to
suddenly expand, the water-vapor will be condensed in a cloud by the
cooling. Conversely, if a gas is compressed it becomes heated. Rapid
compression of air containing an inflammable substance such as carbon-
bisulphide vapor will ignite the latter. If the gas expands against pressure,
thus doing work, we must supply heat to maintain the temperature; and
if we wish to elevate the temperature by a given amount it will take more
heat than if the gas were kept at constant volume. A difference should,
therefore, be expected between the specific heat of a gas at constant
volume and that at constant pressure. This is found to be the case; in
fact, there are an infinite number of specific heats of a gas, depending on
the conditions of pressure and volume. If r p and c v be the two principal
specific heats, the difference between them will be the value in heat units
of the work done on unit mass in expanding against the constant pressure
p while the temperature rises 1.
In general, considering m grams of gas and calling the mechanical
( ' ' Joule's ' ' ) equivalent of heat /, if the gas expands a distance d in a tube
of section a, against a pressure/,
(106) (.,-..) (T,- T^= W ork=-= .
z^ t = mR T^ pv Q = mR T , which gives
(107) *~*~T
The difference between the two specific "heats in any gas is equal to the
gas constant R divided by the mechanical equivalent of heat (4.19 X 10 7
ergs).
70 HEAT.
If a gas expands into a vacuum, no energy is expended in external
work, and if a change of temperature follows it means that molecular
attractions or constraints are overcome (if temperature falls), or that
potential energy of repulsion has been transformed into kinetic energy (if
the temperature rises). In a "perfect" gas following Boyle's law, there
should be no change of temperature.
Joule and Thomson found that in all the gases investigated by them,
except hydrogen, there was a very slight cooling, indicating a feeble
molecular attraction. W
Regnault determined the specific heats of gases at constant pressure by
forcing them from a large reservoir in a water bath through a spiral tube
immersed in a calorimeter. The mass of gas passing in a given time can
be determined, and from the rise of temperature in the calorimeter the
specific heat calculated. The specific heat at constant volume may be
calculated from the above relation, or it may be measured directly by
Jolly's steam calorimeter. A considerable mass of the gas is forced by
pressure into a copper globe, which is suspended in the steam chest. To
obviate doubtful corrections, an exactly similar but exhausted globe is
suspended from the other balance-arm and exposed to the steam.
Air 0.23741 0.1741
Oxygen 21751 .1544
Nitrogen " .24380 .1735
Hydrogen 3.4090 2.4263
It will be observed that the ratio of the two specific heats is practically
constant for all gases and equal to 1.4. This relation will be explained
later (see section 185). The difference of the two specific heats will also
be found to correspond closely to equation (107).
135. Change of Specific Heat with Temperature. In most cases
there is an appreciable change in specific heat with temperature, generally
an increase, although in the case of water there is a minimum value at
about 30. The specific heat of diamond is about three times as great at
200 as it is at 0. Marked changes accompany change of state, as shown
in the case of water. It must not be inferred that the ' ' total heat " in a
body is equal to its thermal capacity multiplied by its absolute tempera-
ture. Owing not only to variation of the capacity with temperature, but
to possibility of transformation into other internal forms of energy, the
expression would be as meaningless as ' ' the total amount of sound in a
horn."
The high specific heat of water causes it to change in temperature
very slowly. This fact has a great influence on climate, regions on the
ocean being more equable than those inland especially if they are
adjacent to warm ocean currents, from which the prevailing winds blow.
136. Latent Heat. The amount of heat absorbed or evolved with-
out change of temperature when a unit mass of a substance changes its
state by fusion, vaporization, solution, etc., is called latent heat, although
it is not latent, but expended or given out by work done upon or by
molecular forces. The latent heat of a substance is usually determined by
the method of mixtures. If m grams of ice be melted in water in a calorim-
(1) See Ames, The Free Expansion of Gases.
MECHANICAL EQUIVALENT OF HEAT. 71
eter whose total thermal capacity is M and its initial and final tempera-
tures T and /
m (L + t)
If m grams of steam be condensed in the calorimeter,
(108) m (L + 100 - T) = M( T- f).
The latent heat of ice is 80 calories ; that of steam, 536.5.
The Bunsen ice calorimeter may be used to determine the latent heat
of ice, knowing the densities d^ and d of ice and of water at 0. If Q
calories of heat be supplied,
(109)
Q(\ 1 \ , ,
-7-1-7 I =v, the change in volume.
137. Mechanical Equivalent of Heat. In absolute scientific
measurements quantities of heat may be expressed in foot-pounds, kilogram-
meters, or ergs. In the first case the practical heat unit used in com-
parison is the amount which will raise the temperature of one pound of
water 1 Fahrenheit ; in the second, the amount which will raise the
temperature of one kilogram 1 centigrade (sometimes called the greater
calorie), and in the latter case the calorie. The first attempt to discover
a definite relation between heat and work was made by Rumford, from his
experiments on boring cannon. His result was much too high, being 847
foot-pounds, or 559.4 kilogram-meters. Dr. Robert Mayer, of Heil-
bronn, in 1842 used the relation c v c v =p (v^ v^lj (equation 107),
substituting the values for the specific heats determined by Regnault.
His result was 367 k / m. The validity of this result has been questioned
by English physicists on the ground that he had no right to assume that
no heat was expended in internal work. Joule had not then proved this,
but it seems that Gay-Lussac had performed a similar experiment, the
results of which were known to Mayer.
Joule about 1842 began a classic series of experiments for deter-
mining the mechanical equivalent. The principal method used by him
was to cause a falling weight to rotate a paddle in a calorimeter filled with
water or mercury. From the number of foot-pounds of work done and
the rise of temperature of the calorimeter the ratio 7 Wl ff could be
determined. The value of a foot-pound or kilogram -meter depends on
the local value of g. The mean result of his work reduced to the
latitude of Greenwich was 772.5 foot-pounds or 424 k j m. The same
method on a more elaborate scale was tried by Professor Rowland, of
Johns Hopkins University in 1880. The paddle-wheel was driven by a
steam-engine and made n revolutions per second. The work done was
measured by a torsion wire, a weight w being placed at a distance r from
the axis to balance the force of friction. The work done was 2 tnwr, and
if C was the thermal capacity of the calorimeter and its contents,
(110) / = = **nwr = 42 7.3>E>=-4.188 X 10 7 ergs per calorie.
heat L(/, r )
Joule's value corrected for Baltimore gives 7~ 426. 75 / /;/.
For Berkeley the value 4. 19 X 10 7 ergs per calorie may be used. -
Joule and others have also determined the mechanical equivalent for
the heat developed by a given quantity of electrical energy in overcoming
r
HEAT.
the resistance of a wire. The best determination by this method was made
by Griffiths, with a result
(111) /=427.45 kjm for Greenwich = 4.193 x 10 7 ergs.
Other less satisfactory methods have also been employed, such as the
measurement of the heat developed by friction or percussion of metals,
work done in expansion, etc.
PROBLEMS.
66. How many ergs are there in a foot-pound ?
67. One hundred grams of lead at 99 are dropped into 100 grams of water
at 15 in a copper calorimeter weighing 40 grams. What is the final temperature
of the mixture ?
68. A mass of lead falls 100 meters on a plate of steel. How much will the
temperature of the lead rise ?
69. Calculate the value of J from the given specific heats of hydrogen.
70 . Ten grams of compressed air expands against a pressure of 5 atmospheres
until its volume is increased by lOOc.c. How much is it cooled ?
The effects of heat other than change of temperature will now be taken
up in detail.
138. Change of Yolume of Solids. Change of temperature is
usually attended by change of volume generally dilatation with increase
of temperature.
Measurement of Expansion. Except in the case of gases, the change in
dimensions is usually small and requires special methods of measurement.
In measurements of lengths there are two principal methods that of
Lavoisier and Laplace, in which one end of a bar of the substance is kept
in a fixed position and the other end in contact with a magnifying lever.
The displacement of the lever may be directly observed or determined
from the angular displacement of an attached telescope and the elongation
of the bar computed; Roy and Ramsderi s method, in which the elon-
gation is directly measured by a micrometer microscope.
Coefficient of Expansion. If the temperature of a bar is raised to /,
its initial and final lengths being L Q and L and its average increase in
length per degree /
L-L = U
L = L = (l+ T -t)=L.(l + af),
in which a is called the mean coefficient of linear expansion between the
limiting temperatures, or the average increase of unit length for one degree.
Isotropic (non-crystalline) bodies expand uniformly in all directions.
If we consider a rectangular surface
S S = L (1 + at) t (1 + O LL l = S (2 at + a*t*}.
When a is very small the last term may be neglected, and
(113) 5- S (1 + 2 of) = S (1 + 6f).
where b is the mean coefficient of superficial expansion.
Similarly, the mean coefficient of cubical expansion is c = 3 a.
(114) V= F
CHANGE OF VOLUME OF SOLIDS. 73
In general, these coefficients are functions of the temperature, although
the change in value within ordinary ranges is very small.
Below are given the linear coefficients of expansion of some substances.
The superficial and cubical coefficients are obtained from these by multi-
plying by 2 and by 3 :
Glass, 0-100, 0.0000086 Brass at 40, 0.0000186
0-200, 92 Diamond, 5 i2
0-300, 101 Ebonite, 770
Platinum at 40, 89 Paraffin, 2785
Iron 122
Glass and platinum have practically the same coefficient, so that when
wires have to be sealed in glass that metal is generally used.
As a rule, coefficients of expansion increase with temperature as
measured by the gas themometer. A general expression for the volume
at a given temperature would be
(115) V= V (l + At+Bt*+ 3 +- ),
in which A, B, and C are small constants to be determined from experiment.
Great force is exerted by thermal changes of length of volume. A rod
contracting a given length when cooled exerts the same force that would
be required to stretch it the same amount, as determined by Young's
modulus.
The coefficient of expansion of a given substance is not always the
same for different specimens; it depends somewhat on temper, im-
purity, etc.
Owing to their different rates of expansion, bars of different materials
riveted together will bend one way or another with change of tempera-
ture. Breguet's thermometer is such a compound ribbon in spiral form,
which twists or untwists with change of temperature.
In Harrison's compensated gridiron pendulum the length of the
pendulum is constant if Z, I? L 2 , and L 3 conform to the condition,
since these quantities represent changes of length in opposite directions; or
139. Anomalous Expansion. Rubber under tension contracts in the
direction of tension when heated, but it expands more at right angles; so that
on the whole there is an increase of volume. Iodide of silver is found to
contract when heated between 10 and 70, although chloride and
bromide of silver expand normally. In the case of the iodide some of the
coefficients in equation (115) are negative, so that the expression changes
sign about 60, showing that below that temperature expansion is normal.
Similar equations indicate that diamond and emerald have maximum
densities at 41. 7 and 4. 2 respectively; below those temperatures
they may expand when cooled; but we must be careful in drawing
inferences beyond the range of experiment. Such anomalous cases are,
no doubt, the result of crystalline structure.
Alloys often show anomalous expansion. Nickel-steel (36$ nickel)
has a coefficient of only 0.000001, which makes it useful for measuring
instruments.
74 HEAT.
Expansion of Crystals. In general, the coefficients of expansion of
crystals differ along different axes, being the same in all directions only in
the isometric system ; consequently a sphere cut from a crystal will in
general assume an ellipsoidal shape with change of temperature. In some
cases the coefficient along one axis may be negative ; for example :
Beryl Iceland spar
Along principal axis + 0.000001722 + 0.0000263
Perpendicular to principal axis - 0.000000134 0.0000031
In every case (with the exception of iodide of silver) there is increase
of volume of crystals with rise of temperature.
140. Expansion of Liquids. As a rule, the rate of expansion of
liquids is greater than that of solids. They must be contained in solid
vessels, so that allowance must be made for the expansion of the vessel in
determining the coefficient. This is avoided in the method of balanced
columns, used by Dulong and Petit and by Regnault. One form of
apparatus used by the latter consists essentially of a pair of U-tubes, the
short arms of which are in communication with an air chamber in which
any desired pressure may be maintained, while a small horizontal tube
connecting the longer arms keeps the mercury in them at the same level.
The two tubes are placed in separate baths of water or other liquid which
may be kept at any desired temperatures. When in equilibrium,
h,P, = h 2 P 2 =p /g.
But P.Cl + 'O^a+'O^o-
Therefore, r (l + cQ = /i 2 (l + rf x ) .
Regnault found for the mean coefficient of dilatation of mercury
between and f
c = 0.0001791 + 0.000000025/,
which is practically constant between and 100.
The Weight Thermometer. A glass bulb is filled with M grams of a
liquid at 0. The bulb is then placed in a steam or water bath at temper-
ature t and m grams of the liquid are driven out by expansion. Assuming
no change in the volume of the bulb and calling the apparent coefficient of
expansion a
F = M/P O = (M- ni)/P-= (M ~ nt} (1 + at}.
^o
Therefore, a =
(M m) t
But if the glass has a coefficient g, and the true coefficient of expan-
sion of the liquid is c
(118) V- V Q (1 + cf) -=^I^L (1 + a f) (!+/)
and c = a +*. approximately.
Weighing. Hallstrom and Matthiessen determined the density of
liquids by weighing in them at different temperatures a cube of glass of
known coefficient. The loss of weight of the glass being equal to the
EXPANSION OF LIQUIDS EXPANSION OF GASES. 75
weight of an equal volume of the liquid, its density can be determined at
any temperature, and the coefficient of expansion computed.
The expansion of water is anomalous. It contracts when cooled down
to about 4 C. ; below that point it expands until it freezes, after which it
contracts normally with reduction of temperature. The density of water
at various temperatures was determined by Hallstrom by the method of
weighing a piece of glass in it. He found the following to express the
density at any temperature / in terms of the density at zero :
(119) P = P O (! +0.000052939* 0.00000653/ 2 + 0.00000001445/ 3 ),
which gives a maximum density at 4. 118.
Despretz determined the density directly by a water thermometer. A
curve is drawn with apparent volumes and temperatures as coordinates.
Another curve is drawn showing the change in volume of the glass. A
third curve, whose ordinates are the sum of those of the first two, gives
the true expansion of water, and a minimum is found at 4. 007. In
another experiment Despretz used four thermometers, placed at different
depths in a vessel of water, which was then cooled. Curves were drawn
for each thermometer, with temperatures and times as coordinates. The
lower thermometer came to a stationary condition when the water around
it came to the maximum density ; likewise with the others, until the tem-
perature of the top thermometer reached the same point, when all began
to fall. The common intersection of the curves shows the temperature of
maximum density, which was found to be 3. 974.
The expansion of water in freezing plays an important part in disin-
tegrating rocks and soil. The fact that the temperature of maximum
density is above the freezing point prevents bodies of water from freezing
solid. It has been suggested that the expansion of water below 4 is due
to the gradual formation of ice crystals at that temperature. The
readjustment of the molecules in the crystalline configuration requires
greater space. Increase of pressure (which would naturally retard the
formation of these crystals) lowers the temperature of maximum density.
For 93 atmospheres the temperature is 2; at 145 atmospheres, 0.6.
In saline solutions the point of maximum density (as well as the
freezing-point) is lowered proportionally to the amount of salt in solution.
Some results for sodium chloride solution are given below :
Percentage of salt Maximum density Freezing-point
4
1 10.77 0.65
4 5. 63 2. 60
8 16. 62 5. 12
The behavior of bismuth is similar to that of water.
141. Expansion of Oases. It has been shown that all gases at
constant temperature very closely conform to Boyle's law: pv = constant.
It has also been demonstrated that the pressure of a gas may be explained
as the result of molecular impact, and that it is directly proportional to
molecular kinetic energy. If either the pressure or the volume of a gas
be kept constant, the other factor in the above expression varies in the
same arithmetical progression ; therefore the volume of a gas, if the
pressure be kept constant varies as the molecular kinetic energy. Experi-
ment also shows that changes in the kinetic energy of a gas (measured
76 HEAT.
by pressure) are proportional to the increments of heat. Joule and
Thomson proved that when a gas expands into a vacuum (thus doing no
external work) its temperature is very slightly changed, showing thai
there is practically no internal molecular work. Nevertheless, we can
not say that the increase in molecular kinetic energy is equal to the
amount of heat absorbed, for the reason that some work may be done
inside the molecules, on the atoms. Since, then, the changes in volume
of a gas at constant pressure are proportional to the amount of heat added,
a gas thermometer has a decided advantage over one of any other sub-
stance, in which changes in volume are not proportional to the absorption
of heat. Moreover, all gases have practically the same rate of expansion,
and we can, if we choose, use variations in pressure at constant volume to
determine temperatures.
The statement of Boyle's and Charles' laws combined is :
If p be kept constant,
(120) a = (coefficient of expansion).
^o^
If v be kept constant,
(121) a 2 = j (coefficient of increase of pressure) .
Variations from the gaseous laws are indicated by differences in these
coefficients. In a perfect gas they would be the same. Measurements of
the coefficients have been made by Gay-Lussac, Dalton, Magnus,
Regnault, Amagat, and others. Some of Regnault's results are given
below :
a x (atmospheric p.). a 2 (in neighborhood of at. p.).
Air .................. 0.0036706 0.0036650
Hydrogen ......... 36613 36678
Carbon dioxide... 37099 36896
Sulphurous oxide 3903 3845
Regnault's experiments show that the more highly rarefied the gas
and the higher the temperature, the nearer a^ and a 2 approach the same
value for all gases that is, the more nearly Boyle's law is obeyed.
142. Absolute Zero. In the expression pv=p v (\+ af) the
second term becomes zero at a temperature t = 1 / a centigrade. This
may be interpreted to mean that if a gas strictly obeyed Boyle's law at all
temperatures, the pressure of a finite volume of the gas would become
zero. It is impossible to conceive of a lower temperature than this,
which is accordingly called the absolute zero of temperature. Substi-
tuting values of a determined by experiment at different temperatures in
the above expression, Regnault found the following values for the absolute
zero: Air at 76 cm. pressure, t = 272. 85; at 149 cm. pressure,
^ = 272. 70. Corrections for deviations from Boyle's law make the
value of / about 273. Reckoning temperatures from the absolute zero,
we have:
(122) pv = ap v (\ / + = ap v T= mR T,
in which T represents temperature on the absolute scale.
INTERNAL WORK FUSION AND SOLIDIFICATION. 77
Gas thermometers are of two kinds constant pressure, in which
variations of volume, and constant volume, in which variations of pressure,
are proportional to the absolute temperature. As a rule gas thermometers
are used only for correcting mercurial thermometers, which are directly
employed in scientific work.
The coefficient of expansion of liquids near the point of vaporization
may exceed that in the gaseous state. For example, Thilorier found for
carbonic dioxide between and 30 C. for liquid, #=.016; for gas, .0037.
Wroblewski found for liquid oxygen at 139, a = .017; liquid nitrogen at
154, # = .0311, these values greatly exceeding those for the gases.
Dewar found for solid hydrogen a coefficient of expansion ten times that
of the gas.
QUESTIONS AND PROBLEMS.
71. Calculate the change in length of the railroad tracks between Berkeley
and Oakland (say 5 miles) caused by a temperature change of 20.
72. What change of period of a brass seconds pendulum is caused by a change
of 10 in temperature ?
73. In a compound pendulum of brass and iron the iron grids are in the aggre-
gate 100 cm. long. How long must the brass grids be to secure compensation?
74. How is the principle of expansion utilized in putting on wagon tires; in
fitting together the concentric steel cylinders of modern cannon; in heating rivets
before hammering them in ?
75. What is the volume of a gram of air at 20 and under 100 cm. pressure ?
76. A mass of aluminum is counterbalanced by a brass kilogram weight when
the temperature is 20 and the pressure 76 cm. What is the weight of the alumi-
num in vacuum ?
CHANGE OF STATE.
143. Internal Work. In the case of gases the absorption of heat
results merely in increase of temperature if no external work is done, no
energy being expended in overcoming molecular forces. In the case of
liquids and solids heat is expended not only in increasing kinetic energy
of the molecules, but in overcoming their constraints. In general, at a
certain temperature depending on the substance and the conditions, a
point is reached beyond which no increase of molecular agitation can take
place without disruption; or, if the substance is cooling, the diminished
motion of the molecules allows them to yield to their mutual attractions.
At this point a change of state takes place, heat being absorbed or given
out by the internal work done upon or by the molecular forces. As a
rule, during a change of state there is no change of temperature; hence
the erroneous name ' ' latent ' ' applied to the heat transformed to work or
conversely. The name was first used by Dr. Black about 1757, on the
assumption that heat is a material fluid.
144. Kinds of Change. Fusion or solution, passage from solid
to liquid state; converse operations, freezing and crystallization. Vapor-
ization, liquid to gas, or sublimation, solid to gas; converse, condensation.
Other less definite modifications due to heat are molecular dissociation
and ' ' allotropic ' ' changes of structure.
145. Fusion and Solidification. All known elementary sub-
stances except carbon and molybdenum have been melted. There are
two classes of substances with regard to condition of melting. The first
class, comprising all substances of crystalline structure, have a definite
melting-point, the temperature remaining constant until the process is
78 HEAT.
complete (ice, sulphur). The second class comprises most non-crystalline
substances, such as wax, glass, iron. They gradually soften on the appli-
cation of heat, the temperature rising continuously until fusion is com-
plete. The following laws apply to the first class:
Laws of Fusion. 1. All crystalline substances have a definite melting
(freezing) point. 2. Until the process is completed the temperature
remains constant. 3. Each unit mass of the substance in melting (or
freezing) absorbs (or evolves) a definite amount of heat, called its latent
heat of fusion.
146. Surfusion. Under certain conditions liquids may be cooled
below their normal freezing-points without freezing. Fahrenheit cooled
pure water in a bulb to 13; it froze on breaking the bulb. Gay-Lussac
reduced the temperature of water beneath a layer of oil to 12; agita-
tion caused immediate solidification. Melted phosphorus may be cooled
far below its freezing-point, but immediately solidifies on introducing a
particle of solid phosphorus. In all cases of surfusion, agitation or the
addition of a particle of the solid causes solidification. Perhaps small ice
crystals are already in suspension, which are thus brought together.
When freezing sets in, the liberation of latent heat at once brings the
temperature up to the normal.
Melting-point Latent h ,f r at j n calories
per >rcirrl.
Hydrogen 258
Chlorine 102
Mercury 39 2.8
Water 80.0
Phosphorus 44 4.7
Sulphur !.. 115 9.4
Bismuth 264 12.4
Lead 328 5.3
Antimony 425
Copper 1082
Cast iron 1000-12001
Steel 1300-1400 I Gradual
Iron (wrought) 1500-1800 [softening.
Platinum 1800-2200 j
Iridium 1950
The possibility of welding depends upon plastic condition before
fusion, as in the case of iron, platinum, and glass.
147. Alloys. Many alloys, like amorphous solids, soften gradually,,
the most fusible constituent melting first. Others have a fairly definite
melting-point, usually lower than that of any of their constituents. For
example, Wood's fusible metal composed of bismuth, 4 parts; lead, 2;
cadmium and tin, 1 each melts at about 60. Rose's fusible metal melts
at 94. Steel and cast iron, containing carbon, melt at a much lower
temperature than wrought iron. Alloys may be considered as solutions of
solids in solids. In all solutions there is a lowering of the freezing-point.
148. Change of Volume. In most cases fusion is attended by an
increase of volume; but there are a few exceptions, such as ice, bismuth,
antimony, brass, iron. Metals like these are best for casting, as they
give sharp impressions. For this reason bismuth and antimony are used
in type metal. In general, the rate of expansion of the liquid is greater
than that of the solid.
SOLUTION AND CRYSTALLIZATION. 79
149. Effect of Pressure. A change of pressure in general affects
the freezing-point slightly. Water expands on freezing ; therefore an
increase of pressure opposes solidification and lowers the freezing-point.
The lowering due to 1 atmosphere pressure is 0.0072. Mousson inclosed
water in a stout steel tube, with a piece of copper at the bottom. The
water was frozen at 20. The tube was inverted and a pressure of
13 * 000 atmospheres applied. On removing the pressure and opening the
tube the water was frozen; but the piece of copper was at the end opposite
to that originally occupied, showing that the ice had melted. Pressure
has an opposite effect on substances which expand when melted. In such
cases increase of pressure tends to maintain the solid state, and the melting-
point is raised, Paraffin melts at 46 under ordinary conditions and at
50 under 100 atmospheres pressure. It is believed that the rocks in the
interior of the earth may for this reason remain solid even at temperatures
far above their normal melting-point.
150. Regelation. When two pieces of ice are pressed together they
are liquefied at the bounding surface ; removing the pressure, they freeze
together. Faraday called this regelation. This is illustrated by snow-
balls, which can be squeezed into compact masses of ice, or by attaching
weights to a wire loop surrounding a block of ice. The ice melts under
the wire, flows around, and freezes, so that when the wire has worked its
way through, the block is still solid. The flow of glaciers is partly due
to the same cause. At any point where pressure exists, such as at the
bottom or sides, the ice is melted, flows around the obstacle, and again
freezes. Their motion is also partly due to expansion downward by the
sun's heat, followed by contraction in the same direction.
151. Solution and Crystallization. The liquefaction of a salt
by solution is essentially the same process as fusion, and is accompanied
by an absorption or evolution of latent heat, although this result may be
masked by thermal changes due to chemical action. The apparent latent
heat may thus be either positive or negative, but the true latent heat is
always negative (absorbed on solution).
152. Freezing-points of Solutions. It is found that the freezing-
points of solutions are lowered in proportion to their concentrations (so
long as these are small), the forces between dissimilar molecules prevent-
ing the formation of crystals. Raoult found about 1882 that in most
cases, especially in solutions of organic substances, such as sugar, the
lowering of the freezing-point was the same when concentrations of the
solutions of different substances were made proportional to their molec-
ular weights, (equimolecular). This shows that all molecules, whatever
their nature, have the same effect in lowering the freezing-point. This is
approximately true for some alloys. The divergences from this law in the
case of solutions of inorganic salts or acids have been explained as the
result of partial dissociation, each atom having the same effect as an entire
molecule. These deviations are of the same magnitude as the deviations
from normal osmotic pressure ascribed to the same cause. (See section
103.) This principle is useful in comparing molecular weights. W Gen-
erally when a solution is frozen the ice excludes the salt, except small traces
mechanically suspended. Ice from a colored solution is colorless.
Freezing Mixtures. If a salt be mixed with broken ice, a solution
will be formed owing to the breaking down of the crystals by the attrac-
(1) See Jones, Theory of Solutions; Ostwald, Solutions ; Whetham, Solutions.
VAPORIZATION AND CONDENSATION. 81
that point. At a certain temperature the vapor pressure will equal this
sum, and from that time there will be formation of vapor bubbles through-
out the mass. This distinguishes boiling or ebullition from evaporation.
Latent Heat. When a liquid begins to boil, the heat which is supplied
to it is wholly used in doing work against molecular forces and the
temperature of the liquid remains constant. Of course some heat dis-
appears in this way in evaporation at lower temperatures.
156. Laws of Vaporization. 1. Liquids evaporate at all tempera-
tures. Every liquid begins to boil when the temperature is such that the
pressure of its vapor begins to exceed the external pressure 2. So long
as the pressure remains constant the temperature remains constant.
3. To change a unit mass of liquid to vapor (or conversely) a definite
amount of heat is absorbed (or evolved) , called latent heat of vapori-
zation.
The normal boiling-point of a liquid corresponds to a pressure of 76
cm. of mercury. Below are some values of the normal boiling-point.
Hydrogen -258 Water 100
Oxygen -181.4 Mercury 357.25
CO 2 -80 Sulphur 448.4
SO 2 -10 Zinc 891-1040
Ether +35 Lead 1500
Alcohol 78
157. Distillation. Mixed substances having different boiling-points
may be separated by maintaining the temperature of the mixture just
above the boiling-point of one constituent until it has all been vaporized ;
then raising the temperature just above the boiling-point of the next and
so on. Some of the substances having higher boiling-points pass off by
ordinary evaporation. At each repetition, however, the fraction is reduced ;
hence the term fractional distillation. In similar chemical compounds
there is often a definite relation between boiling-points. For example, in
certain homologous carbon compounds there is a rise of about 19 for
each CH 2 radical added to the molecule.
158. Superheating. Distilled water free from air may have its
temperature raised above the boiling-point without ebullition. Mechanical
disturbance or the introduction of impurities cause boiling with explosive
violence, and the temperature then drops down to the normal. This is
illustrated in the "bumping" of pure water in a clean glass beaker.
The introduction of sand or broken glass causes normal boiling. Air
bubbles or sharp points seem to act as nuclei for the aggregation of vapor
molecules into bubbles. Drops of water in a mixture of linseed oil and
oil of cloves have been raised to 178 without boiling. In the last case
surface tension plays an important part. In addition to the external
pressure, the vapor must overcome the pressure p = 2 T j r. (Surface
tension promotes surface evaporation from a convex surface, but boiling
takes place internally. See section 166.) The steam from boiling water
always has the same temperature under the same pressure ; hence in
calibrating thermometers they are suspended in steam.
159. Supersaturation. Aitken has shown that if a vapor be entirely
free from suspended dust particles or water drops, it may be cooled
considerably below its ordinary condensing point without condensation.
Such particles seem to act as nuclei for the formation of drops. Surface
tension plays some part in this. (See section 166.)
82 HEAT.
160. Boiling-point of Solutions. It is found that the boiling-point
of solutions is raised in proportion to their concentrations. The same
attraction between dissimilar molecules which promotes solution of solids
resists their separation by vaporization. Raoult found the same law to
apply as in the case of lowering of the freezing-point. In solutions of
different salts of concentrations proportional to their molecular weights
the raising of the boiling-point is the same, or each molecule has the same
effect, whatever the substance. Since only pure-water vapor passes off,
with a pressure corresponding to the temperature, we see that for a given
temperature of solution there is a lowering of the vapor pressure propor-
tional directly to the concentration (so long as this is not great) and
inversely to the molecular weight. This principle is used in the com-
parison of molecular weights. It applies also to the vapor pressure of
solutions of some metals in mercury (amalgams) . W
161. Spheroidal State. A small quantity of liquid placed on a
surface heated considerably above its boiling-point does not immediately
boil away, but assumes a spheroidal shape, vibrating and rolling on the
surface. This phenomenon was first investigated by Leidenfrost and later
by Boutigny. Close observation shows that the drop is separated from
the surface by a cushion of vapor. It has been found by introducing the
bulb of a small thermometer into a drop that the temperature of the lower
part did not exceed 98 and that of the upper about 90, showing that the
conduct of the substance is not due to the assumption of a new state, as
once supposed. If liquid sulphur dioxide be placed in red-hot crucible it
will assume the spheroidal state at a temperature below the freezing
temperature of water. If a small quantity of the latter be poured above
it, the water will be frozen. In the same way the hand, if wet, may be
plunged into melted metal without injury, being protected by the vapor.
162. Sublimation. Ice and snow in the arctic regions evaporate with-
out melting ; in this case the temperature is too low for the liquid to exist.
Under ordinary conditions arsenic, camphor and iodine will pass directly
into vapor ; by increasing the pressure they may be made to pass through
the liquid state. Carbon will vaporize in the electric arc, but has never
been liquefied. In general, sublimation occurs when the pressure of the
vapor at the freezing-point is greater than the external pressure. Under
such conditions the liquid cannot exist. Substances which sublime, such
as salammoniac, show phenomena resembling those of the spheroidal state.
163. Latent heat of vaporization is usually determined by the
method of mixture. It is found to be variable with the temperature at
which boiling occurs. For water Regnault found that the expression
(123) Z, = 606.5 0.695 1 0.00002 1 2 0.0000003 1*
holds up to t 230. This expression vanishes at 706, indicating a
direct passage from fluid to gaseous state without expenditure of work
(see section 172). Some values for latent heat of vaporization under
ordinary conditions are: Water, 536; alcohol, 202; ether, 90.4.
164:. Cooling Due to Evaporation. When evaporation takes
place without a supply of heat from an external source, cooling of the
liquid and its surroundings takes place. Water in porous earthenware
vessels, shallow dishes, or canvas-covered jars (ollas) is cooler than if kept
inclosed. Water will freeze under the exhausted receiver of an air purnp^
(' ) S_c w -rks on Solutions by Jones, Ostwald and Whet>am.
84 HEAT.
of radius r, and shows that the equilibrium pressure around a drop is
greater than for a plane surface ; hence a drop, if very small, will evapo-
rate even into saturated space. This shows why condensation is aided by
dust particles, etc. , (the moisture spreading over the surface forming a
larger effective drop), and why air bubbles promote ebullition (the equilib-
rium pressure of the vapor within them being less than that corresponding
to the existing temperature). It is to be noted that although evaporation
is promoted by a convex surface, boiling is not, the inclosed bubbles being
subjected to a total pressure of P + 2 Tl r, P being the external pressure.
(See section 158.)
167. Daltori s law applies to mixtures of saturated vapors and gases
when they exert no chemical action on each other, so long as the pressure
is not very great. The pressure of aqueous vapor, for example, is not
appreciably greater in a vacuum than in air at ordinary pressure. The
process of vaporization is somewhat delayed by air, diffusion being
hindered.
168. Fog) Clouds^ Rain, Dew. If a superheated vapor be cooled
below its point of saturation, condensation will occur. When a mass of
air is cooled by passing from above warm ocean currents to colder land
or water, or from heated land to cold water, or by expanding when it
rises to regions of lower pressure, fogs or rain are caused by the precipi-
tation of the suspended water vapor. The formation of fogs is aided by
suspended particles of dust or smoke, which act as nuclei of condensation.
This is the principal cause of the dense fogs of London. Clouds are formed
in the same way as fogs, but at greater altitudes. The particles of con-
densed vapor slowly descend, their progress being' impeded by viscosity; but
as they reach warmer air strata they again vaporize, so that a cloud appears
stationary, although constantly falling and reforming. When a great quan-
tity of water vapor is in suspension and the cooling very great, the particles
form drops and fall as rain. Mountain ranges promote rainfall on the ocean
side in two ways: 1, If they are colder than the air coming from the
ocean the water-vapor is condensed; 2, the deflection of the air-currents
upward to regions of lower pressure causes expansion, followed by cool-
ing and precipitation (see section 135). The climate on the inland side is
usually dry. It was once supposed that dew was a gentle rain, but Dr.
Wells, a London physician, who carefully studied the matter about 1818,
showed this to be a mistake. He pointed out that dew was most abun-
dant on clear nights; that some classes of objects would be covered by
dew, while neighboring objects would be dry, and that the objects collect-
ing most dew were invariably those having great radiative power and low
specific heat that is, those objects which cool most rapidly at night.
These facts show that it is the result of local precipitation on objects
cooled below the point of saturation of the surrounding space. Screens,
smoke, and clouds hinder the formation of dew by preventing radiation
into free space, and wind by preventing the air from remaining in contact
with the objects long enough to be cooled below the point of saturation
"dew-point" Grass has great radiative power, thus collecting quantities
of dew. A thermometer placed on grass may indicate a temperature 10
lower than one a few feet above it. The formation of dew is illustrated
by the condensation on pitchers containing cold water. (1 ^
169. Hygrometry is the measurement of the amount of moisture in
(1) See works on Meteorology by Davis, Greeley, Waldo, Russell.
ISOTHERMAL CURVES. 85
the atmosphere. The absolute humidity, or the mass of water vapor
present in a given space, is determined by drawing a known volume of
air' through calcium -chloride tubes and noting the increase of weight.
Our perception of moisture in the atmosphere does not depend, however,
upon absolute humidity, but upon relative humidity* which is measured by
the ratio of the pressure of the vapor present to that which accompanies
saturation at the same temperature. The dew-point may be determined
with various forms of hygrometer, the best of which is Regnault's. A
polished silver tube contains ether and a thermometer. The ether is
caused to evaporate by blowing air through it, and the temperature noted
at which dew forms on the tube and when it disappears. The mean is the
dew-point. If p is the saturation pressure corresponding to this tempera-
ture and P that corresponding to the temperature of the atmosphere, the
relative humidity is pjP. The wet and dry bulb thermometer is also
much used. One thermometer has a wet cloth around its bulb and is
cooled by evaporation, the rate of which depends on the amount of
aqueous vapor present. The amount of aqueous vapor is then determined
by an empirical calibration.
Sultry and oppressive weather is due to great humidity, preventing
evaporation of perspiration.
170. Vapor Density The density of a vapor is, strictly speaking,
the mass of unit volume under standard conditions. The name is often
applied, however, to the specific gravity of a vapor, with dry air at same
pressure and temperature taken as the standard. If the density of the air
at and 76 cm. pressure is P O under other conditions
(125) P
7760
7" being the absolute temperature. In the various methods of determin-
ing vapor density a vessel of known volumn is filled with superheated
vapor and its mass m determined ; then D = vpjm. This gives the relative
vapor density at the temperature and pressure of the experiment. If the
vapor obeys the laws of Boyle and Charles (which most vapors approxi-
mately do) above the saturation-point, the same value holds for all other
conditions. The density of water vapor is about $/% that of air, so that it
is necessary to make corrections for the hygrometric state in hysometrical
observations.
PROBLEMS.
77 .yu-What is the weight of a liter of water vapor at 20 and at 10 cm. pressure ?
78. The dew point of air at 18 and 76 cm. pressure is 5. What is the weight
of water vapor per liter ?
79. At what temperature will water boil on Mount Tamalpais (3000 feet) ; on
Mount Whitney, (15,000 feet)?
171. Isothermal Curves. The locus of the points of intersection
of rectangular coordinates representing pressures and volumes (tempera-
tures remaining constant) is called an isothermal curve. For a perfect
gas, Boyle's law shows us that the isothermals would be rectangular
hyperbolas ; for saturated vapors they are straight horizontal lines ( / con-
stant) ; for unsaturated vapors they approximate rectangular hyperbolas ;
for solids and liquids they are nearly vertical lines (v nearly constant).
86 HEAT.
V
A set of isothermals showing the pv relations for a wide range of tempera-
tures gives us a complete history of the conduct of the substance with
regard to changes of state.
(Lines of equal temperature on the earth's surface are also called
isothermals).
172. Continuity of the Liquid and the Gaseous States
Critical Temperature. In 1822 Cagniard de la Tour heated ether,
alcohol, and other liquids in sealed glass tubes to very high temperatures.
Under such circumstances ebullition is impossible, as any bubbles formed
within the mass are exposed to a pressure greater than that of the vapor
within them. There was a gradual evaporation, and at a certain tempera-
ture (about 200 for ether, 259 for alcohol) the meniscus suddenly
disappeared, leaving the tube apparently filled with a homogeneous mass.
De la Tour concluded that the liquid was entirely vaporized.
In 18.34 Thilorier liquified carbon dioxide by cold and pressure, and
in 1863 Dr. Andrews plotted its isothermal curves through a wide range
of temperature. The gas was compressed in a narrow glass tube by a
column of mercury, which also communicated with an air manometer
registering the pressure. These curves throw great light on the nature ot
gases and liquids. At 13. 1 (ordinary temperature) the substance dimin-
ishes in volume almost inversely as the pressure. When the latter reaches
about 50 atmospheres it remains constant until the volume is reduced to
about one-fifth, after which increase of pressure produces little cha nge in
volume. At 50 atmospheres the gas begins to condense to a liquid, this
being the saturation pressure corresponding to that temperature. The
curve for 21. 5 likewise has two sharp discontinuties and a horizontal
portion, but that for 31. 1 is quite different. It shows two inflections, but
there is no point at which the curve is horizontal, as would be the case
with a saturated vapor. At a pressure of about 73-75 atmospheres the
meniscus disappeared as in De la Tour's experiments. The curves above
31. 5 show less evidence of irregularity, and that for 48. 1 does not
differ greatly from the isothermal of an almost perfect gas such as air.
Andrews concluded that above the temperature of 33, which he called
the critical temperature, carbon dioxide cannot exist as a liquid. Later
researches by Amagat give the critical constants for this gas as 31. 35
and 72.9 at. Andrews investigated other gases in the same manner
with similar results. He inferred that there was "continuity of
state," or gradual passage from a liquid to a gas. About 1870
Cailletet and Collardeau studied the subject and concluded that the
disappearance of the meniscus did not prove that no liquid persisted.
Iodine dissolves in liquid carbon dioxide, but not in the gas , and they
showed that on partly liquefying the gas in a tube containing iodine the
latter was dissolved in the liquid ; on heating above the critical tempera-
ture, the line of demarcation remained unchanged. They concluded
that liquid was still present, although its surface tension had vanished.
It seems evident, however, that above the critical temperature the sub-
stance has properties different from those of either a liquid or a gas. It
is possible to change a substance from an undoubtedly gaseous to an
undoubtedly liquid state without any visible change. If carbon dioxide
is heated to 50, compressed under 100 atmospheres pressures, then
cooled under constant pressure to 25, it is certainly a liquid ; but no
visible liquefaction has occurred.
Surface tension diminishes as the temperature increases, vanishing at
LIQUEFACTION OF GASES. 87
the critical temperature. This means that the forces between the
molecules of the liquid and those of the gas are the same. The latent
heat of vaporization also vanishes, for the density of the saturated vapor
approaches that of the liquid near the critical point; so that no work is to
be done in separating the molecules. Regnault's equation (123) indicates
that the latent heat of water vanishes at 706, while its critical tempera-
ture is about 365; but the formula was deduced from observations
within a limited range and cannot be expected to hold for high tem-
peratures.
Distinction Between Gas and Vapor. A gas may be defined as a
vapor above its critical temperature; a vapor as a gas between its critical
temperature and the point of liquefaction. A vapor shows very marked
deviations from Boyle's law, but van der Waals' equation holds fairly
well, giving curves resembling those of carbon dioxide above the critical
temperature. Superheated water vapor approximately obeys the law
^17/16 = constant (Rankine).
173. Liquefaction of Gases. Faraday, between 1823 and 1849,
liquefied all the gas known to him except hydrogen, nitrogen, oxygen,
carbon monoxide, nitrous oxide, and methane. His method was to place
the gas in solution, or a substance from which the gas could be evolved by
heat, in one arm of a sealed U-tube, place the other arm (inverted) in a
freezing mixture, and evolve the gas by applying heat, thus producing
great pressure. The gas condensed in the cooled arm. It is evident
from Andrews' work that in order to liquefy a gas it is necessary to
reduce it below the critical temperature. The method adopted is based
upon the principle of freezing-machines. About 1877 Cailletet in Paris
and Pictet in Geneva both succeeded in liquefying oxygen, nitrogen, and
carbon monoxide. In Pictet' s apparatus liquid sulphur dioxide was allowed
to evaporate under low pressure around a tube containing carbon dioxide
under pressure. The latter was thus liquefied at a temperature of 70.
This was allowed to evaporate around another tube containing oxygen
under 275 atmospheres pressure, and the latter liquefied, at a tempera-
ture of 130. Still lower temperatures were produced by Wroblewski
and Olszewski, who began their investigations together about 1880, but
finally worked independently. They used apparatus similar in principle
to that of Pictet, but secured lower temperatures by evaporating liquid
ethylene ( 152), the resulting liquid oxygen evaporating at low
pressure condensed nitrogen at 200, the latter evaporating under low
pressure produced a temperature estimated at 225 and froze. Up
to 1895 hydrogen alone remained unliquefied, although Pictet and
Wroblewski, by using the greatest cold and pressure obtainable and
allowing it to expand suddenly, had produced a temporary mist. In
1895 Olszewski reduced this gas to the lowest temperature obtainable by
the evaporation of liquid nitrogen, and, allowing it to expand suddenly,
observed evidences of ebullition. Dewar, of the Royal Institution, has
recently done much work in liquefying gases. In 1898 he succeeded in
liquefying hydrogen. He found the constants given for this gas in the
table below. When liquid, its density was one-fourteenth and its surface
tension one-thirty-fifth that of water. At 258 it froze to a solid, with
a coefficient of expansion ten times that of gas, and a specific heat five
times that of water. The solid hydrogen in a vacuum is reduced by
evaporation to a temperature of about 260 or 13 absolute. It is
an ice-like solid having a density one-eleventh that of water.
88 HEAT.
. . Boiling P. at Freezing- Density
- nt - 1 Cnt - * atmos. P. point. at B. P.
H 246 15 252 258 0.070
N 146 35 194 214 0.885
O 115 50.8 181.4 1.124
Air 140 39 191.4
Argon 121 50.6 187 189. 6
CO 2 -f 31 73 80
Ether -f 194 35.6 + 35
All the above when liquefied are colorless except oxygen, which is
light blue.
These temperatures are to be regarded only as approximations. They
were determined either from a hydrogen or helium thermometer (which
probably deviates widely from Charles' law at these low temperatures) or
by a thermopile calibrated by a hydrogen thermometer.
Liquid air is now generally made by a purely mechanical regenerative
process,based on the fact referred to in Sec. 13y thalfj^ases on expanding
againsrvpressure become cooled. Air is first compressed by a steam
engine, and is then allowed to expand into a second reservoir, after being
cooled by passing through a cold-water jacket. In expanding it is further
cooled ; it is then pumped back into the first reservoir, through a pipe
concentric with and surrounding that through which it entered the second
reservoir. It then cools the second installment of air to a lower tempera-
ture that it had itself ; and thus by successive steps the air is cooled until
a part of it becomes liquefied. This method was first applied by Linde,
of Munich.
Helium is the only known gas which has not been liquefied. Dewar
believes that its critical temperature lies near 5 absolute.
References. Dewar, Scientific Uses of Liquid Air, London Electrician, July
12, 1895 ; Presidential Address at British Association, Nature, Vol. 66, p. 462, 1902,
and Chemical News, Vol. 86, p. 127, 1902 ; Harden, Liquefaction of Gases ;
Sloane, Liquid Air. For a description of Linde's machine, see Watson's Physics.
OTHER EFFECTS OF HEAT.
174. Changes in Molecular Constraints and Structure.
Certain continuous changes, less sharply defined than those of state, are
produced by heat. As temperature rises
Elasticity diminishes. The percentage diminution in Young's modulus
between 0and 100 is: Platinum, .89; iron, 2.35; brass, 4.4; aluminum,
19.5. Dewar observed a great increase of elasticity and strength in metals
at the low temperatures produced by evaporating liquid air.
The viscosity of liquids diminishes, owing to greater molecular free-
dom ; for water at 50 it is less than one-third that at 0.
The viscosity of gases increases from the same cause, since the motion
of one stratum over another is retarded by interchange of molecules.
Surface tension diminishes. According to the results of Frankenheim,
if h be the height in a capillary tube at
(126) h>-=h (l-ct).
Some values for c are: Water, 0.0019; alcohol, 0.0024; ether,
0.0047. Assuming these results to hold for any temperature (which is
not quite legitimate, as they were deduced between and 100), the
critical temperature = 1 / c would be : Water, 526; alcohol, 417; ether,
213, which shows a close agreement with facts only in the case of ether.
CHEMICAL COMBINATION AND DISSOCIATION. 89
Gradual or abrupt changes in molecular structure may be caused by
heat. Sulphur heated beyond 250 and suddenly cooled by pouring in
water becomes a viscous tenacious mass. It slowly returns to its normal
form, or if heated to 100 it returns suddenly with the evolution of heat.
If iron is heated to white heat and allowed to cool slowly, at a dull red it
begins to glow more brightly, showing an evolution of heat due to some
internal change (recalescence).
175. Chemical Combination and Dissociation. Often chem-
ical combination takes place at a high temperature. At a still higher
temperature dissociation may result, as in the case of iodine, sulphur and
other elements. This is shown by a gradual decrease in the density due
to the splitting of molecules into atoms. According to the modern
theories a fraction of the molecules of a gas or an electrolyte in solution is
always dissociated, the proportion increasing with the temperature. At
any given temperature, recombination balances dissociation due to
impact, etc., producing a state of kinetic equilibrium as in evaporation. In
this way abnormal gaseous and osmotic pressures, lowering of the freezing-
point, raising of the boiling-point, and electrolytic conduction can be
explained.
Latent Heat of Combination. Dissociation involves work done upon
molecules, or an absorption of heat ; combination, work done by them in
falling together, or an evolution of heat. The heat in calories evolved by
one gram of some substances burning in oxygen is as follows : Hydrogen,
34,000; marsh gas, 13,000; ether, 9,000; coal, 8,000; wood, 4,000;
iron, 1,500. These figures area measure of thermal efficiency, but not of
economic efficiency, which depends on prices. In the transformation of
oxygen and hydrogen to one gram of ice, the amount of heat given out is
3,777 + 536 + 80 = 4,393 calories, and the absorption of the same
amount will return the elements to their original state.
All the effects of heat thus far described are in accordance with and are
best explained by the dynamical theory of heat.
176. Electrical Effects. If a circuit be made of two or more
different metals soldered together, any difference of temperature between
the junctions will cause a flow of electricity, called a thermo-electric
current. If the temperature of an electrical conductor changes, its
resistance also changes, usually increasing with the temperature.
177. Measurement of High and Low Temperatures. The
expansion of a substance (Wedgewood pyrometer) is a doubtful measure
of high temperature on account of the unknown rate of expansion ; so are
methods depending on specific heat. Violle determined the temperature
of the electric arc by knocking a button of carbon from the hot portion
into a calorimeter and calculating the original temperature from the
specific heat and change in temperature. The specific heat of carbons at
high temperatures is unknown; consequently the result, 3,500, is
uncertain. This is probably the highest temperature attained by us,
except, perhaps, that of the electric spark. The most accurate methods
for very low or very high temperatures are gas thermometers in porcelain
bulbs ; platinum thermometers based on changes of electric resistance,
which read accurately up to 1,300; thermo-electric couples of platinum -
iridium, with which Barus read up to 1,700.
Differential Measurements. With a sensitive galvanometer and a
thermopile (thermal couple) differences of .001 may be detected, and by
90 HEAT.
Langley's bolometer, which utilizes the change of resistance in a thin plati-
num wire, differences as small as .000001. Equally sensitive is Boys'
radiomicrometer, made by suspending a thermo-electric circuit in a mag-
netic field. The most sensitive instrument, however, is a modification of
the Crooke's radiometer.
TRANSFER OF HEAT.
Experience shows that if two or more bodies are at different temper-
atures, there is a tendency toward equalization of temperatures, whether
they are in contact or not. The transfer of heat takes place by three
distinct but generally cooperating methods radiation, convection, and
conduction.
178. Radiation. It is found that isolated bodies at a higher tem-
perature than their surroundings lose heat even in a vacuum. It is
assumed that the vibratory motion of the molecules is imparted to the
surrounding ether, and these waves, striking other objects, set their
molecules also in vibration. Although radiant energy is capable of trans-
formation to heat, it is not itself heat. The ether or other transparent
medium is not warmed by it. Radiation is therefore more properly
discussed under the head of Light.
179. Convection. If a region within a fluid acquires a higher tem-
perature than that of the surrounding fluid, its density becomes less
(except in a few cases such as that of water below 4), and the heated mass
rises. Its place is taken by the colder fluid around it. By this process
of convection circulation is thus established, motion being from the colder
to warmer regions below and from warmer to colder above. Topler's
"schlieren" method of visibly projecting such phenomena on a screen by
passing light from a point source through the region of circulation shows
these convection currents rising from flames or warm objects in air or
through fluids heated at the bottom of the containing vessel, and the
diffusion currents in solutions.
Convection phenomena play an important part in modifying climate.
The heated air at the equator rises, flows toward the poles, and descends
in temperate latitudes; its place is taken by colder air passing to the equa-
tor. Such currents, modified in direction by the rotation of the earth,
give rise to the trade winds. Cyclones, tornadoes, and lesser wind disturb-
ances are caused in a similar way. The water of the oceans behaves in
the same manner, expanding several feet higher than its normal level at
the equator, and flowing down hill to the poles in various currents, which
are modified in their directions by the conformation of continents and
islands, as well as by the rotation of. the earth. The air above these warm
currents becomes healed, and may be carried by the wind to land, thus
modifying temperatures near the coast.
Convection has its practical applications in ventilation and the heating
of houses by hot water and air.
In the case of convection, masses of heated fluid are transported
bodily, carrying the heat with them. The transfer of heat (kinetic
energy) from one molecule to another is by radiation or conduction.
180. Conduction. If a solid such as a metal rod have one end
inserted in a flame, that end will rapidly rise in temperature. The whole
rod will in time become heated, but more and more slowly and with a
CONDUCTION. 91
lower final temperature, as the distance from the heated end increases.
We may imagine that the nearer molecules have their kinetic energy
increased, and transfer a portion of this energy, by impact or otherwise,
to adjacent molecules until the effect is felt at the other end of the rod.
Each molecule passes on only a fraction of its gain in energy, so that the
temperature diminishes in going from the source. After a time the loss at
a given point due to radiation and conduction balances the gain, and a
condition of steady flow is reached. If a curve be plotted with tempera-
tures and distances along the rod as coordinates, and \idt and dx be small
increments of / and x, the ratio dt / dx = tan is called the temperature
gradient.
Therrnometric Conductivity. If uniform rods of different materials be
covered with wax and exposed to the same source of heat, the wax will
melt at different rates rapidly on the metal rods and very slowly or not
at all on wood and glass rods. The attainment of a given temperature at
a given point depends both on the amount of heat reaching that point and
the specific heat of the substance.
Thermal Conductivity is defined as the number of heat units which
flow per second through a unit cube of the substance when its opposite
faces are maintained at a temperature difference of 1. Experiment shows
that the quantity is, within small limits, directly proportional to the tem-
perature difference and inversely to the thickness. Considering a slab of
area A and thickness dx, with a small temperature difference dt, if K be
the thermal conductivity and H the amount of heat flow per second,
(127) H=KA^>
A similar equation applies to diffusion, osmotic pressure taking the
place of temperature. Ohm's law for electric flow is also of the same
form.
Some values of Km C. G. S. units are: Copper, 0.96; iron, 0.16;
lead, 0.11; sandstone, 0.003; water, 0.0015; wood, 0.0004 (the values
parallel to the grain and at right angles to it are different) ; flannel, 0.00023;
air, 0.00005; hydrogen, 0.00033.
If k is the thermometric conductivity, p the density, and s the specific
heat, A"= psk. For iron ^ = 0.86, for lead, 0.35; consequently k\\ k\=
16/87 : 11/35, or the rate at which lead is heated by conduction is greater
than that of iron, although its thermal conductivity is less.
The conductivity of crystals differs along different axes is shown by
touching a hot needle to a layer of wax on a quartz crystal cut parallel to
the principal axis. The wax will melt in elliptical shape, with axes in the
ratio 289 : 158.
The conductivity of liquids (except melted metals) and gases is usually
small and difficult to determine on account of disturbances due to convec-
tion and radiation. Hydrogen has a conductivity about seven times as
great as air, as shown by its cooling effect on a wire heated by electricity.
It is probably impossible to determine the pure conductivity (transfer
of energy by collision) in gases, because of simultaneous diffusion and
convection. The value of flannel as clothing is shown by its poor conduc-
tivity, which is largely due to the imprisoned air. Double walls, with the
space between entirely exhausted, form a still better screen, as heat can
then escape only by radiation. In order to protect liquid air from heat,
92 HEAT.
Dewar devised glass test-tubes and flasks with double walls, with the inter-
space exhausted to prevent conduction, and silvered outside to reflect
incident radiation. This reduced the loss to about one-fifth what it would
otherwise have been.
A flame will not penetrate the meshes of wire gauze, because its heat
is so rapidly conducted away that the temperature of combustion cannot
be maintained. This fact is utilized in Davy's safety lamp, which can be
used in the inflammable gases found in mines.
Owing to the small conductivity of heat by the soil, the maxima and
minima of temperature below ground occur much later than at the surface,
and the fluctuations rapidly get less and less. The daily w r ave penetrates
2 or 3 feet, the yearly wave 40 or 50 feet, in temperate latitudes. At
greater depths the temperature is approximately constant.
QUESTIONS.
80. Can any amount of pressure liquefy argon at a temperature of 100 ?
81. What conditions will prevent smoky chimneys?
82. In heating houses by hot air or water, would it be satisfactory to place the
furnace on the top floor?
83. How does a chimney assist ventilation ?
84. Why does a lamp chimney make the flame brighter and free from smoke ?
85. If a room has a single window (ordinary double-sash) how must it be
opened to secure best ventilation ?
86. Why do winds usually blow down mountain canons at night ?
87. The prevailing winds blow from the ocean in California and Western
Europe, from the land in the North Atlantic States. How is the climate of these
regions affected thereby ?
88. Why does so little rain fall between the Sierras and the Rocky Mountains ?
89. Which feels colder to the touch, copper or iron, when both are at the
temperature of the body ?
90. State four objections to the use of water as a thermometric substance.
THERMODYNAMICS.
181. The two fundamental principles of Thermodynamics, or the
study of the mechanical effects of heat, are based on the following experi-
mental facts : (1) When no other forms of energy are involved, the
performance of a definite amount of mechanical work by a heat engine is
accompanied by the disappearance of a definite amount of heat and vice
versa; (2) A heat engine can be continuously operated only by the
passage of heat from a hotter to a colder body heat cannot run "up
hill." About 1824 Sadi Carnot laid the foundation of the theory. By
supposing any material substance to undergo a cycle of changes in
pressure and volume by the application of heat, finally returning to its
original condition, thus eliminating any energy losses due to change of
condition, he sought to obtain a definite relation between heat and \vork,
although at the time he held the material idea of heat. We may use his
process, remembering; the equivalence of heat and energy. In order to
do this we must consider two kinds of change isothermal, in which the
substance remains at constant temperature, and adiabatic, in which no
heat is given to or taken from it, the substance being cooled or heated in
doing work or having work done on it.
CARNOT'S REVERSIBLE CYCLE ABSOLUTE TEMPERATURE. 93
182. Carnot's Reversible Cycle. The substance may be sup-
posed to be contained in a non-conducting cylinder, with perfectly
conducting bottom, and provided with a piston. Place the cylinder on a
non-conducting support and compress the substance adiabatically from
A to B, its temperature rising from T to Z 1 ,. Allow it to expand
isothermally to C, keeping its temperature constant by placing it on a
source at temperature T t . The heat absorbed from this source is H^.
Place it again on the non-conducting stand and compress it adiabatically
until it comes to its original temperature T at D. Compress it
isothermally to A, the heat given to a refrigerator at temperature T Q
being H . The external work done by the substance is W= H r H
(//'being measured in ergs), for since it is in its original condition the
total internal work must be zero. The efficiency is
(128) Wj //; = (//, - /O / H, = (7; - T ) I r x .
The external work done is evidently equal to the area A BCD
(indicator diagram) \{ p and v are given in C. G. S. units. If the cylinder
is always at the same temperature as the source or the refrigerator when
in contact with them, it is evident that the process is reversible; the work
W being done on the substance, H units of heat are taken from the
refrigerator and H^ units returned to the source. This imaginary arrange-
ment is called Carnot's perfect reversible engine. No heat is supposed to
be lost by friction, etc. All such engines must be equally efficient, what-
ever the working substance, else by running one with another of different
efficiency a surplus of useful work might be indefinitely obtained, which
may be shown to involve a continuous transfer of heat from the refrig-
erator to the source at higher temperature, in opposition ~ to one of the
fundamental principles of thermodynamics.
183. Absolute Temperature. Thomson (Lord Kelvin) adopted
an absolute scale of temperature based on the assumption that in a perfect
engine the temperatures of the source and the refrigerator are proportional
to the amounts of heat absorbed from the one and returned to the other,
or TJHj_ = TJH . If the thermodynamic substance be a gas obeying
Boyle's and Charles' laws, it may be shown that the absolute temperature
as determined from the expansion of the gas coincides with that defined
by the above relation.
Equation (128) shows that efficiency depends only on the temper-
atures of the source and the refrigerator, and no engine could be perfectly
efficient unless the refrigerator be at absolute zero. Evidently no lower
temperature is possible. Ether has a smaller latent heat than water, boils
at a lower temperature, and has a higher vapor pressure ; yet it has no
advantage over water, because it cools so much more rapidly on expan-
sion (thus reducing the pressure) that all its other advantages are
neutralized.
184. Change of Freezing-point Determined by Cyclic
Process. This is an instructive application of Carnot's method. Let
AB represent the change of volume of m grams due to change of state,
the temperature and pressure both remaining constant. If z\ and z> 2 are
the volumes of unit mass in the first and second states respectively,
m = AB'iiy^ z/ r ) and the heat obsorbed is
(129) H= Lm = LAB!(v 2 - z/ f ).
94 HEAT.
If the temperature be lowered adiabatically to T, and the mass m
condensed by pressure, the heat evolved may be represented by a similar
equation ; or, more simply, calculated from the efficiency equation which
gives W= H(T 2 ~- T^)/T 2 ; also, since AB CD is approximately a par-
allelogram, W=ABI(p 2 /> r ), which gives
(130) 7;-r I = ( n -
This expression is negative for v 2 v t ; hence the
freezing-point is lowered in the first case, raised in the second, by increase
of pressure. The lowering of the freezing-point of water calculated from
this expression is .0075; the observed value is .0072 (see section 149).
This equation, of course, applies also to vaporization, in which case
Z' 2 v l is always positive.
185. Adiafoatic Expansion of Gases. When heat is applied to
a gas, if dT and dv are the small changes in T and
(131) H=
This is the algebraic expression of the first law of Thermodynamics. In
case of adiabatic expansion this expression equals zero; also pv= mRT
and (p + dp) (v + dv) = mR (T+dT), from which dT= P ' dv + '* ' dp .
m JT\.
From this and the relation
(132) ^ + k ^ = (putting k = c, I O-
Assume the adiabatic relation of pressure and volume to be pv n ~
constant = (p + dp) (v-{- dv) n =pv n + npv*~ l dv + v . dp (neglecting the
vanishing product of infinitesimals), from which
(133) ^. + w *L = o.
p v
Comparing with (132) we find the relation to be
(134) pv*=C
Elasticity is stress I strain or v . dp I dv. From (134) e = kp } or the
elasticity in adiabatic transformations is k times as great as in isothermal
transformations. This is important in the theory of sound, since it is the
adiabatic elasticity which determines the velocity of sound waves.
The ratio k is nearly the same for all the permanent gases and equals
1.4 (see section 135). It may be determined experimentally by the
method of Jamin and Richartz. A calibrated glass vessel communicating
with a mercury manometer is heated by a wire carrying an electric current
and filled with the gas. If the pressure be kept constant for a time /,
. (135) H^(7- T ^t-
ORIGIN AND MAINTENANCE OF SUN'S HEAT. 95
If the volume be kept constant and the heat applied for an equal time,
From which, assuming H^ = H z ,
(137) k= 2 _ = -, 2 _ .
The pressures are determined by the manometer, and the volumes
from the calibration of the vessel.
186. Origin and Maintenance of Sun's Heat. There has been
no great change in the amount of heat received by the earth from the sun
in thousands of years. Langley has shown that the earth receives energy
from the sun at the rate of about 2.5 horse-power per square meter, or
the latter must radiate from its surface more than 100,000 horse-power per
square meter. If the sun were solid coal it could not maintain this supply
by combustion for 3000 years. There are two dynamical theories to
account for this heat: Mayer's meteoric theory is that the heat is main-
tained by the fall of meteors on the sun. The earth, if it fell into the
sun, would generate as much heat as would be furnished by the combus-
tion of 5600 globes of coal. Nevertheless this theory is hardly sufficient.
Helmholtz's contraction theory is that the heat is evolved by the con-
traction of the sun, considered as a partly gaseous body. From thermo-
dynamics it may be shown that a contraction of diameter of 250 meters
per year would maintain a constant temperature, and such a shrinkage
would not be observed for ten thousand years. When the sun is mostly
liquefied or solidified this action will cease and it will grow cooler. The
interior heat of the earth was probably originally produced by a similar
process, which has now ceased, and it is constantly being lost by con-
duction to the surface and radiation into space.
187. Dissipation and Degradation of Energy. From the effi-
ciency equation (128), it is evident that only a fraction of a given
quantity of heat can ever be converted into work. The remainder is lost
by conduction or radiation to cooler bodies. There is thus a constant
tendency for the universe to come to a uniform temperature. This has
been called by Kelvin the dissipation or degradation of energy. When
universal-temperature equilibrium exists there can be no available energy,
unless by some process (Maxwell's "demon") the individual molecules
possessing greater kinetic energy than the average can be sorted from
those having less.
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