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According to the (lis.sociiition theory of electrolysis, it is jM.ssible to calciiJute the conductivity of complex solutions of electrolytes, provided we know the quantities of the electrolytes present in a given volume of the solution, tlieir states of dissociation, and their specific molecular con- ductivities at infinite dilution. If r is the volume of a solution which contains .V,, i\T^. X.j. etc., gramme-equivalents of the respective electro- lytes ])rcsent, if n^. a.^, ^r,, etc., are their coeflicients of dissociation or ionisation, and if /< qq,; M 'xa- /' xa' ®tc., avo their specific conductivities per gramme-equivalent at infinite dilution, then, according to the above theory, the specific conductivity of tlie solution is equal to 7, ( '•'i -"^'1/' 001 + 'h^\^* oc.. + a a / 5 66 ROYAL t?OCIETY OF CANADA cules uiideri^o dirtsociiUioti is e([iiul to tho iVo([»ency with whicli frco ions ro-combitu!. The {iini)Uiil of an t'U'ctrolyto which is dissoeiiUecl when the cqiiilibvium condition lias been attained is, therefore, to bo determined by the aiiplication of the law of chemical equilibrium, which ex])resses the equality of the two frequencies just mentioned. As any electrolyte winch is in dissociational equilibrium is to l)e re<;arded as being in this state iu)t only throughout the whole volume of the solution, but also tliroughout any tinite part of it, the law of equili- brium may be applied to any such i)art. As each electrolyte in a comjjlex solution, with its undissociated and dissociated parts, though disseminated throughout the wliole volume, may be regarded as occupying a derinite portion of the volume, wliich we may speak of us its region, the law of equilibrium may be applied either to one sucli region or to the regions of two or more electrolytes which have ions in common. Wo shall consider, tii-st, solutions containing electrolytes which have all a common ion, and, next, the more complex cases of solutions contain- ing two or more electrolytes having no common ion. Cask T. — Solutions roNTAiNiNO Two Electrolytes w[th a Common Ion. In such a case the two electrolytes added to the solvent, in preparing the solution, are the only electrolytes present. The numbers of gramme- equivalents (A^, and AT^) in any given volume of the solution are thus known. Call the electrolj'tes 1 and i, respectively. Let (>„ h.^ be the numbers of undi«sociated. and //,, /?, the numbers of dissociated gramme- equivalents of 1 and 2 in the given volume v of the solution, and let r,, r.,, be the volumes of the regions occupied by them, respectively. Ap])lying the law of equilibrium to electrolytes 1 and 2 throughout their own regions, respectively, we obtain ; (1) ^ _ (i, /^. (2) /^, /i. here (\ and c^ are constants. Applying the law throughout the whole ')lu:ne. we have (3) CO l\ + V: r, -f v.. /i, + /A r, -f- V.2 til + H, + *'; f^ >I3 y, 4- v.: Ik .'.S [MACORKdORj CONDUCT' VITY OF ELECTROLYTES From (1) and (H) we ohtuiii : 67 Hon CO :ui(i /A /A /A. I f-iy + H, '•, + '•/ II" we fomliiiie (2) and (4), we olttaiii llie .sanie result. Uenco the Hole condition of eiiuililiriiim is that tiie )iumhers of dissociati I i;ranime-equi- valents of the two clt'etroiytes, per unit volume of tin; rei^ions (Kcupietl l»y them, or the eoiieentrations of tlie ions of the two electrol>'tes, hIuiII be equal. Arrhenius has hhowji ' (and the above is but a slii^htly modified form of his reasoniuif ) that two simple .solution.s of electrolytes, having a com- mon ion, which undergo no c hange of volume on l)ein) from the volume relation. .Y, F, + X T', = r. Other two equations are furnished by our knowledge that, at a definite temperature, the ionisation coefticients depend upon dilution alone, and ' Ztschr. f. phy.sikal. Cliemie, vol. ii., p. 2Hf (1H8^I). <'2 t^ 68 HOYAL SOCIKTY OF CANADA tliat, llioivf'oiv, tlic cM)iirc'iit rations of ions aro functions of tho dilution only, llcnie- \vi' have ('•) fi'om tho rolution of dissociation to ililiition. r, '\(V^)■ Wiial tiic functions involved in those equations are, may be determined IVom measuroinents of the specitic molecular conductivity of sufflcicntly extended series of simjde solutions of the n.'spective electrolytes. We have thus four equations for determininn- the four unknown quaniities involved, viz.. ^r,. (if.^, Vi, ]'^.. These eqiuition.'i may be most conveniently solved by a graphical process. Draw ciirv ;s from the experimental data just referred to, having as abscissa' the concentrations of ions for simple solutions of 1 and 2, res]»ectively, and as ordinatcs JVi and X times the corresponding values of the respective dilutions. Then select two points, one on each curve, which have the same abscissa, and which have ordinates tho sum of which is equal to r. Multiply the values of the ordinates of these points, read off on the scale on which they would re[)re.sent dilutions, by the common value of the abscissa, and we have the values of «, and a.,- If the solution have been formed by the mixing of two simple solutions, and especially if the constituent solutions had equal volumes and were so dilute as to undergo no appreciable change of volume on mixing, the graphical process is very easily carried out.' Arrhenius- has shown that the conductivity of a complex solution, containing two electrolytes with one ion in common, may be calculated by, first, making sufficiently extended series of observations to determine what simple solutions of tho two electrolytes are isohydric with one an- othei- (/.('., do not change in their state of dissociation on being mixed), and. secondly, finding by the aid of these observations, and by a series of approximations, of what two isohydric solutions the complex solution might be formed by mixture. Isohydric solutions were recognized as such by the equality of the specific conductivity of a mixture of equal volumes of them, to the mean of thoii- specific conductivities. The iso- hydric constituents of the (complex .solution having been determined, its conductivity was the mean of their conduct! viiies. The method was ap- ])licable only to solutions so dilute that there was no appreciable change of volume on mixing. ' For a detailed account of this graphical process see Trans. Nova Scotian Inst. Sci., vol. ix., p. 107. - Wied. Ann., xxx., p. 73(1887). I 1 on [MACflHWiOlt] CONDUCTIVITY OF KI^KCTUOLYTKS 09 Cask 11 —Solutions fONXAiM.No anv mmhkk ok Ki,ectii<»i,vtks HA VINO A Common Ion. In this cuHc also tlu- electrolytes jul(l:'<| to tlio solvent in tlio prepar- atioii of llie solution ai'e the only ones present in it, and tho ^V's are thus known. We may indicate the electrolytes in the solnlion hy the numerals 1.2 //. Applyiii^f the law of eciiiililirinm to the ditl'crent eleelr(»lytes throui^h- out tho regions occupied hy th'-mselves oidy. and usin^ the sanio symhols as ahove, we ohtain a set of e(|uation-« siniihir to (1) and (2) of Case I. Applying tho law to the various electrolytes throughout the ri'i^ions occu- pied hy themselves and one otlu'r electrolyte, wu ohtain a set of «'quations similar to (:J) and (4) of Case I. Ooinhining these equations, us in that case, wo find that the}' reduce to (6; 1^. A, p. If Wi- now apjily the law to ca( h electrolyte throughout the region otcu- pied hy it and two othei- electrolytes, we ohtain a series of equations, such as ^'1 '•i + v.. + /•, These equations, however, are not iinlependent of those already ohtained. For from (1) and (5) we have r, — ! |■i^ Hence '■I + v., -f /'. fh±J- + f^-' ._. ''l + f'-l + '"li ' '"l + >'■> + '■;)' ft^ Similar!}' the equations ohtained hy ap})lying the law to the various elec- ti'olytes throughout the regions oocu])ied by themselves and other three, foui', etc., electrolytes, may he deduced from such equations as those given above. Hence the sole condition of e(iuilihrium is expresaed in equations (5). For the determination of the coeiRcients of ionisation we have thus : (a) from the conditions of i-quilibrium. Of,. l> — 1 equations. (/() from the volume relation, 1 equation, 70 ROYAL SOCIKTY OF CANADA (c) from till' ivliidnii of tliMHociutioii to dilution. = /■l cr,). IX, v. = ''.. (!'.)• ot '. // eqimtions. in ill! 2/* equations, for the (U'tcrmiiuition of y^ /v's, anuKno.O CONDUrTIVITY OV KLUCTHCLVTKS 71 ll,n.uKlu...t th.- .vgions .u.cuim.a l.y it a.wl il. and l.y it un.l 4, roHpec. livfly, i,'iv»'H tlu" «'(iimti(»nH, , _Jm__ _ /A 4- /i, _Jh___ . ''(•, + '•, ~ r, 4- /•. '', + '■« and u similar a,.,.li.nti.,M to llu, oil,..,- olrHrolytos ^iv.s six otlu-r Him.lur nivialions. Tl.cs.' (w.-lv. .•.,.uiti..us luay be reduml. a> ... ( asc I, to the Ihnv indfiK'ivltMit t'liuations, a) . . . . ^ = -, It is thftvtb.v, a i.cc*'ssai-v coiuliti.... of cM,uiiii.riu.n that tl.e coiu-ontia- lioi.sofions.i.. the ro-ioivs omiim'd l.yH.o to...- dertn.lytos. irspoctivdy, shall l>o t-qnal. We may ol.tain othci' (Miuatious oloqiiilihriiuu hy oxprossiiig tlu' Inci that each electrolyte is in e^iuilihriuni thn.iighout th<' wlw.le solution. Thus we obtain tor electrolyte 1. /I /^. /^. '•;> ''i (8) /', /A + /^:, /A + /^, und si.nilar eqiuitions for 2, 3 aud 4. Fi'<)iu (i!) we have, Px and from (8), hi'i = (/A 4- P.) (/^i + /^.) Jleiice From the other equations similar to (6) and (S) we obtain : A, (A + Pi)_(A±_P^, ^'3 ~ /^3'- From these four equations and (7), we obtain ; A iPy + Z^'a) (/A + /A) = /A (A. 4- MO (/^-^ + A). /^;-, (/«. + A) iP^ + /A) = Ml (M + ^i) CM + M2), M, (Ml + M) (M. + M*) = M. (M. + M.) (M + M.) ; 78 ROYAL SOriKTY OF CANADA 1111*1 tlit'sc llirrc, wlii'ii iiiiiiliiiu'd, iH'diic*' to It is. tluTflbrc, aUo ii luci'Msiiiy condition oC t'i|iiilil)i'iiiin that lln' pro- ducts of tlu- iiunilK'rs of tlu' diss(»('iatt'd ^I'ainine ('(|uivult'iitK of tho two |mirs ol't'li'ftrolylt's liaviny; no common ion, sliali Itc equal. This condition may lie otherwise i-xprcsscd. For, liy (7), llcnci', (J>) /.(.. lor c(|iiilil)ri>im. tiic Mroducts of the volumes of tlie irgions occupied liy the two pairs of electrolytes haviim no common ion, must bo e(|uul. In the case of a solution contaiilflig two electrolytes with no common ion, ihi're are. therefore, four necessary eoiulitions of ei^uiiibrium, ex- pressiid in eciuations (") and (!t). Arrhenius' has shown that if a solution containing two electrolytes with no common ion have Ijeen prepart'd as a mixture of four simple solutions of these electrolytes and of the products of tlieir double deeom- j)osition, if the sim])le solutions before mixture hail equal concontrutionH of ions, had such volumes that the products of the volumes of tho solu- tions containing electrolytes with no common ion were equal, and were so dilute that no change of volume occurs on mixing, and if no change occur in the state of dissociation on mixing, the mixture will satisfy the conditions of the equilibrium. I'\)r the determination of the coerticients of ionisution and tho num- bers of gramme-e(|uivalents of the four electrolytes in any volume c o( the solution, we have, theretore. the fcdlowing equatiDUs: (_(() Irom the conditions of equilibrium, '••i a., ^f.. i^. \\ - V, - V, - V, N,V, . N,V,-^^\V, . N:V, . . . . (6) from the volume relation, ((•) from the relation of ionisation to dilution. ii equations; 1 equation ; 1 equation ; 1 T- 1 ~pj = /i ( ^'^i)' L 4 equat ions. etc. Although in tliis case we do not know the values of JV,. N^, N^, iV"^, wo ii-< tor tlic lU'trriiiiiiatinn of ili iiii- kiiowii (jiiantitii's, vi/,.. I as, 4 iV>, ami 4 I' s. To Holv«' tln'Sf t'i|iiatiuiis liy a ifra|iliii'al iiiflhod. \vi' may tirst rrdiu'o {\w second, tliird and last ), — V-iti., (V V \ anil Now draw i-nrvi's for all foiir i-U-ct rolytcs. from I'xiK'rini' tjil data oh- tuiiK'd from ili.siTvations on tlu'ir sinipli' solutions, wit' values ol tho coiicontrutioii of ions as aliscissa' and tlu' I'orrt'sitoiiiliiig vidui's* of tho dilution as oi-dinatus. St'U-ct what sci'ins a iirohahlc value of tho common conri'iitration of ions in tin- i-(im|ilox solution, ami nuid ott' from tho curves tho corros|tondinif values of Fj. K, F,. V^ Dotormino A', from tho ttrst of tho last two uquutioiis liy suhstitutinjjf those vahu'S in it, and sec wliolhor this value of iVy is. tirst, a iiossililo value, and. secondly, one that will satisty the last equation. If not 've must make another shot )it the common value of the euncentration ol ions; and so on until a value of N;i is ohtained which does satisfy tho last equation. 8ueh a value having heen foiuid, the common value of a/ V, and tho values of Vi, ^'i? ^111 ^v hocomo known, and ay, a.^, «'a, tx^ may he determinod by mul- tipli(!atioii. Also /tj, ii-j and .'V, heinyj known, A',. X.j and 3^^ may bo determined from eijuations (d) above. And thus all tlie data arc avail- able for calculating the ■conductivity. The a|i|)lieation of this method would require tlu'.t the curves based on tho observations on simple solu- tions should bo very accurately drawn. For tl e denominator of the above expression for J\\ is tho ditt'erenco between the sums of two dilu- tions, and this ditforonce ma}' be small. Ilenco oven a small error in their determination may load to a great error in tho value of i\\, which is found. If wo wish not to calculate the conductivity of a given solution, but merely to tost tho dissociation theory by comparing tho observed and Sec. II r., I8H6. 6. T 74 ROYAL SOCIETY OF CANADA calculattHl values of any soluUon, it luij^ht bo bettor to proceod other- wiso. viz., by detorminin^ the constitution of a comitjex solution of the electrolyto.H 1, 2, 3, 4 with a known common concentration of ions. For this purpose select any value of the concentration of ions and read off from the experimental curves referred to above the correspondin/i( values of the dilutions Fj. V.,, V.^, V^, of simple solutions of 1, 2, 3, 4, selecting a concentration of ions characteristic of dilute solutions so as to avoid the complication which would be introduced by change of volume on mixing. If simple solutions of these dilutions are mixed in proper proportions as to volume, there will be no change of ionisation on mixing. To find the proper [troportions, select arbitrarily any value of v^ the volume of the solution of 4 which is to be mixed with the others. It will contain JV^ = v^/Vi gramme-equivalents of 4. From equations (d) above we must have iVg = iV,. Hence the volume of 3 to be mixed with the others will bo )•;; = F3V4/ F4. Next select arbitrarilj'^ any value of Uj. Then in order that there may be no change of ionisation on mixing we must have Vi = VsVi/Vi = V.,vl/ V,v. The volumes of the simp'e solutions of dilutions Vi. V2, Fi, ^ir which must be mixed in order to form a complex solution with the selected concen- tration of ions, are thus known. The solution may therefore be prepared and its conductivity experimentally determined. The conductivity may also bo calculated. For the concentrations of the simple solutions and the volumes of them which are mixed being known, the N's may be found ; and the common concentration of ions and the dilutions being known, the a'n may be found. If the densities of the simple solutions are known, either from the data of published tables or from preliminary experiments, the numbers, Wi, /ij, of gramme-equivalents of 1 and 2 which must be added, say, to a kilogramme of water in order to produce the required comjilex solution, may be calculated, and the solution may thus bo pre- pared by three weighings, the errors due to measurements of volume being thus avoided. Case IV. —Solutions containing Three Electrolytes having no Common Ion, In this case thei-e will in general be nine electrolytes in the solution, as illustrated in the diagram, in which the numbers are given by which we shall indicate the various electrolytes. The application of the law of equilibrium to each electrolyte throughout its own region gives nine equations, such as, NaCl (1) Nal (5) HCl (4) KCl m (2) KI (8) NaBr 0) HBr (9) KBr (3) ^1 - = Vl m. Applied to each electrolyte throughout the j:> ■ , .I , 5%J [MACORKOOn] CONDUCTIVITY OF ELECTROLYTES 78 region oucupietl by itHelf and one *)ther olectrolyto having a common ion, it gives thirty-six «>quation8 siicli us, t'l + i\ ' V, + V,' These forty-five equations, however, aie not independent, but may be reduced to eigiit, viz. : (10) A' ^8 /^» V,' which are tlius necessary conditi«»ns of equilibrium. Applied to each electrolyte througliout the i-ei:;ion occupied l)y itself and two other electrolytes having a common ion, we obtain 18 equations, such as, ' ii + '^« + Vb «'i + t\ + i)g ■ f, + v^ + y/ These, however, as in Case III., may be deduced from those given above. Applied to each electrolyte throughout the region occupied by it, another electrolyte having no ion in comtnon with it, and the products of their double decomposition, the equilibrium law gives 36 equatiims, such as, ,., ^L_ ^ 0^1 + /^JCA + A) '.'l + A\' + Vt + <-5 ( t'l + fi + V^ + ('-)'■ The.se equati(ms, when combined, give 9 equation, such as, which, however, are not independent, but reduce to four, say. ai) V1V2 = iui\, whicii thus form additional necessary conditions of equilibrium. Applied to each electrolyte throughout the whole volume of the solution, the equilibrium law gives 9 equations, such as ■^- 0, ^ = (!l±A±A . /^- + h±A^ which, however, may be deduced from the twelve equatitws (10) and (11). These twelve equations thus express the necessaiy conditions of equili- brium. 76 ROYAL SOCIETY OF CANADA For the (Ictermimvtion of the ionisation coefficients and the numbers of gramme-equivalents of the nine elcclrolytos in a sohition of this com- plexity, we luive, therefore, the following equations : (a) from the conditions of equilibrium, v.. etc., 8 equations, 4 equations. (b) from the volume relation, iV^i ^i + K,V2+ . ■ . . 4- iV„ Fa = y . . . .1 equation, (c) from the relation of ionisation to dilution, T; = /i(^o, I etc., / (d) from the conservation of mass, «, = iVi + iX, + -^T, ?io = M, + ^\ + jv^, 9 equations, 5 equations, since each of these last equations may be deduced from the other five. We have thus in all twenty-seven equations for the determination of nine a's, nine iV's, and nine F's. The solution of these equations by the aid of the graphical process referred to above would, of course, be ideally possible ; but it would probably be found Impossible, with the data as to the conductivity of simple solutions at present at our disposal, to draw the requisite curves, showing the relation of concentration of ions to dilution, with sufficient accuracy for the purpose. ^■•^ [mACX3RE()0K] CONDUCTIVITY OF ELECTROLYTES 77 ler five, tion of Case V. — Solutions containing any Xumber op Klectrolytes with NO Common Ion. If the solution is formed by the addition of /> such eloctrolytos to the solvent, there will be p^ 5I0C trolytes in the solution, us i8 illustrated in the diagram. It is unnecessary to write down the equations which may be obtained from the application of the law of equilibrium to the various electrolytes throughout ditt'ereut regions of the solution For, as each electrolyte is in equilibrium in its own region, and each also in the region oc- cupied by it and every other electrolyte liaving one ion in common with it, we have, as in Nam (1) HCl (4) KGl (<>) -— Nal (5) HI (2) Kl (H) NaBr 0) HBr (9) KBr former cases. ftp forming a set of p^ — 1 conditions of equilibrium. Also, as each electro- lyte is in equilibi'iura in the region occupied by itself, any other electro- lyte having no ion in common with it, and the products of their double decomposition, wo have, etc., forming a set of (/> — 1)- conditions of equilibrium. These 2p (p — 1) conditions are also the only conditions. For if we apply the law of equilibrium to each electrolyte through the whole volume of the solution, we obtain p' equations, such as, Ci ^' = /A + ^4 + /^H + etc. ^ A + A + A + etc. V V y ' v/hich, as in Case IV., will be found to be deducible from the equations gi%'en above. For determining the ionisation coefficients, and the numbei"s of gramme-equivalents of tbe various electrolytes present in the solution, we have therefore : (rt) from the conditions of equilibrium. Oil a., ' -^ = -p^ = etc., iViF, . N,V, = N,\\ . N,V,, etc. } p' — 1 equations, (p — 1)^ equations, « 78 ROYAL SOCIETY OF CANADA 1 equation, p- equations, (6) from the volume relation, T, r, + X,V., + etc. = (; ((•) from the relation of ionisation to dilution, • p = .^ui:),l :... etc., /' ((/) trom the conservation of nui»H, •«, = T, + X, + X -f etc., \ „, = i\^, 4- X^ + X,. 4- etc., 2;>—l equations, etc., J in all, therefore, 3p^ equationj for determining //' a's, p" X's, and p'V'H. Case VI. — Solutions containing any number of Klectbolytes with NO Common Ion, the products of their Double Decomposition, AND ANY OTHER ElECTROLVTE.S. It is obvious that if in preparing such a solution there are added to the solvent, p electro- lytes with no common ion and — 1) (p + q — 1) equations. That the equations of this set are (p — 1) (p -f- '/ — in number is obvious from the fact that if such c([uations be written down for each group of four electrolytes i-epresented in the above diagram as occupj'ing contiguous squares, of NaCl (1) Nil I (5) J/ri (4) HI (2) KCl (G) KI («) (10) IMijh (10 NaBr (7) J/Br KBr ^M.jBr., (12) [MACOREcoR] CONDTTCTI VITY OF ELECTROLYTES 79 which thove are (p — 1) (p 4-7 — 0) »^iiiiihir oqmitions may then bo deduced from them lor all groups of the electrolytes consisting of two with no common ion and the products of their double decomposition. That the above 2yy (// -(-7 — — '/ c(|uations form the only condi- tions may be shown by di^ducing from thom the equations obtainetl by applying the law of equilibrium to each electrolyte througiiout the whole volume, as. for example, <■, ''• ^ /A + /:^4 + /^,, + fK + »tc. /A + /i, + /I + etc. i: V V ' or, without this trouble, by showing, as is done below, that I best' ecpui- lions are sutHcient to determine the state of dissociation. Arrhenius ' saj's with regard to this general case : •' It is easy to see that if we mix an}' mnnber //(/( of combinations of m positive and // neg- ative ions, e(piilibnuiM will not be disturljed, ])rovi(le(l these ))iti electro- lytes formed isohydric solutions among one another before the mixing, and the product (*f the active masses [i.e., (luantities of dissociated ions] of two electrolytes /,„./„, /„./^, is eqiuil to the product of the active masses of f„,J„ and /„./„. This will be the ca.se if the volumes of the isohydric solu- tions are related to one another as the surfaces in tlie anne.Kcd diagram," in which diagram the surliices, so far as the eye can judge, ai"e equal. It is obvious that if the volumes occupied by the various electrolytes in the solutions were equal, the second set of equilibrium conditions would be satistieil ; but equality of volumes is not necessary tor their .satisfaction. For determining the quantities of the various electrolytes present in the solution, and their ionisation coetttcients, we have the following ('([luitions : (a) from the equilibrium conditions. V ~ r' ~ ^'^^■•' /' (V + 7) — 1 ^qiiiitions. iV'iFi • Xl\, = X, I', • X r,(^ _ ^^^ _^ ^^ _ j^ ^^^ _ ^^J equations, etc. ( (h) from the volume relation, NiVi + N.,\\ -f etc. = V 1 equation, (r) from the relation of ioni.sation to dilution, F = fi ( ''')' 1 I' (p-\- '/) •■qnations, etc., / . (r/) from the conservation of mass, /', + "10 -f ftc. = N, + K, + etc. + Nn> + *'tt'-] • /',u - iN'.o + iN'i. + A',. + I'tc. ■ . . (2y; + Y — 1) eqns., etc., J ill all 3/) {[) 4- 7) equations for determining p {p -|- 7") a's. p{p-\- if) iN"s, and p (j) + q) T's. 1 Ztschr. f. physikal. Clu'tuie, vol. ii., p. 284 (188H). 80 ROYAL SOCIETY OF CANADA The possibility of culeulating the conductivity of complex solutions of electrolytes is probiihly of no pmctical value ; but it affords a means of subjecting the dissociation theory of electrolytes to severe tests, tbo greater the complexity of the solution the greater being the severity of the test. With the object of testing this theory, I have calculated ' the con- ductivities of a series of solutions containing sodium and potassium chlorides, the conductivities of which had been observed by Bender. The following table gives the restilts : ! Constituent solutions. (iraninie-e<|iiivalcnts pi'i- lili-f. Conductivity of Mixture. NaCl. KCl. Bender's ob- served values (corrected). Calculated values. Dilference per cent. 0-5 0-5 0-5 0-5 0-5 1-0 10 1-0 1-0 1-0 1-0 0-1 H75 0-375 0-5 0-75 1-5 201 370 430 540 mi 2HO-5 373-1 426-1 537-6 8.58-3 461-4 540 •« 701-1 800-2 1015-2 1200-6 1 — 0-52 + 0-84 — 0-90 — 0-44 — 0-54 O-IKZS 0-375 0-75 1-0 1-5 2-0 4m 541 7o;» 80H 1015 1200 + 0-52 — 0-07 — 0-27 -1 0-15 + 002 — om + 0-12 + 0-s(!rvc'(| and tin' culculated valuoH in vary satis- i'actory for tho mort' diiiilf Holutioiis. Am j)ointod out above, tho inoro oonc'ontratod tlu' .solutions, the inorc must the ionic velocities of oaeh eU-etrolylc he nioditied by the presence of other electrolytes in the solu- tion, and the greater, thcri'l'ore, must be the error invojved in taking the specilic molecular conductivity at inHnite dilution, dotermine