# Scientific Papers of Josiah Willard Gibbs, Volume 1/Chapter IX

IX.
UNPUBLISHED FRAGMENTS.
[Being portions of a supplement to the "Equilibrium of Heterogeneous Substances" in preparation at the time of the author's death, and intended to accompany a proposed reprint of his thermodynamic papers]
[A list of subjects found with the manuscript and printed below appears to indicate the scope of the supplementary chapters as planned by Professor Gibbs. As will be observed, however, the authors unfinished manuscript, except for a number of disconnected notes, relates to only two of these subjects, the first and fourth in the list]
On the values of potentials in liquids for small components. (Temperature coefficients.)
On the fundamental equations of molecules with latent differences.
On the fundamental equations for vanishing components.
On the equations of electric motion.
On the liquid state, $p=0$.
On entropy as mixed-up-ness.
Geometrical illustrations.
On similarity in thermodynamics.
On the Values of Potentials in Liquids for Substances which form but a Small Part of the whole Mass.
The value of a potential for a volatile substance in a liquid may be measured in a coexistent gaseous phase, and so far as the latter may be treated as an ideal gas or gas-mixture, the value of the potential will be given by the equation (276), ["Equilib. Het. Subs."] which may be briefly written
| $\mu ={\text{func}}(t)+at\log \gamma _{\text{gas}},$ | [1] |
$\mu$
$t$
$\gamma _{\text{gas}}$
* $a={\frac {A}{M}},$
$M$
$A$
| $\mu ={\text{func}}(t)+{\frac {At}{M}}\log \gamma _{\text{gas}},$ | [2] |
The validity of this equation, it is to be observed, is only limited by the applicability of the laws of ideal gases to the gaseous phase; there is no limitation in regard to the proportion of the substance in question to the whole liquid mass. Thus at 20° Cent. the equation may be determined by the potential for water or for alcohol in a mixture of the two substances in any proportions, since the vapor of the mixture may be regarded as an ideal gas-mixture. But at a temperature at which we approach the critical state, the same is not true without limitation, since the coexistent gaseous phase cannot be treated as an ideal gas-mixture. At the same temperature however, if we limit ourselves to cases in which the proportion of water does not exceed ${\tfrac {1}{10}}$ of one per cent., and suppose the density of the ​water-vapor, $\gamma _{\text{gas}}$, to be measured in a space containing only water-vapor and separated from the liquid by a diaphragm permeable to water and not to alcohol, then the above equation would probably be applicable, since then the water-vapor might probably be treated as an ideal gas. The same would be true (mutatis mutandis) of the potential for alcohol in a mixture of alcohol and water containing not more than ${\tfrac {1}{10}}$ of one per cent, of alcohol.
This law, however, which makes the potential in a liquid depend upon the density of the substance in some other phase is manifestly not convenient for use. We may get over this difficulty most simply by the law of Henry according to which the ratio of the densities of a substance in coexistent liquid and gaseous phases is (in cases to which the law applies) constant. If $\gamma$ be the density in the liquid phase and $\gamma _{\text{gas}}$ in the gas, we have
| $\gamma _{\text{gas}}=c\gamma ,$ | [3] |
* $\mu ={\text{func}}(t)+{\frac {At}{M}}\log c\gamma ,$
| or$\mu ={\text{func}}(t)+{\frac {At}{M}}\log \gamma ,$ | [4] |
${\frac {At}{M}}\log c$
With this value of the potential, which is manifestly demonstrated only to be used so far as the law of Henry applies, in connection with the general equation (98), ["Equilib. Het. Subs."] viz.,
* $dp={\frac {\eta }{v}}dt+{\frac {m_{1}}{v}}d\mu _{1}+{\frac {m_{2}}{v}}d\mu _{2}...+{\frac {m_{n}}{v}}d\mu _{n},$
I. Osmotic pressure.
II. Lowering freezing point.
III. Diminishing pressure of other gas.
IIIa. Effect on total pressure.
IIII. Raising boiling point with one pressure.
IIIIa. Raising boiling point with two pressures.
V. Interpolation formula for mixtures of liquids.
In fact, when $\gamma _{\text{D}}$ is small, we have approximately
| $\gamma _{\text{D}}d\mu _{\text{D}}={\frac {At}{M_{\text{D}}}}d\gamma _{\text{D}}=Atd{\frac {\eta _{\text{D}}}{v}},$ | [5] |
$n_{\text{D}}$
$D$
| $dp={\frac {\eta }{v}}dt+\gamma _{\text{S}}d\mu _{\text{S}}+At\,d{\frac {n_{\text{D}}}{v}}\cdot$ | [6] |
If $t$ is constant, and also $\mu _{\text{S}}$,—a condition realized in equilibrium, when the solution is separated from the pure solvent by a diaphragm permeable to the solvent but not to the solutum, the equation reduces to
* $dp=At\,d{\frac {n{\text{D}}}{v}}\cdot$
| Whence$p-p'={\frac {At}{M_{\text{D}}}}\gamma _{\text{D}}=At{\frac {n_{\text{D}}}{v}},$ | [7] |
$p'$
$\gamma _{\text{D}}=0$
$p-p'$
${\frac {At}{M_{\text{D}}}}\gamma _{\text{D}}$
For a coexistent solid phase of the solvent, with constant pressure, the general equation gives
* $0=\eta dt+m_{\text{S}}d\mu _{\text{S}}+v\,At\,d\gamma _{\text{D}}$
* $0=\eta 'dt+m_{1}d\mu _{\text{S}}$
$t$
$\mu _{\text{S}}$
$m_{\text{S}}$
| $(\eta '-\eta )dt=v{\frac {At}{M_{\text{D}}}}d\gamma _{\text{D}}.$ | [8] |
In integrating from $\gamma _{\text{D}}=0$ to any small value of $\gamma _{\text{D}}$, we may treat the coefficients of $dt$ and $d\gamma _{\text{D}}$ as having the same constant values as when $\gamma _{\text{D}}=0$. This gives
* $(\eta '-\eta )\Delta t=v{\frac {At}{M_{\text{D}}}}d\gamma _{\text{D}}={\frac {At}{M_{\text{D}}}}m_{\text{D}}.$
$Q_{\text{S}}$
${\frac {t(\eta '-\eta )}{m_{\text{S}}}}$
* $-\Delta t={\frac {At}{M_{\text{D}}}}m_{\text{D}}{\frac {t}{Q_{\text{S}}m_{\text{S}}}},$
| or$-\Delta t={\frac {\frac {m_{\text{D}}}{M_{\text{D}}}}{\frac {m_{\text{S}}}{M_{\text{S}}}}}{\frac {At^{2}}{Q_{\text{S}}M_{\text{S}}}}={\frac {m_{\text{D}}}{M_{\text{D}}}}{\frac {At^{2}}{Q_{\text{S}}m_{\text{S}}}}\cdot$ | [9] |
$m_{\text{S}}Q_{\text{S}}$
$m_{\text{S}}=1$
Raoult makes $\Delta t\propto {\frac {m_{\text{D}}}{M_{\text{D}}}}$, with exceptions.
With a coexistent gaseous phase of the solvent (the solutum being not volatile), we have for the solution
* $dp=\gamma _{\text{S}}d\mu _{\text{S}}+{\frac {At}{M_{\text{D}}}}d\gamma _{\text{D}},$
* $dp=\gamma _{\text{S}}'d\mu _{\text{S}}.$
$p$
$\mu _{\text{S}}$
$dp$
$d\mu _{\text{S}}$
* $dp={\frac {\gamma _{\text{S}}}{\gamma _{\text{S}}'}}dp+{\frac {At}{M_{\text{D}}}}d\gamma _{\text{D}}.$ $-dp{\frac {(\gamma _{\text{S}}-\gamma _{\text{S}})}{\gamma _{\text{S}}}}={\frac {At}{M_{\text{D}}}}d\gamma _{\text{D}}.$
* Say$-dp={\frac {\gamma _{\text{S}}'}{\gamma _{\text{S}}}}{\frac {At}{M_{\text{D}}}}d\gamma _{\text{D}},$
| $p-P={\frac {\gamma _{\text{S}}'}{\gamma _{\text{S}}}}{\frac {At}{M_{\text{D}}}}d\gamma _{\text{D}}.$ | [10] |
* or$-{\frac {dp}{p}}={\frac {M_{\text{S}}}{\gamma _{\text{S}}}}{\frac {d\gamma _{\text{D}}}{M_{\text{D}}}},$
| ${\frac {p-P}{p}}={\frac {M_{\text{S}}}{\gamma _{\text{S}}}}{\frac {\gamma _{\text{D}}}{M_{\text{D}}}}\cdot$ | [11] |
$M_{\text{D}}$
$M_{\text{S}}$ is the molecular weight [of solvent] in vapor.
But the foregoing equation suggests a generalization which is not confined to cases in which the law of Henry has been proved. The letter $M$ in the equation has been defined as the molecular weight of the substance in the form of gas. Now the molecular weight which figures in the relation between the potential and the density of a substance in a liquid would naturally be the molecular weight of the substance as it exists in the liquid. It is therefore a natural supposition suggested by the equation that, in the case where Henry's law holds good, and consequently eq. [4], the molecular weight of the solutum is the same in the liquid and in the gaseous phase; that in ​case the law of Henry and eq. [4] do not hold, it may be on account of a difference in the molecular weight in the gas and the liquid, and that the eq. [4] may still hold if we give the proper value to $M$ in that equation, viz., the molecular weight in the liquid.
But as these considerations, although natural, fall somewhat short of a rigorous demonstration, let us scrutinize the case more carefully. It is easy to give an a priori demonstration of Henry's law and equation [4] in cases in which there is only one molecular formula for the solutum in liquid and in gas, so long as the density both in liquid and in gas is so small that we may neglect the mutual action of the molecules of the solutum. In such a case the molecules of the solutum will be divided between the liquid and the gas in a (sensibly) constant ratio (the volume of the liquid and gas being kept constant), simply because every molecule, moving as if there were no others, would spend the same part of its time in the vapor and in the liquid as if the others were absent, and the number of the molecules being large, this would make the division sensibly constant. This proof will apply in cases in which the law of Henry can hardly be experimentally demonstrated, because the density of the solutum as gas is so small as to escape our power of measurement. Also in cases in which a semi-permeable diaphragm is necessary, an arrangement very convenient for theoretical demonstrations, but imperfectly realizable in practice. (Also in cases in [which a] difference of level is necessary, with or without diaphragm.) But in every case when the law of Henry is demonstrably untrue for dilute solutions, we may be sure that there is more than one value of the molecular weight of the solutum in the phases considered.
This theoretical proof will apply to cases in which experimental proof is impossible:
(1) When the density in gas is too small to measure.
(2) When the density in gas is too great, either the total density or the partial. (Diaphragm or vertical column.)
(3) When the liquid (or other phase) is sensitive to pressure and not in equilibrium with the gas.
Will the various theorems exist in these cases?
If one or both appear in a larger molecular form, the densities of $\gamma _{\text{M}}$ and $\gamma _{\text{M}}'$ are proportional and
| $\gamma _{2{\text{M}}}\propto \gamma _{\text{M}}^{2},$$\gamma _{\text{2M}}'\propto \gamma _{\text{M}}^{'2},$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right\}\,$ [12] |
| $\gamma _{\text{2M}}=\gamma _{\text{M}}=\gamma _{\text{M}}'=\gamma _{\text{2M}}',$ | $\scriptstyle {\left.{\begin{matrix}\ \\\\\ \\\ \ \end{matrix}}\right\}\,$ [12] |
$\mu _{\text{M}}={\frac {At}{M}}\log \gamma _{\text{M}}$
​Let us next consider the case in which the solutum appears with more than one molecular formula in the liquid or gas or both. Now there are two cases, that in which the quantities of the substance with the different molecular formulæ are independently variable, and that in which they are not. In the [first] case there is no question. If, for example, hydrogen appears with the molecular formula H$_2$O and also in molecules with the molecular formula H$_2$O, these are to be treated as separate substances, and we have the two equations
* $\mu _{{\text{H}}_{2}}={\text{func}}(t)+{\frac {At}{M_{{\text{H}}_{2}}}}\log \gamma _{{\text{H}}_{2}},$
* $\mu _{{\text{H}}_{2}{\text{O}}}={\text{func}}(t)+{\frac {At}{M_{{\text{H}}_{2}{\text{O}}}}}\log \gamma _{{\text{H}}_{2}{\text{O}}},$
* $\mu _{{\text{O}}_{2}}={\text{func}}(t)+{\frac {At}{M_{{\text{O}}_{2}}}}\log \gamma _{{\text{O}}_{2}}.$
| $M_{1}\mu _{1}+M_{2}\mu _{2}=M_{12}\mu _{12},$ | [13] |
$p$
It would therefore appear that we may regard the equation
* $\mu ={\text{func}}(t)+{\frac {At}{M}}\log \gamma$
$M$
$\gamma$
$\gamma$
$\gamma$
$M$
​In case of one molecular formula in liquid and none in gas, we may give the molecules repelling forces which will make the gas possible. (?) [See p. 417.]
Deduce Ostwald's law in more general form.
Deduce interpolation formula.
What use can we make of Latent Differences? $\mu _{\text{A}},\mu _{\text{AA}},\mu _{\text{B}},\mu _{\text{BB}},\mu _{\text{AB}}$ all conform to law, I think.
[On the Equations of Electric Motion.]
[A somewhat abbreviated copy of a letter written four years earlier (in May 1899) to Professor W. D. Bancroft of Cornell University had been placed by Professor Gibbs between the pages of the manuscript, and was evidently intended to serve as a basis for the chapter "On the equations of electric motion" mentioned in the list on page 418.
Through the courtesy of Professor Bancroft the original letter has been placed at the disposal of the editors and is here given in full. The major portion of this letter was incorporated by Professor Bancroft in an article entitled "Chemical Potential and Electromotive Force" published after the death of Professor Gibbs, in the Journal of Physical Chemistry, vol. vii.,p. 416, June 1903.]
My dear Prof. Bancroft:
A working theory of galvanic cells requires (as you suggest) that we should be able to evaluate the (intrinsic or chemical) potentials involved, and your formula
* $d\mu =Rt\,d\log p,$
| $\mu _{\text{D}}=B+{\frac {At}{M_{\text{D}}}}\log \gamma _{\text{D}},$ | (1) |
| or$\gamma _{\text{D}}d\mu _{\text{D}}={\frac {At}{M_{\text{D}}}}d\gamma _{\text{D}},$ | (2) |
$\gamma _{\text{D}}$
$\gamma _{\text{D}}$
$M_{\text{D}}$
$A$
$\left({\frac {pv}{mt}}={\frac {A}{M}}\right),$
$B$
$\gamma _{\text{D}}$
​We may avoid 'hedging' in regard to $B$ by using the differential equation (2). We may simply say that this equation holds for changes produced by varying the quantity of ($D$), when $\gamma _{\text{D}}$ is small. It is not limited to changes in which t is constant, for the change in $\mu _{\text{D}}$ due to $t$ appearing in (1) (both explicitly, and implicitly in $B$) becomes negligible when multiplied by the small quantity $\gamma _{\text{D}}$.
The formula contains the molecular weight $M_{\text{D}}$, and if all the solutum has not the same molecular formula, the $\gamma _{\text{D}}$ must be understood as relating only to a single kind of molecule.
Thus if a salt ($_{12}$) is partly dissociated into the ions ($_1$) and ($_2$), we will have the three equations
* $\mu _{1}=B_{1}+{\frac {At}{M_{1}}}\log \gamma _{1},$ $\mu _{2}=B_{2}+{\frac {At}{M_{2}}}\log \gamma _{2},$ $\mu _{12}=B_{12}+{\frac {At}{M_{12}}}\log \gamma _{12}.$
* $M_{1}\mu _{1}+M_{2}\mu _{2}=M_{12}\mu _{12},$
* $M_{1}B_{1}+M_{2}B_{2}-M_{12}B_{12}+At\log {\frac {\gamma _{1}\gamma _{2}}{\gamma _{12}}}=0,$
I may observe in passing that this relation, eq. (1) or (2), which is so fundamental in the modern theory of solutions, is somewhat vaguely indicated in my "Equilib. Het. Subs." (See [this volume] pp. 135–138, 156, and 164–165.) I say vaguely, because the coefficient of the logarithm is only given (in the general case) as constant for a given solvent and temperature. The generalization that this coefficient is in all cases of exactly the same form as for gases, even to the details which arise in cases of dissociation, is due to van't Hoff in connection with Arrhenius, who suggested that the "discords" are but "harmonies not understood," and that exceptions vanish when we use the true molecular weights. At all events, eq. (2) with (98) (E.H.S.) gives for a solvent ($S$) with one dissolved substance ($D$),
* $dp={\frac {\eta }{v}}dt+{\frac {m_{\text{S}}}{v}}d\mu _{\text{S}}+{\frac {At}{M_{\text{D}}}}d\gamma _{\text{D}}.$
$t$
$\mu _{\text{S}}$
* $p-p'={\frac {At}{M_{\text{D}}}}\gamma _{\text{D}}.$
* $p-p_{1}=At\left({\frac {\gamma _{1}}{M_{1}}}+{\frac {\gamma _{2}}{M_{2}}}+{\frac {\gamma _{3}}{M_{3}}}\right)\cdot$
* $0=\eta dt+md\mu _{\text{S}}+v{\frac {At}{M_{\text{D}}}}d\gamma _{\text{D}},$ $0=\eta 'dt+md\mu _{\text{S}},$
$m_{\text{S}}$
$(\eta '-\eta )dt=v{\frac {At}{M_{\text{D}}}}d\gamma _{\text{D}}.$
$\gamma _{\text{D}}$
$dt$
$d\gamma _{\text{D}}$
* $(\eta '-\eta )\Delta t=v{\frac {At}{M_{\text{D}}}}d\gamma _{\text{D}}=At{\frac {m_{\text{D}}}{M_{\text{D}}}},$
$Q_{\text{S}}$
* $-\Delta t={\frac {At^{2}m_{\text{D}}}{Q_{\text{S}}M_{\text{D}}m_{\text{S}}}}\cdot$
* $-\Delta t{\frac {m_{\text{S}}}{m_{\text{D}}}}{\frac {M_{\text{D}}}{M_{\text{S}}}}={\frac {At^{2}}{Q_{\text{S}}M_{\text{S}}}}\cdot$
$M_{\text{S}}$
* $-{\frac {\Delta t}{t}}{\frac {m_{\text{S}}}{m_{\text{D}}}}{\frac {M_{\text{D}}}{M_{\text{S}}}}={\frac {At}{Q_{\text{S}}M_{\text{S}}}},$
With a coexistent vapor phase of the solvent, we have
* ${\begin{aligned}v\,dp&=m_{\text{S}}d\mu _{\text{S}}+v{\frac {At}{M_{\text{D}}}}d\gamma _{\text{D}},\\v'\,dp&=m_{\text{S}}d\mu _{\text{S}},\\(v-v')dp&=v{\frac {At}{M_{\text{D}}}}d\gamma _{\text{D}},\\-dp&={\frac {v}{v'-v}}{\frac {At}{M_{\text{D}}}}d\gamma _{\text{D}}.\end{aligned}}$
${\frac {v}{v'-v}}$
$\gamma _{\text{D}}$
* $P-p={\frac {v}{v'-v}}At{\frac {\gamma _{\text{D}}}{M_{\text{D}}}}\cdot$
${\frac {At}{v'-v}}={\frac {At}{v'}}={\frac {PM_{\text{S}}}{m_{\text{S}}}}$
${\frac {P-p}{P}}={\frac {m_{\text{D}}}{m_{\text{S}}}}{\frac {M_{\text{S}}}{M_{\text{D}}}},$
${\frac {At}{v'-v}}$
${\frac {PM_{\text{S}}}{m_{\text{S}}}}\cdot$
$M_{\text{D}}$
$M_{\text{S}}$
* ${\begin{aligned}\mu &=B+{\frac {At}{M_{\text{D}}}}\log \gamma _{\text{D}},\\\mu &=B'+{\frac {At}{M_{\text{D}}}}\log \gamma _{\text{D}}',\\{\frac {B'-B}{At}}M_{\text{D}}&=\log {\frac {\gamma _{\text{D}}}{\gamma _{\text{D}}'}},\end{aligned}}$
${\frac {\gamma _{\text{D}}}{\gamma _{\text{D}}'}}$
So for the galvanic cell which you first consider, I should write
* $V''-V'=a_{a}(\mu '-\mu '')=a_{a}{\frac {At}{M_{a}}}\log {\frac {\gamma _{a}'}{\gamma _{a}''}},$
$\gamma _{a}',\gamma _{a}''$
$a_{a}$
$M_{a}$
$E_{a},a_{\text{H}}$
$a_{\text{H}}$
$E_{a}$
$E_{a}$
$a_{a}$
* $V''-V'=a_{\text{H}}At{\frac {E_{a}}{M_{a}}}\log {\frac {\gamma _{a}'}{\gamma _{a}''}},$
$a_{\text{H}}A$
$R$
${\frac {M_{a}}{E_{a}}}$
$v,v'.$
​The meagreness of the results obtained in my E.H.S. in the matter of electrolysis has a deeper reason than the difficulty of the evaluation of the potentials.
In the first place, cases of true equilibrium (even for open circuit) are quite exceptional. Thus the single case of unequal concentration of the electrolyte cannot be one of equilibrium since the process of diffusion cannot be stopped. Cases in which equilibrium does not subsist were formally excluded by my subject, and indeed could not be satisfactorily treated without the introduction of new ideas quite foreign to those necessary for the treatment of equilibrium.
Again, the consideration of the electrical potential in the electrolyte, and especially the consideration of the difference of potential in electrolyte and electrode, involves the consideration of quantities of which we have no apparent means of physical measurement, while the difference of potential in "pieces of metal of the same kind attached to the electrodes" is exactly one of the things which we can and do measure.
Nevertheless, with some hedging in regard to the definition of the electrical potential, we may apply
* $V''-V'=a_{a}(\mu _{a}'-\mu _{a}'')$
This gives
* $V''-V'=a_{a}\left(B+{\frac {At}{M_{a}}}\log \gamma _{a}-\mu _{a}''\right),$
$V''-V'={\frac {a_{a}At}{M_{a}}}\log {\frac {\gamma _{a}}{G}}\cdot$
$G$
$P$
$a$
The case of unequal concentration, or, in general, cases in which the electrolyte is not homogeneous, I should treat as follows; Let us suppose for convenience that the cell is in form of a rectangular parallelepiped with edge parallel to axis of x and cross section of unit area. The electrolyte is supposed homogeneous in planes parallel to the ends, which are formed by the electrodes.
Of course we should have equilibrium if proper forces could be applied to prevent the migration of the ions and also of the part of the solutum which is not dissociated. What would these forces be? For the molecules ($_{12}$) which are not dissociated, the force per unit of mass would be ${\frac {d\mu _{12}}{dx}}\cdot$ (The problem is practically the same as that discussed in E.H.S. [this volume], pp. 144 ff.) If the unit of mass of ​the cation ($_1$) has the charge $c_{1}$, the force necessary to prevent its migration would be
* ${\frac {d\mu _{1}}{dx}}+c_{1}{\frac {dV}{dx}}\cdot$
* ${\frac {d\mu _{2}}{dx}}-c_{2}{\frac {dV}{dx}}\cdot$
* $-{\frac {k_{1}}{c_{1}}}\left({\frac {d\mu _{1}}{dx}}+c_{1}{\frac {dV}{dx}}\right),$
| $\phi _{1}=-{\frac {k_{1}}{c_{1}}}\gamma _{1}\left({\frac {d\mu _{1}}{dx}}+c_{1}{\frac {dV}{dx}}\right)=-At{\frac {k_{1}}{c_{1}M_{1}}}{\frac {d\gamma _{1}}{dx}}-k_{1}{\frac {dV}{dx}}\gamma _{1};$ | (3) |
| $\phi _{2}=-{\frac {k_{2}}{c_{2}}}\gamma _{2}\left({\frac {d\mu _{2}}{dx}}+c_{2}{\frac {dV}{dx}}\right)=-At{\frac {k_{2}}{c_{2}M_{2}}}{\frac {d\gamma _{2}}{dx}}+k_{2}{\frac {dV}{dx}}\gamma _{2},$ | (4) |
$k_{1},k_{2}$
* $\phi =\textstyle \sum \displaystyle \pm c_{1}\phi _{1},$
* $\phi =At\textstyle \sum \displaystyle \mp {\frac {k_{1}}{M_{1}}}{\frac {d\gamma _{1}}{dx}}-{\frac {dV}{dx}}\textstyle \sum \displaystyle c_{1}k_{1}\gamma _{1}.$
* $\phi {\frac {dx}{\textstyle \sum \displaystyle c_{1}k_{1}\gamma _{1}}}=At{\frac {\textstyle \sum \displaystyle \mp {\frac {k_{1}}{M_{1}}}d\gamma _{1}}{\textstyle \sum \displaystyle c_{1}k_{1}\gamma _{1}}}-dV.$
$\phi$
${\frac {dx}{\textstyle \sum \displaystyle c_{1}k_{1}\gamma _{1}}}$
​Integrating from one point to another in the electrolyte,
* $\phi \int {\frac {dx}{\textstyle \sum \displaystyle c_{1}k_{1}\gamma _{1}}}=At\int {\frac {\textstyle \sum \displaystyle \mp {\frac {k_{1}}{M_{1}}}d\gamma _{1}}{\textstyle \sum \displaystyle c_{1}k_{1}\gamma _{1}}}+V'-V''.$
* $\phi {\frac {dx}{c_{1}k_{1}\gamma _{1}+c_{2}k_{2}\gamma _{2}}}=At{\frac {-{\frac {k_{1}}{M_{1}}}d\gamma _{1}+{\frac {k_{2}}{M_{2}}}d\gamma _{2}}{c_{1}k_{1}\gamma _{1}+c_{2}k_{2}\gamma _{2}}}-dV,$
$c_{1}\gamma _{1}=c_{2}\gamma _{2}$
$c_{1}d\gamma _{1}=c_{2}d\gamma _{2}$
* $\phi {\frac {dx}{c_{1}k_{1}\gamma _{1}+c_{2}k_{2}\gamma _{2}}}=At{\frac {-{\frac {k_{1}}{c_{1}M_{1}}}+{\frac {k_{2}}{c_{2}M_{2}}}}{k_{1}+k_{2}}}{\frac {d\gamma _{1}}{\gamma _{1}}}-dV,$
* $\phi {\frac {dx}{c_{1}k_{1}\gamma _{1}+c_{2}k_{2}\gamma _{2}}}=At{\frac {-{\frac {k_{1}}{c_{1}M_{1}}}+{\frac {k_{2}}{c_{2}M_{2}}}}{k_{1}+k_{2}}}\log {\frac {\gamma _{1}''}{\gamma _{1}'}}+V'-V''.$
For $c_{1}M_{1}$ and $c_{2}M_{2}$ we may write ${\frac {v_{1}}{a_{\text{H}}}}$ and ${\frac {v_{2}}{a_{\text{H}}}}$, where $v_{1}$ and $v_{2}$ are the "valencies" of the molecules. This gives
* $V''-V'=a_{\text{H}}At{\frac {{\frac {k_{1}}{v_{1}}}-{\frac {k_{1}}{v_{1}}}}{k_{1}+k_{2}}}\log {\frac {\gamma _{1}'}{\gamma _{1}''}},$for$\phi =0$(circuit open).
$V$
Planck's problem is less simple. We may regard it as relating to a tube connecting the two great reservoirs filled with different electrolytes of same concentration, i.e., $\textstyle \sum _{0}\displaystyle c_{0}\gamma _{0}'=\textstyle \sum _{0}\displaystyle c_{0}\gamma _{0}''$. I use ($_0$) for an y ion, ($_1$) for any cation, ($_2$) for any anion. [The accents ($'$) and ($''$) refer to the two reservoirs.]
The tube is supposed to have reached a stationary state and dissociation is complete. The number of ions is immaterial, but they all must have the same valency $v$.
Now by equations (3) and (4), since $c_{0}M_{0}={\frac {v}{a_{\text{H}}}}$,
* $\phi _{0}=-{\frac {aAt}{v}}k_{0}{\frac {d\gamma _{0}}{dx}}\mp k_{0}{\frac {dV}{dx}}\gamma _{0},$
$T$
${\frac {aAt}{v}},$
* ${\begin{aligned}\phi _{0}&=-Tk_{0}{\frac {d\gamma _{0}}{dx}}\mp k_{0}\gamma _{0}{\frac {dV}{dx}},\\{\frac {c_{0}\phi _{0}}{k_{0}}}&=-Tc_{0}{\frac {d\gamma _{0}}{dx}}\mp c_{0}\gamma _{0}{\frac {dV}{dx}},\\\textstyle \sum _{0}\displaystyle {\frac {c_{0}\phi _{0}}{k_{0}}}&=-T\textstyle \sum _{0}\displaystyle {\frac {c_{0}d\gamma _{0}}{dx}}\end{aligned}}$
$\mp {\frac {dV}{dx}}k_{0}\gamma _{0}$
$\textstyle \sum \displaystyle c_{1}\gamma _{1}=\textstyle \sum \displaystyle c_{2}\gamma _{2}$
* $\textstyle \sum _{0}\displaystyle \mp {\frac {c_{0}\phi _{0}}{k_{0}}}=-{\frac {dV}{dx}}\textstyle \sum _{0}\displaystyle c_{0}k_{0}\gamma _{0}.$
${\frac {\textstyle \sum _{0}\displaystyle c_{0}\gamma _{0}}{dx}}$
$\textstyle \sum _{0}\displaystyle c_{0}k_{0}\gamma _{0}''=\textstyle \sum _{0}\displaystyle c_{0}k_{0}\gamma _{0}',$
$\textstyle \sum _{0}\displaystyle c_{0}k_{0}\gamma _{0}$
${\frac {dV}{dx}}$
$X=-{\frac {dV}{dx}}\cdot$
Our original equation is
* $\phi _{0}=-Tk_{0}{\frac {d\gamma _{0}}{dx}}\pm Xk_{0}\gamma _{0}.$
Now with $X$ constant this is easily integrated.
* ${\begin{aligned}{\frac {\phi _{0}}{Xk}}&=-{\frac {T}{X}}{\frac {d\gamma _{0}}{dx}}\pm \gamma _{0},\\\pm {\frac {T}{X}}{\frac {d\gamma _{0}}{dx}}&=\gamma _{0}\mp {\frac {\phi _{0}}{Xk_{0}}},\\{\frac {d\gamma _{0}}{\gamma _{0}\mp {\frac {\phi _{0}}{Xk_{0}}}}}&=\pm {\frac {X}{T}}dx,\\\log \left(\gamma _{0}\mp {\frac {\phi _{0}}{Xk_{0}}}\right)&=\pm {\frac {X}{T}}x+\log H_{0},\\\gamma _{0}\mp {\frac {\phi _{0}}{Xk_{0}}}&=H_{0}e^{\pm {\frac {X}{T}}x}.\end{aligned}}$
$H_{0}$
* $\gamma _{0}''-\gamma _{0}'=H_{0}\left(e^{\mp {\frac {X}{T}}x''}-e^{\pm {\frac {X}{T}}x'}\right).$
* $x'=-x''.$
* Let$P=e^{{\frac {X}{T}}x''},$$\gamma _{0}''-\gamma _{0}'=\pm H_{0}(P-P^{-1}).$ Let$\Delta _{0}=\gamma _{0}''-\gamma _{0}',$$\gamma _{0}\mp {\frac {\phi _{0}}{Xk_{0}}}=\pm \Delta _{0}{\frac {e^{\pm {\frac {X}{T}}x}}{P-P^{-1}}},$
* $c_{0}k_{0}\gamma _{0}\mp {\frac {c_{0}\gamma _{0}}{X}}=\pm c_{0}k_{0}\Delta _{0}{\frac {e^{\pm {\frac {X}{T}}x}}{P-P^{-1}}}\cdot$
* $\textstyle \sum _{0}\displaystyle c_{0}k_{0}\gamma _{0}=\textstyle \sum _{0}\displaystyle \pm c_{0}k_{0}\Delta _{0}{\frac {e^{\pm {\frac {X}{T}}x}}{P-P^{-1}}}\cdot$
* ${\begin{aligned}\textstyle \sum _{0}\displaystyle c_{0}k_{0}\gamma _{0}''+\textstyle \sum _{0}\displaystyle c_{0}k_{0}\gamma _{0}'&=\textstyle \sum _{0}\displaystyle \pm c_{0}k_{0}\Delta _{0}{\frac {e^{\pm {\frac {X}{T}}x''}+e^{\pm {\frac {X}{T}}x'}}{P-P^{-1}}},\\&=\textstyle \sum _{0}\displaystyle \pm c_{0}k_{0}\Delta _{0}{\frac {P+P^{-1}}{P-P^{-1}}}\cdot \end{aligned}}$
* ${\begin{aligned}K_{1}'=\textstyle \sum _{1}\displaystyle c_{1}k_{1}\gamma _{1}',&K_{1}''=\textstyle \sum _{1}\displaystyle c_{1}k_{1}\gamma _{1}'',\\K_{2}'=\textstyle \sum _{2}\displaystyle c_{2}k_{2}\gamma _{2}',&K_{2}''=\textstyle \sum _{2}\displaystyle c_{2}k_{2}\gamma _{2}''.\end{aligned}}$
* $K_{1}''+K_{2}''+K_{1}'+K_{2}'={\frac {P+P^{-1}}{P-P^{-1}}}(K_{1}''-K_{1}'-K_{2}''+K_{2}'),$
* which gives$P^{2}={\frac {K_{1}''+K_{2}'}{K_{2}''+K_{1}'}}\cdot$
| Now | $\log P$ | $={\frac {X}{T}}x''={\frac {-X}{T}}x',$ | $2\log P$ | $={\frac {X(x''-x')}{T}}={\frac {V'-V''}{T}}$ |
| | ${\frac {V'-V''}{T}}$ | $=\log {\frac {K_{1}''+K_{2}'}{K_{1}'+K_{2}''}},$ | $V''-V'$ | $={\frac {Aat}{v}}\log {\frac {K_{1}'+K_{2}''}{K_{1}''+K_{2}'}}\cdot$ |
$K_{1}'$ is the part of the conductivity of the first electrolyte which is due to the cations.
If the first electrolyte contains only one cation ($_1$) and one anion ($_2$), and the second only one cation ($_3$) and one anion ($_4$), we have
| | $V''-V'$ | $={\frac {Aat}{v}}\log {\frac {c_{1}k_{1}\gamma _{1}'+c_{4}k_{4}\gamma _{4}''}{c_{3}k_{3}\gamma _{3}''+c_{2}k_{2}\gamma _{2}'}},$ |
| or, since | $c_{1}\gamma _{1}'$ | $=c_{2}\gamma _{2}'=c_{3}\gamma _{3}''=c_{4}\gamma _{4}'',$ |
| | $V''-V'$ | $={\frac {Aat}{v}}\log {\frac {k_{1}+k_{4}}{k_{3}+k_{2}}},$ |
​I regret that I have been obliged to delay my writing so long. I presume that you would have preferred to have me reply more promptly and more briefly. But the matter did not seem to be capable of being dispatched in few words.
One might easily economize in letters in the formulae by referring densities ($\gamma$) and potentials ($\mu$) to equivalent or molecular weights, as you have done, but I thought I was more sure to be understood with the notations which I have used. Moreover, since the molecular weight is often the doubtful point in the whole problem, there is a certain advantage in bringing it in explicitly rather than implicitly, so that we can see at a glance how a change in our assumptions in regard to the molecules will affect the measurable quantities.
Yours, very sincerely,
J. Willard Gibbs.
* ↑ [See Preface.]
* ↑ The object of this chapter is to show the relation of the doctrine of potentials to van't Hoff's Law (what form van't Hoff's Law takes from the standpoint of the potentials); and to the modern theory of dilute solutions as developed by van't Hoff and Arrhenius. "Equilib. Het. Subs." [this volume], pp. 135–138, 138–144, 164–165, 168–172, 172–184.
* ↑ For the definition of this term see p. 93, also pp. 92–96.
* ↑ In some cases a semi-permeable membrane may be necessary. (Enlarge.) (Is the term coexistent right in this case?)
* ↑ Definition. (Enlarge.)
* ↑ ${\frac {pvM}{mt}}=A,$$pv={\frac {m}{M}}At.$ Is absolute used correctly?
* ↑ Also the potentials of water and alcohol in a mixture may be measured in a vertical tube of sufficient height. [See p. 413.]
* ↑ [In the following discussion, $D$ indicates the dissolved substance, or solutum, and $S$ the solvent.]
* ↑ Not experimentally found.
* ↑ [$p$ is the vapor pressure of the pure solvent, $P$ that of the solution.]
* ↑ Assuming that the vapor behaves like an ideal gas, we have $\gamma _{\text{S}}'={\frac {pM_{\text{S}}}{At}}\cdot$
* ↑ [$\gamma$ refers to the liquid, and $\gamma '$ to the gaseous phase.]
* ↑ [Although left blank in the MS., this probably refers to p. 423.]
* ↑ [Probably equation [12].]
* ↑ [Probably equation [13].]
* ↑ [The valence of the ion].
* ↑ [$c_{2}$ is a positive number equal numerically to the negative charge on unit mass of the anion.]
* ↑ [The positive direction for both these fluxes is the direction of increasing $x$.]
* ↑ [The sign of the charge is not included in $c$. Honce the double sign is necessary.]
* ↑ [Planck, Wied. Ann., vol. xl (1890), p. 561.]