Scientific Papers of Josiah Willard Gibbs, Volume 1/Chapter VII

by Josiah Willard Gibbs

VII.

ELECTROCHEMICAL THERMODYNAMICS.

Two letters to the Secretary of the Electrolysis Committee of the British Association for the Advancement of Science.

[Report Brit. Asso. Adv. Sci, 1886, pp. 388, 389; and 1888, pp. 343–346.]

New Haven, January 8, 1887.

Professor Oliver J. Lodge,

Dear Sir,—Please accept my thanks for the proof copy of your "Report on Electrolysis in its Physical and Chemical Bearings," which I received a few days ago with the invitation, as I understand it, to comment thereon.

I do not know that I have anything to say on the subjects more specifically discussed in this report, but I hope I shall not do violence to the spirit of your kind invitation or too much presume on your patience if I shall say a few words on that part of the general subject which you discussed with great clearness in your last report on pages 745 ff. (Aberdeen). To be more readily understood, I shall use your notation and terminology, and consider the most simple case possible.

Suppose that two radicles unite in a galvanic cell during the passage of a unit of electricity, and suppose that the same quantities of the radicles would give $\theta \epsilon$ units of heat in uniting directly, that is, without production of current; will the union of the radicles in the galvanic cell give $J\theta \epsilon$ units of electrical work? Certainly not, unless the radicles can produce the heat at an infinitely high temperature, which is not, so far as we know, the usual case. Suppose the highest temperature at which the heat can be produced is $t''$, so that at this temperature the union of the radicles with evolution of heat is a reversible process; and let $t'$ be the temperature of the cell, both temperatures being measured on the absolute scale. Now $\theta \epsilon$ units of heat at the temperature $t''$ are equivalent to $\theta \epsilon {\frac {t'}{t''}}$ units of heat at the temperature $t'$, together with $J\theta \epsilon {\frac {t''-t'}{t''}}$ units of mechanical or electrical work. (I use the term "equivalent" strictly to denote reciprocal convertibility, and not in the loose and often misleading sense in which we speak of heat and work as equivalent when there is only a one-sided convertibility.) Therefore the rendement of a perfect or reversible galvanic cell would be ${\text{J}}\theta \epsilon {\frac {t''-t'}{t''}}$ units of electrical work, with $\theta \epsilon {\frac {t'}{t''}}$ units of (reversible) heat, for each unit of electricity which passes.

You will observe that we have thus solved a very different problem from that which finds its answer in the Joule-Helmholtz-Thomson equation with term for reversible heat. That equation gives a relation between the e.m.f. and the reversible heat and certain other quantities, so that if we set up the cell and measure the reversible heat, we may determine the e.m.f. without direct measurement, or vice versâ. But the considerations just adduced enable us to predict both the electromotive force and the reversible heat without setting up the cell at all. Only in the case that the reversible heat is zero does this distinction vanish, and not then unless we have some way of knowing á priori that this is the case.

From this point of view it will appear, I think, that the production of reversible heat is by no means anything accidental, or superposed, or separable, but that it belongs to the very essence of the operation.

The thermochemical data on which such a prediction of e.m.f. and reversible heat is based must be something more than the heat of union of the radicles. They must give information on the more delicate question of the temperature at which that heat can be obtained. In the terminology of Clausius they must relate to entropy as well as to energy—a field of inquiry which has been far too much neglected.

Essentially the same view of the subject I have given in a form more general and more analytical, and, I fear, less easily intelligible, in the closing pages of a somewhat lengthy paper on the "Equilibrium of Heterogeneous Substances" (Conn. Acad. Trans., vol. iii, 1878), of which I send you the Second Part, which contains the passage in question. My separate edition of the First Part has long been exhausted. The question whether the "reversible heat" is a negligible quantity is discussed somewhat at length on pages 510–519.^[1] On page 503^[2] is shown the connection between the electromotive force of a cell and the difference in the value of (what I call) the potential for one of the ions at the electrodes. The definition of the potential for a material substance, in the sense in which I use the term, will be found on page 443^[3] of the synopsis from the Am. Jour. Sci., vol. xvi, which I enclose. I cannot say that the term has been adopted by physicists. It has, however, received the unqualified commendation of Professor Maxwell (although not with reference to this particular application—see his lecture on the "Equilibrium of Heterogeneous Substances," in the science conferences at South Kensington, 1876); and I do not see how we can do very well without the idea in certain kinds of investigations.

Hoping that the importance of the subject will excuse the length of this letter,

I remain,

Yours faithfully,

J. Willard Gibbs.

New Haven, November 21, 1887.

Professor Oliver J. Lodge,

Dear Sir,—As the letter which I wrote you some time since concerning the rendement of a perfect or reversible galvanic cell seems to have occasioned some discussion, I should like to express my views a little more fully.

It is easy to put the matter in the canonical form of a Carnot's cycle. Let a unit of electricity pass through the cell producing certain changes. We may suppose the cell brought back to its original condition by some reversible chemical process, involving a certain expenditure (positive or negative) of work and heat, but involving no electrical current nor any permanent changes in other bodies except the supply of this work and heat.

Now the first law of thermodynamics requires that the algebraic sum of all the work and, heat (measured in "equivalent" units) supplied by external bodies during the passage of the electricity through the cell, and the subsequent processes by which the cell is restored to its original condition, shall be zero.

And the second law requires that the algebraic sum of all the heat received from external bodies, divided, each portion thereof, by the absolute temperature at which it is received, shall be zero.

Let us write ${\text{W}}$ for the work and ${\text{Q}}$ for the heat supplied by external bodies during the passage of the electricity, and $[{\text{W}}],[{\text{Q}}]$ for the work and heat supplied in the subsequent processes.

Then${\text{W}}+{\text{Q}}+[{\text{W}}]+[{\text{Q}}]=0,$

(1)

and${\frac {\text{Q}}{t'}}+\int {\frac {d[{\text{Q}}]}{t}}=0,$

(2)

where $t$ under the integral sign denotes the temperature at which the element of heat $d[{\text{Q}}]$ is supplied, and tf the temperature of the cell, which we may suppose constant.

Now the work ${\text{W}}$ includes that required to carry a unit of electricity from the cathode having the potential ${\text{V}}''$ to the anode having the potential ${\text{V}}'$. (These potentials are to be measured in masses of the same kind of metal attached to the electrodes.) When there is any change of volume, a part of the work will be done by the atmosphere or other body enclosing the cell. Let this part be denoted by ${\text{W}}_{\text{P}}$. In some cases it may be necessary to add a term relating to gravity, but as such considerations are somewhat foreign to the essential nature of the problem which we are considering, we may set such cases aside. We have then

${\text{W}}={\text{V}}'-{\text{V}}''+{\text{W}}_{\text{P}}$

(3)

Combining these equations we obtain

${\text{V}}''-{\text{V}}'={\text{W}}_{\text{P}}+[{\text{W}}]+[{\text{Q}}]-t'\int {\frac {d[{\text{Q}}]}{t}}\cdot$

(4)

It will be observed that this equation gives the electromotive force in terms of quantities which may be determined without setting up the cell.

Now $[{\text{W}}]+[{\text{Q}}]$ represents the increase of the intrinsic energy of the substances in the cell during the processes to which the brackets relate, and ${\frac {d[{\text{Q}}]}{t}}$ represents their increase of entropy during the same processes. The same expressions, therefore, with the contrary signs, will represent the increase of energy and entropy in the cell during the passage of the current. We may therefore write

${\text{V}}''-{\text{V}}'=-\Delta \epsilon +t'\Delta \eta +{\text{W}}_{\text{P}},$

(5)

where $\Delta \epsilon$ and $\Delta \eta$ denote respectively the increase of energy and entropy in the cell during the passage of a unit of electricity. This equation is identical in meaning, and nearly so in form, with equation (694) of the paper cited in my former letter, except that the latter contains the term relating to gravity. See Trans. Conn. Acad., iii (1878), p. 509.^[4] The matter is thus reduced to a question of energy and entropy. Thus, if we knew the energy and entropy of oxygen and hydrogen at the temperature and pressure at which they are disengaged in an electrolytic cell, and also the energy and entropy of the acidulated water from which they are set free (the latter, in strictness, as functions of the degree of concentration of the acid), we could at once determine the electromotive force for a reversible cell. This would be a limit below which the electromotive force required in an actual cell used electrolytically could not fall, and above which the electromotive force of any such cell used to produce a current (as in a Grove's gas battery) could not reach.

Returning to equation (4), we may observe that if $t$ under the integral sign has a constant value, say $t''$, the equation will reduce to

${\text{V}}''-{\text{V}}'={\frac {t''-t'}{t''}}[{\text{Q}}]+[{\text{W}}]+{\text{W}}_{\text{P}}.$

(6)

Such would be the case if we should suppose that at the temperature $t''$ the chemical processes to which the brackets relate take place reversibly with evolution or absorption of heat, and that the heat required to bring the substances from the temperature of the cell to the temperature $t''$, and that obtained in bringing them back again to the temperature of the cell, may be neglected as counterbalancing each other. This is the point of view of my former letter. I do not know that it is necessary to discuss the question whether any such case has a real existence. It appears to me that in supposing such a case we do not exceed the liberty usually allowed in theoretical discussions. But if this should appear doubtful, I would observe that the equation (6) must hold in all cases if we give a slightly different definition to $t''$, viz., if $t''$ be defined as a temperature determined so that

${\frac {[{\text{Q}}]}{t}}=\int {\frac {d[{\text{Q}}]}{t}}\cdot$

(7)

The temperature $t''$, thus defined, will have an important physical meaning. For by means of perfect thermo-dynamic engines we may change a supply of heat $[{\text{Q}}]$ at the constant temperature $t''$ into a supply distributed among the various temperatures represented by $t$ in the manner implied in the integral, or vice versâ. We may, therefore, while vastly complicating the experimental operations involved, obtain a theoretical result which may be very simply stated and discussed. For we now see that after the passage of the current we may (theoretically) by reversible processes bring back the cell to its original state simply by the expenditure of the heat $[{\text{Q}}]$ supplied at the temperature $t''$, with perhaps a certain amount of work represented by $[{\text{W}}]$, and that the electromotive force of the cell is determined by these quantities in the manner indicated by equation (6), which may sometimes be further simplified by the vanishing of $[{\text{W}}]$ and ${\text{W}}_{\text{P}}$.

If the current causes a separation of radicles, which are afterwards united with evolution of heat, $[{\text{Q}}]$ being in this case negative, $t''$ represents the highest temperature at which this heat can be obtained. I do not mean the highest at which any part of the heat can be obtained—that would be quite indefinite—but the highest at which the whole can be obtained. I should add that if the effect of the union of the radicles is obtained partly in work—$[{\text{W}}]$, and partly in heat—$[{\text{Q}}]$, we may vary the proportion of work and heat; and $t''$ will then vary directly as $[{\text{Q}}]$. But if the effect is obtained entirely in heat, $t''$ will have a perfectly definite value.

It is easy to show that these results are in complete accordance with Helmholtz's differential equation. We have only to differentiate the value which we have found for the electromotive force. For this purpose equation (5) is most suitable. It will be convenient to write ${\text{E}}$ for the electromotive force ${\text{V}}'-{\text{V}}''$, and for the differences $\Delta \epsilon ,\Delta \eta$ to write the fuller forms $\epsilon ''-\epsilon ',\eta ''-\eta '$, where the single and double accents distinguish the values before and after the passage of the current. We may also set $p(v'-v'')$ for ${\text{W}}_{\text{P}}$, where $p$ is the pressure (supposed uniform) to which the cell is subjected, and $v''-v'$ is the increase of volume due to the passage of the current. If we also omit the accent on the $t$, which is no longer required, the equation will read

${\text{E}}=\epsilon ''-\epsilon '-t(\eta ''-\eta ')+p(v''-v').$

(8)

If we suppose the temperature to vary, the pressure remaining constant, we have

$d{\text{E}}=d\epsilon ''.d\epsilon '-td\eta ''+td\eta '-(\eta ''-\eta ')dt+pdv''-pdv'.$

(9)

Now, the increase of energy de is equal to the heat required to increase the temperature of the cell by dt diminished by the work done by the cell in expanding. Since $d\eta '$ is the heat imparted divided by the temperature, the heat imparted is $td\eta '$, and the work is obviously $pdv'$. Hence

$d\epsilon '=td\eta '-pdv',$

and in like manner

$d\epsilon ''=td\eta ''-pdv''.$

If we substitute these values, the equation becomes

$d{\text{E}}=(\eta '-\eta '')dt.$

(10)

We have already seen that $\eta '-\eta ''$ represents the integral $\int {\frac {d[{\text{Q}}]}{t}}$ of equations (2) and (4), which by equation (2) is equal to the reversible heat evolved, Q, divided by the temperature of the cell, which we now call $t$. Substitution of this value gives

${\frac {d{\text{E}}}{dt}}=-{\frac {\text{Q}}{t}},$

(11)

which is Helmholtz's equation.

These results of the second law of thermodynamics are of course not to be applied to any real cells, except so far as they approach the condition of reversible action. They give, however, in many cases limits on one side of which the actual values must lie. Thus, if we set $\leqq$ for $=$ in equations (2), (4), (5), (6), and $\geqq$ for $=$ in (8), the formula will there hold true without the limitation of reversibility. But we cannot get anything by differentiating an inequality, and it does not appear à priori which side of (10) is the greater when the condition of reversibility is not satisfied. The term ${\frac {\text{Q}}{t}}$ in (11) is certainly not greater than $\eta ''-\eta '$, for which it was substituted. But this does not determine which side of (11) is the greater in case of irreversibility. It is the same with Helmholtz's method of proof, which is quite different from that here given, but indicates nothing except so far as the condition of reversibility is fulfilled. (See Sitzungsberidite Berl. Acad., 1882, pp. 24, 25.)

I fear that it is a poor requital for the kind wish which you expressed at Manchester, that I were present to explain and support my position, for me to impose so long a letter upon you. Trusting, however, in your forbearance, I remain, yours faithfully,

J. Willard Gibbs.

1. [This vol., pp. 339–347.]
2. [Ibid., p. 333.]
3. [Ibid., p. 356.]
4. [This volume, p. 388.]