Page:Scientific Papers of Josiah Willard Gibbs.djvu/445

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.^[1] 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.

1. [This volume, p. 388.]