Page:Scientific Papers of Josiah Willard Gibbs.djvu/374

General Properties of a Perfect Electro-chemical Apparatus.

When an electrical current passes through a galvanic or electrolytic cell, the state of the cell is altered. If no changes take place in the cell except during the passage of the current, and all changes which accompany the current can be reversed by reversing the current, the cell may be called a perfect electro-chemical apparatus. The electromotive force of the cell may be determined by the equations which have just been given. But some of the general relations to which such an apparatus is subject may be conveniently stated in a form in which the ions are not explicitly mentioned.

In the most general case, we may regard the cell as subject to external action of four different kinds. (1) The supply of electricity at one electrode and the withdrawal of the same quantity at the other. (2) The supply or withdrawal of a certain quantity of heat. (3) The action of gravity. (4) The motion of the surfaces enclosing the apparatus, as when its volume is increased by the liberation of gases.

The increase of the energy in the cell is necessarily equal to that which it receives from external sources. We may express this by the equation

$d\epsilon =(V'-V'')de+dQ+dW_{\text{G}}+dW_{\text{P}}$

(691)

in which $d\epsilon$ denotes the increment of the intrinsic energy of the cell, $de$ the quantity of electricity which passes through it, $V'$ and $V''$ the electrical potentials in masses of the same kind of metal connected with the anode and cathode respectively, $dQ$ the heat received from external bodies, $dW_{\text{G}}$ the work done by gravity, and $dW_{\text{P}}$ the work done by the pressures which act on the external surface of the apparatus.

The conditions under which we suppose the processes to take place are such that the increase of the entropy of the apparatus is equal to the entropy which it receives from external sources. The only external source of entropy is the heat which is communicated to the cell by the surrounding bodies. If we write $d\eta$ for the increment of entropy in the cell, and $t$ for the temperature, we have

$d\eta ={\frac {dQ}{t}}\cdot$

(692)

Eliminating $dQ$, we obtain

$d\epsilon =(V'-V'')de+td\eta +dW_{\text{G}}+dW_{\text{P}},$

(693)

or

$V''-V'=-{\frac {d\epsilon }{de}}+t{\frac {d\eta }{de}}+{\frac {dW_{\text{G}}}{de}}+{\frac {dW_{\text{P}}}{de}}\cdot$

(694)

It is worth while to notice that if we give up the condition of the reversibility of the processes, so that the cell is no longer supposed