Page:Scientific Papers of Josiah Willard Gibbs.djvu/385

pressures or by gravity. But the relations expressed by (693), (694), and (696) may be put in a briefer form.

If we set, as on page 89,

$\psi =\epsilon -t\eta ,$

we have, for any constant temperature,

$d\psi =d\epsilon -td\eta ;$

and for any perfect electro-chemical apparatus, the temperature of which is maintained constant,

$V''-V'=-{\frac {d\psi }{de}}+{\frac {dW_{\text{G}}}{de}}+{\frac {dW_{\text{P}}}{de}};$

(697)

and for any cell whatever, when the temperature is maintained uniform and constant,

$(V''-V')de\leqq -d\psi +dW_{\text{G}}+dW_{\text{P}}.$

(698)

In a cell of any ordinary dimensions, the work done by gravity, as well as the inequalities of pressure in different parts of the cell may be neglected. If the pressure as well as the temperature is maintained uniform and constant, and we set, as on page 91,

$\zeta =\epsilon -t\eta +pv,$

where $p$ denotes the pressure in the cell, and $v$ its total volume (including the products of electrolysis), we have

$d\zeta =d\epsilon -td\eta +pdv,$

and for a perfect electro-chemical apparatus,

$V''-V'=-{\frac {d\zeta }{de}},$

(699)

or for any cell,

$(V''-V')de\leqq -d\zeta .$

(700)