THERMODYNAMICS OF THE VOLTAIC CELL. fa 
To explain this relation ; let us suppose, for example, that p, 
denotes a position co-ordinate; then dp, is a displacement, and 
P, the corresponding force. We see at once that I'd — U is 
numerically equal, but of opposite sign, to the potential energy 
of the system. Since this always holds good so long as the 
parameters are really independent—whatever their number or 
nature may be—we thus arrive at the following general law :— 
In any system of bodies the potential energy of isothermal 
transformation is given by U — T ¢. 
If therefore we wish to investigate the direction in which, or 
the extent to which, any particular isothermal change will take 
place, it is with this function U — T¢ that we have to deal, and 
not—or at least, not directly—with the intrinsic energy or entropy 
of the system. 
3. The Application of the General Law to the determination of 
Llectromotive Force. 
Joule’s law tells us that if a quantity dq of positive electrification 
traverse a circuit in which the e. m. f. is #, the energy liberated 
isothermally in the circuit is # dq. By suitably arranging the 
circuit, it is quite imaginably possible—and very nearly so in 
practice—to obtain all this energy in the form of mechanical 
work. Let us suppose the necessary arrangements for this 
effected ; and let us further assume that no external work is done 
except by the ec. m. f. If then the source of e. m.f. be a reversible 
cell—z.e., if the transfer through it of a positive quantity d q in 
the reverse direction undoes all the chemical and physical reactions 
produced by the original transfer—we have at once :— 
te sy . 
B= —y (U—T¢) © (B) 
as the thermodynamic equation of e. m. f. 
This fundamental relation was first obtained by Willard 
Gibbs* ; the more general relation (4 bis) was subsequently—but 
independently—obtained by Helmholtz,+ who developed its 
consequences in a series of memoirs. t 
The quantity U—Tq has received different names from 
different investigators. Gibbs calls it the “force-function for 
constant temperature” ; Helmholtz, the “free energy” ; Duhem, 
the ‘thermodynamic potential at constant volume.” ‘‘ Potential 
energy of isothermal transformation’§ would probably be the best 
*Trans. Conn. Acad. iii., p. 509, 1878. 
+ See first of the memoirs cited in the ensuing note. 
¢{ Sitzungsb. d. Akad. Wiss., Berlin, 1882; Monatsb. d. Akad. Wiss., Berlin, 1883; 
Sitzungsb. d. Akad. Wiss., Berlin, 1887; Wissensechaftliche Abhandlungen von H. Helmholtz, 
Vols. ii and iii ; translated in ‘*‘ Physical Memoirs” of the Physical Society of London, vol. i. 
§ This is really equivalent to Gibbs’s name; but physicists are more likely to talk about 
** potential energies” than about ‘‘ force functions.” 
