Thermodynamics, 5th Edition

(1.12.22)

With ${(\partial E/\partial S)}_{T,V,n_{j\ne i}}=T$, we find that

${(\partial A/\partial n_i)}_{T,V,n_{j\ne i}}={(\partial E/\partial n_i)}_{S,V,n_{j\ne i}}={\mu }_i,$ (1.12.23)

(1.12.23)

showing that the chemical potential for the state function E is the same as for the state function A so long as the various constraining conditions are properly attended to.

Similarly, starting with the defining relation Eq. (1.12.3), we write

${\mu }_i={(\partial G/\partial n_i)}_{T,P,n_{j\ne i}}={(\partial A/\partial n_i)}_{T,P,n_{j\ne i}}+P{(\partial V/\partial n_i)}_{T,V,n_{j\ne i}},$ (1.12.24)

(1.12.24)

and we again introduce ...