Statistical Mechanics · Problem 1.9

Elements of Ensemble Theory — R. K. Pathria

Physics — Statistical Mechanics · entropy-extensivity-euler-relation

Problem

Making use of the fact that the entropy

S(N, V, E)

of a thermodynamic system is an extensive quantity, show that

N\left(\frac{\partial S}{\partial N}\right)_{V,E} + V\left(\frac{\partial S}{\partial V}\right)_{N,E} + E\left(\frac{\partial S}{\partial E}\right)_{V,N} = S.

Note that this result implies:

(-N\mu + PV + E)/T = S

, i.e.

N\mu = E + PV - TS