Callen H.B - Thermodynamics And An Introduction To Thermostatistics · Problem 2.2-9

THE CONDITIONS OF EQUILIBRIUM

Physics — Statistical Mechanics · adiabatic-process-energy

Problem

Show that if a single-component system is such that

PV^k

is constant in an adiabatic process (

k

is a positive constant) the energy is

U = \frac{1}{k-1}PV + Nf\left(PV^k/N^k\right)

where

f

is an arbitrary function. Hint:

PV^k

must be a function of

S

, so that

(\partial U/\partial V)_S = g(S)\cdot V^{-k}

, where

g(S)

is an unspecified function.