Question

Use the master equation for Helmholtz free energy, A, to determine the change in entropy as a function of volume under constant temperature conditions (ds/dv) for a van der waals gas

Answer 1

From the master equation for Helmholtz free energy, A,

A = U - TS

We can take the derivative with respect to volume at constant temperature to obtain:

( dA / dV)_T = (dU / dV)_T - (dS / dV)_T

but, (dU / dV )_T = T(dS / dV)_T - P

We can use one of the Maxwell relations to obtain:

(dU / dV )_T = T ( dP / dT )_V - P

The equation is known as a thermodynamic equation of state. It relates the internal energy dependence on volume only to temperature and pressure.

The calculation of the internal pressure is:

(dU / dV )_T = T ( nR / ( V - nb)) - nRT / ( V - nb) + n²a / V²

   ⁼ n²a / V²

(dS / dV)_{T =}   (dU / dV)_T - ( dA / dV)_T

   ₌ n²a / V² - ( dA / dV)_T