Callen H.B - Thermodynamics And An Introduction To Thermostatistics · Problem 4.8-1

ALTERNATIVE FORMULATIONS AND LEGENDRE TRANSFORMATIONS

Physics — Statistical Mechanics · entropy-measurability-identity

Problem

To corroborate equation 4.22 show that

\left(\frac{\partial P}{\partial s}\right)_T = -\left(\frac{\partial T}{\partial v}\right)_P

First consider the right-hand side, and write generally that

dT = u_{ss}\,ds + u_{vs}\,dv

so that

\left(\frac{\partial T}{\partial v}\right)_P = u_{ss}\left(\frac{\partial s}{\partial v}\right)_P + u_{vs} = -u_{ss}\frac{u_{vv}}{u_{sv}} + u_{vs}

Similarly show that

\left(\frac{\partial P}{\partial s}\right)_T = u_{ss}u_{vv}/u_{sv} - u_{sv}

, establishing the required identity.