(a) Use the Maxwell relations to express the derivatives (∂S/∂V)_T , (∂V/∂S)_p, (∂p/∂S)_V , and (∂V/∂S)_p in terms of the heat capacities, the expansion coefficient α=(1/V)(∂V/∂T)_p, and the isothermal compressibility, κ_T =−(1/V) (∂V/∂p)_T .

(b) The Joule coefficient, μ_J , is defined as μ_J=(∂T/∂V)_U. Show that μ_J C_V =p–αT/κ_T.