We consider h to be a function of the temperature and pressure h(T, p), recalling that for an ideal gas we have h(p). Recall that the Tds equation is given as du = Tds - pdv. Show that this leads to dh = Tds + vdp. Now write s(T, p) and show that dh = T partial differential s/partial differential T dT + {v + T partial differential s/partial differential p} dp. When the pressure is constant, the above equation must reduce to dh = c_pdT. Thus, T(partial differential s/partial differential T)_p = c_p. Now consider the Gibbs function g = h - Ts and show, using dh = Tds + vdp, that dg = -sdT + vdp: from this result, show that -(partial differential s/partial differential p) = (partial differential v/partial differential T) and that therefore produce the equation dh = c_pdT + v(1 - beta T)dp; beta = 1/v partial differential v/partial differential T Integrate this equation to obtain h_2 - h_1 = integral_T_1^T_2 c_p dT + integral_p_1^p_2 v(1 - beta T) dp. Evaluate this result for a perfect gas, pv = RT, and set it up - evaluate, if possible - for a Van der Waals gas, (p + a/v^2) (v - b) = RT.
