Question

c) Show that dVm by working from the total differential of the state function entropy S(T, Vm) and us- ing one of the Maxwell relations and the relationship linking the differential entropy dS and the heat capacity cv d) Using the differential entropy dS given or derived above, show that the reversible 2] isothermal expansion of a gas causes a change of the internal energy of the gas of by working from the differential form of the internal energy, e) Adapt the formula for internal energy change given or derived above to the situation [3] of a van-der-Waals gas by using the p(Vm) relationship established in (b) to prove that

Answer 1

Total differential of state function entropy will be, ds = partial differential s/partial differential T Vertical-bar _vm dT + partial differential s/partial differential v_m Vertical-bar _T dv_m Because dQ = T partial differential s = > partial differential s = partial differential Q/T ds = 1/T partial differential Q/partial differential T Vertical-bar _vm^dT + partial differential s/partial differential v_m Vertical-bar ^dv_m_T Because c_v = partial differential Q/partial differential T Vertical-bar _vm also partial differential s/partial differential v_m Vertical-bar _T = partial differential P/partial differential T Vertical-bar _vm maxwell Relation ds = c_v/T dT + partial differential P/partial differential T Vertical-bar ^dv_m_vm Because du = T ds - P d V_m (1) substituting value of ds in eqn du = T [cv/T d T + partial differential P/partial differential T Vertical-bar _vm dv_m] = pd v_m \r\n du = c_v dT + T partial differential P/partial differential T Vertical-bar _vm dv_m - p dv_m For isothermal expansion dT = 0 dv = Vertical-bar T partial differential P/partial differential T Vertical-bar _vm - p] dv_m (2) using vanderwards equation