Question

11) We know the internal energy of a given quantity of an ideal gas depends only on its temperature. There is no change in internal energy purely due to a change in volume. But what about for a real gas? Does the energy depend on volume and, if so, how important is it to account for this? In Lecture #5 we show that, when a system undergoes an isothermal process, the change in internal energy due to a change in volume is given by: Evaluate this change-as an explicit function of Vand V-for a van der Walls gas. Show that (a) with a van der Walls gas, the internal energy does depend on the volume; and (b) the dependence of the internal energy upon volume is small, such that it is a good approximation to neglect it. You can do part b by comparing the change in internal energy due purely to a volume change (e.g, 1 mole of a typical gas is expanded from 20L to 40L at constant temperature) to that due purely to a temperature change (e.g, 1 mole of a diatomic gas is heated from 200K to 400K).

Answer 1

a) dU = (partial differential U/partial differential T)_V dT + (partial differential U/partial differential V)_T dV And you can integrate for the special case of constant T, but then (partial differential U/partial differential V)_T is not (_P). The value of partial derivative can be obtained from the Helmholtz equation (partial differential U/partial differential V)_T = T^2 [partial differential/partial differential (P/T)]_V Integration gives. U (T, V, N) = U_0 (T, V_0, N) + integral_V_0^V T^2 (partial differential/partial differential T P/T)_V dV. V_0 rightarrow infinity u rightarrow U_ideal U (T, V, N) = U_ideal (T, N) + integral_infinity^V T^2 (partial difference/ partial difference T P/T)_V^dV Substituting P/T = NR/V-Nb - q/T N^2/V^2 and working out yields the internal of the van der wall gas: U(T, V, N) = (I_ideal downarrow nC_v T (T, N) - aN^2/V (partial differential U/partial differential T)_T^van = partial difference U ideal/ partial difference T - partial difference/ partial difference T (q N^2/V) = partial difference U ideal/ partial difference T - a N^2.