Question

12) we also show in Lecture #5 that, for any system, OU where β is the constant-pressure (isobaric) thermal expansivity and κ is the constant temperature (isothermal) compressibility. Use this result, along with typical values for /P and K, to show that it is a good approximation to neglect the dependence on volume of the internal energy ofa liquid or solid. Explicity, show that: TF

Answer 1

A) du = (partial differential u/partial differential T)_v dT + (partial differential u/partial differential v)_T dv & you can integrate for the special case of constant T, but then (partial differential u/partial differential v)_T is not The value of partial derivative can be obtained from the Helmboltz equation (partial differential u/partial differential v)_T = T^2[partial differential/partial differential T(P/T)]_v Integration gives U(T, V, N) = u_0(T, V_0, N) + integral^V_V_0 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^V_infinity T^2 (partial differential/partial differential T P/T)_V dv Substituting P/T = NR/V - Nb - a/T N^2/V^2 & working out yields the internal of the van der wall gas: U(T, V, N) = u_ideal downarrow nc_v T(T, N) - a N^2/V (partial differential u/partial differential T)_T van = partial differential u/partial differential T ideal - partial differential/partial differential T(a N^2/V) = partial differential u/partial differential T ideal - a N^2 partial differential/partial differential N (t) partial differential V/partial differential T = (partial differential u/partial differential T ideal)_T + a N^2/V^2 (partial differential V/partial differential T)_T b) we can see internal energy depend on volume for v rightarrow large (a N^2/V) rightarrow 0
u can skip last part written on first page.