2. Thermodynamics of the Free and Independent Electron Gas (a) Deduce from the thermodynamic identities − (è). – π (²²). (2.96) from Eqs. (2.56) and (2.57), and from the third law of thermodynamics (s 0 as T→ 0) that the entropy density, s S/V is given by: dk -ks Inf+ (1 - ) In (1-)], where f(&(k)) is the Fermi function (Eq. (2.56)). (2.97) (b) Since the pressure P satisfies Eq. (B.5) in Appendix B, P=-(u - Ts - μn), deduce from (2.97) that dk P = k T √ In (1 + exp[-h 4s in (h2k2/2m)-H KBT (2.98) Show that (2.98) implies that P is a homogeneous function of μ and Tof degree 5/2; that is, for any constant 2. P(T) = 25/2P(H, T) (c) Deduce from the thermodynamic relations in Appendix B that (2.99) (on), = n, P = zu = S. (2.100) (2.101) (e) Show that when kgT << &, the ratio of the constant-pressure to constant-volume specific heats satisfies 1 = +0 (f) Show, by retaining further terms in the Sommerfeld expansions of u and n, that correct to order 73 the electronic heat capacity is given by C₁ = -kB²Tg(&F) kT³g(8) 90 'g'(Ep) A+ d) [15 (8))" - 21 (1) g"(&F) 9(&F) (2.102) We have introduced the Fermi function f(E) to emphasize that f, depends on k only through the electronic energy &(k): 1 f(ε) = e-H)/kBT +1 (2.56) If we divide both sides of (2.55) by the volume V, then (2.29) permits us to write the energy density u = U/V as dk 11 = 4π3 ε(k)f(&(k)). (2.57)