The grand-canonical ensemble: general relations in this problem, we want to derive general relations in the grand-canonical ensemble. for this, you can assume that all the functions involved are analytic. the grand-canonical partition function 2(t, v,) is given by (1) bpv 2(17,v ) -σνz(t, v)-epv n-0 a) (3p) in the lecture, we showed that o0(t, v, μ) θμ 1 aln2(t, v, u n) b (2) t, v it, v where 2(t, v, u)=-kbt in2 (t, v, u) is the grand-canonical potential. show in an that the variance of the number of particles is given by analogous manner v a2p (n2)(n) v/n. prove the following relations between the b) (2p) we now introduce the volume per particle v derivatives (4) n out, n av an t, v n ov t, v t. n can be written in terms the volume per particle the free energy f(t, v, n) 5 f(t, v, n) nf(t, u), where the free energy per particle f only depends on v and n implicitly. the chemical potential p and the pressure p can be obtained from the free energy p(t, v,n) avtn (6) (t, v,n) = n t, v c) (3p) show that ap =1 aut. v au t. n d) (3p) use your previous results to show that n (8) kt t, v 1 av is the isothermal compressibility. convince yourself that this means that where kt v bp t. n (n2) (n) a n and thus n2) (n)2/(n) o 1/vn -12