(3) [15 points] Partition function of ideal gas comprising N particles of mass m in potential V (r) at temperature T (or / = 1/AT) is given by the following: 3N/2 m Z = N! 2nh-B Consider ideal gas comprising Mparticles of mass m in equilibrium at temperature 7, and confined in a box 0 < x = 1; 0 < y = L; ho < z s h1, in potential V(x, y, z) = mgz (ie. uniform gravity). (a) Calculate the partition function. (b) Calculate the Helmholtz free energy F(B, L, ho, h,)- (c) Describe that the pressures at the bottom and top of the box are given by 1 OF 1 OF Po = 12 aho P1 = 12 Oh1 and calculate po, P1- (d) Calculate the heat capacity Cy of this gas in the box. Your answer depends on h = h, - ho: so it is different from 3/,R of monoatomic ideal gas. Evaluate the difference between your calculated Cy and 3/-R for a realistic measurement using argon gas (m = 40 [g/mol])