(a) It can be shown that the isothermal and adiabatic compressibilities T and kg of a material with volume V and temperature T fulfils the following equation: KTNS TV B Cp Here 3, is the isobaric expansivity and C, is the isobaric heat capacity i. Derive this equation by expanding dV as function of p and T, and expanding dT as a function of p and S. You will also need to use the help of Maxwell's relations and the chain rule. ii. What does this result show about the relation of KT to ks and why? iii. Show that this equation is correct for the case of an ideal gas where the compress- ibilities are given by Kr=and Ks = with the heat capacity ratio 7. (b) Assume that a substance has a isothermal compressibility of T = a/V and an isobaric expansivity of 3p = 6T2/p with constants a and b. Show that the equation of state is given by V-bT2+ ap = const (c) Assume that the system contains 2 identical particles that can occupy any available state. Let us assume that the system contains 10 single-particle states and that each state has a constant energy value of E=kT. Derive a value for the partition function for the system if we assume the particles are i. two identical fermions ii. two identical bosons