(Solved): Critical point behavior: the pressure P of a gas is related to its density n = N/V , and temperature ...

Critical point behavior: the pressure P of a gas is related to its density n = N/V , and temperature T by the truncated expansion P = kBT n ? b 2 n 2 + c 6 n 3 where b and c are assumed to be positive temperature-independent constants. (a) Locate the critical temperature Tc below which this equation must be invalid, and the corresponding density nc and pressure Pc of the critical point. Hence find the ratio kBTcnc/Pc. (b) Calculate the isothermal compressibility ?T = ? 1 V ?V ?P |T , and sketch its behavior as a function of T for n = nc. (c) On the critical isotherm give an expression for (P ? Pc) as a function of (n ? nc). (d) The instability in the isotherms for T < Tc is avoided by phase separation into a liquid of den- sity n+ and a gas of density n?. For temperatures close to Tc, these densities behave as n± ? nc(1±?). Using a Maxwell construction, or otherwise, find an implicit equation for ?(T), and indicate its behavior for (Tc ? T) ? 0. (Hint. Along an isotherm, variations of chemical potential obey d? = dP/n.)