Spring 2006 PHY 203: Introduction to Statistical Mechanics Assignment #9 Due at the beginning of class on Wednesday, 4/12/06 Problems: 1. Free expansion of a van der Waal’s gas (a) Show that for a van der Waals’ equation of state CP − CV = N k 1 − 2a (v − b)2 kT v 3 −1 . [10 points] (b) Also show that, for a van der Waals gas with constant specific heat CV , an adiabatic process conforms to the equation (v − b) T CV /N k = const. [10 points] (c) Further show that the temperature change resulting from an expansion of the gas (into vacuum) from volume V1 to volume V2 is given by N 2a 1 1 T2 − T1 = − CV V2 V1 [10 points] 2. Dietrici equation of state Assume the Dietrici equation of state, P (v − b) = kT exp (−a/(kT v)) , and (a) evaluate the critical constants Pc , vc and Tc of the given system in terms of the parameters a and b, and show that the quantity kTc /(Pc vc ) = e2 /2 ≈ 3.695; [10 points] (b) show that the EoS yields the same expression for the second virial coefficient a2 as the van der Waals EoS does; [10 points] (c) show that for all values of P and for T ≥ Tc , the EoS yields a unique value of v; [10 points] (d) show that for T < Tc , there are three possible values of v for certain values of P and the critical volume vc is always intermediate between the largest and the smallest of the three volumes. [10 points] 1