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]
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