PHYSICS 313
Problem set 6 2001
Given November 9
Due November 19

Problem 1:
A sealed cylindrical vessel is in contact with a heat bath at constant temperature $T$. A friction-less airtight piston of weight $mg$ divides the container into two volumes $V_1$ and $V_2=V-V_1.$ There are $N$ ideal gas atoms in each partition. The area of the piston is $A$.
(a). Find the equilibrium height of the piston.
(b). The ideal gas is replaced by a real single component gas. At a certain temperature the top partition is found to contain a puddle of liquid coexisting with its vapor. Which of the following statements may be true at equilibrium:

(i). The bottom partition contains a liquid in coexistence with its vapor. (ii). The bottom partition contains only vapor. (iii). The bottom partition contains only liquid.

Problem 2:
The Helmholtz Free energy of a "hard sphere fluid" can be approximated by

$$F(N,V,T)=-Nk_BT \{\ln\left[\frac{V-Nb}{Nv_q}\right]+1\}$$

where

$$v_q=\frac{constant}{T^{5/2}}$$

and $b$ is another positive constant.
a:
Find the pressure as a function of volume and temperature.
b:
Find the chemical potential.
c:
Does this hard sphere fluid have a liquid-gas phase transition?
Problem 3:
$N_A$ molecules of one species and $N_B$ molecules of another species are initially kept separate in two containers each at pressure $P$ and temperature $T$. The Helmholtz free energy of a mixture of the two gases can be approximated by the "hard sphere" expression

$$F(N_A,N_B,V,T)=-(N_A+N_B)k_BT \{\ln\left[{{V-N_Ab_A-N_Bb_B} \over{(N_A+N_B)v_q}}\right]+1\}$$

where $b_A$ and $b_B$ are positive constants and as before

$$v_q=\frac{constant}{T^{5/2}}$$

a: If the two gases are mixed at constant temperature $T$ and pressure $P$ what is the volume of the mixture?
b: What is the entropy the mixture?