**Problem 1 ** [20 points total]
In lecture we discussed the canonical ensemble, in which the temperature T, volume V, and number of particle N of a system are fixed, and energy E is allowed to fluctuate. Suppose now that you wish to describe the system by a new ensemble in which the pressure p is fixed, and the volume V is allowed to fluctuate.

a) Define the appropriate partition function Z(T, p, N) of the system in this new constant pressure ensemble. [5 points]

b) If you defined Z properly in part (a), then the Gibbs free energy should be given by

G(T, p, N) = -k_{B}T ln Z(T, p, N)

To demonstrate this, show that using G defined from Z as above, the average volume of the system is correctly given by,

<V> = (∂G/∂p)_{T,N} [5 points]

c) Derive a relation, in this constant pressure ensemble, between the isothermal compressibility κ_{T} and fluctuations in the volume V of the system. Show from this relation that the relative fluctuation in V vanishes in the thermodynamic limit. [5 points]

d) Consider an ideal gas of non-relativistic, non-interacting, point particles of mass m. Explicitly compute the partition function Z(T,p,N) of this gas. Use Z to compute G(T,p,N), and then from G compute the specific heat at constant pressure, C_{p}. Show that you get the correct answer for the ideal gas. [5 points]

**Problem 2** [20 points total]
Consider a three dimensional classical ideal gas of atoms of mass m, moving in a potential energy

V(x) = | { |
½ αx^{2} +∞ | x≥0 x<0 |

The infinite potential for x < 0 may be viewed as a rigid wall filling the y-z plane at x=0. The atoms, therefore, are attracted to this wall, but they move freely in the y and z directions. Let T be the temperature and n = N/A be the total number of atoms per unit area of the wall.

a) calculate the local atomic density n(x) (number of atoms per unit volume) as a function of the distance x from the wall. Note that

n= |
0 | n(x) dx |

[5 pts]
b) Show that the ideal gas law (relating pressure, temperature, and density) continues to hold locally everywhere. [5 pts]

c) Find the pressure that the atoms exert on the wall. How does this pressure vary with temperature? [5 pts]

d) Calculate the energy and specific heat per unit area of the wall. [5 pts]