Problem 1 [20 points total]
Consider a three dimensional classical ideal gas of atoms of mass m, moving in a potential energy
| V(x) = | { |
½ αx2 +∞ | 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) [5 pts] 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 the number density n(x) is related to the probability to find a particle at x, and that
b) [5 pts] Show that the ideal gas law (relating pressure, temperature, and density) continues to hold locally everywhere.
c) [5 pts] Find the pressure that the atoms exert on the wall. How does this pressure vary with temperature?
d) [5 pts] Calculate the energy and specific heat of the entire gas per unit area of the wall.
Hint: Unlike the ideal gas discussed in class, here the particles are in an external potential. But for some parts of this problem it might be useful to think of the subsystem consisting of the slab of small thickness Δx at position x, for which the potential V(x) can regarded as being approximately constant over this slab.
Problem 2 [20 points]
In lecture we discussed the canonical ensemble, in which the temperature T, volume V, and number of particle N of a system are fixed, while the energy E is allowed to fluctuate. Suppose now that you wish to describe a system in which the temperature T, number of particles N, and pressure p are fixed, while the volume V is allowed to fluctuate. This would describe a system in contact with a thermal and mechanical reservior, in which the wall separating the system and the reservior is heat conducting and moveable. We can call this case the constant pressure ensemble.
a) [5 pts] Define the appropriate partition function Z(T, p, N) of the system in this new constant pressure ensemble.
b) [5 pts] If you defined Z properly in part (a), then the Gibbs free energy should be given by
G(T, p, N) = -kBT 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
c) [5 pts] Derive a relation, in this constant pressure ensemble, between the isothermal compressibility κT and fluctuations in the volume V of the system. Show from this relation that the relative fluctuation in V vanishes in the thermodynamic limit.
d) [5 pts] Consider an ideal gas of indistinguishable, 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, Cp. Show that you get the correct answer for the ideal gas.