PHYSICS 313
Problem set 7 2001
Given November 19
Due November 28

Problem 1:
a: Calculate the variance in internal energy

$$\sigma^2=<U^2>-<U>^2$$

for N oscillators of frequency $\omega$ constituting an Einstein solid. The variance of $N$ oscillators is $N$ times the variance of a single oscillator.
b: Use the formula

$$<U^2>-<U>^2=k_BT^2C$$

to calculate the specific heat $C$ of the above Einstein solid.
c: The specific heat can also be computed from

$$C=\frac{\partial U}{\partial T}$$

Check that the results from the two methods agree.
Problem 2: Assume that the atmosphere at ground level contains $20$ atomic% oxygen and $80$ atomic % nitrogen. How much has the oxygen/nitrogen ratio changed at an altitude of 5000 m? Assume an isothermal atmosphere at $0^0$ C and atomic masses of 32 g/mol and 28 g/mol, respectively, for oxygen and nitrogen.
Problem 3:
A system is made up of two distinguishable particles. Each particle can be in any one of three possible states , labeled 1,2,3. If the two particles are in states with the same label, the energy of the system is $-\epsilon$. If the labels are different the energy is $0$. The temperature is $T=\frac{1}{\beta k_B}$.
a: Find the partittion function of the system.
b: Find the mean energy of the system.
c: Find the entropy of the system as a function of temperature.
d: Give a physical interpretation of the entropy in the limits $T\Rightarrow 0$, $T\Rightarrow \infty$.