PHYSICS 313
Second midterm 2000.

Problem 1:
The multiplicity of an Einstein solid with $N$ oscillators and $q$ vibrational quanta is

$$\Omega=\frac{(N+q-1)!}{q!(N-1)!}$$

a:
Show using Stirlings formula

$$\ln N!\approx N\ln N-N$$

that when $N$ is large the entropy of the Einstein solid can be written

$$S=k_B[{(N+q)\ln(N+q)-q \ln q-N\ln N}]$$

b:
The internal energy is $U=\hbar\omega q$ , where $\hbar\omega$ is the energy of a single quantum. Use the thermodynamic identity

$$dS=\frac{dU}{T}-\frac{\mu}{T}dN$$

to find the temperature $T$ and chemical potential/oscillator $\mu$, as a function of $U$ and $N$.
c: Evaluate the entropy per particle (in natural units) $s=S/Nk$ at the "Einstein temperature"

$$T_E=\frac{\hbar\omega}{k_B}$$

Problem 2:
One container contains 2 mol of helium while another contains 1 mol of oxygen. Initially both gases have temperature $T_i$ and volume $V_i$ . Assume the gases are ideal.
a:
The partition between the two containers is broken so that they spontaneously mix with a total volume $V_f=2 V_i$. How does the entropy of the system change?
b:
The mixture is compressed isothermally until the volume is $V_i$. How much work has to be done on the gas? What happens to the entropy of the system?
c:
Answer question a: if both containers initially contain helium.
The entropy of an ideal gas can be written

$$S=Ns_i(T)+Nk_B[\ln\frac{V}{Nv_q(T)}+\frac{5}{2}]$$

where the entropy per molecule $s_i$ due to internal degrees of freedom such as vibration and rotation and the quantum volume $v_q$ depends only on the temperature and not on the volume $V$ or number of particles $N$.