PHYSICS 313
$2^{nd}$ optional midterm review problem set 2003
Given Oct. 29 2003
These problems will be discussed in class Monday November 3. You should hand in problems before class if you want to obtain credit.

Problem 1:
Wool fibers are polymers made up of keratin monomers. In an idealized model each monomer can be either in an $\alpha$ or a $\gamma$ state. The internal energy of the $\alpha$ and $\gamma$ states are respectively $\epsilon_\alpha$ or $\epsilon_\gamma$, respectively. The projection of the molecule in the two states in the general direction of the fiber is, respectively, $a$ and $g$ so that the enthalpy of the polymer is

$$H=N_\alpha(\epsilon_\alpha-fa)+N_\gamma(\epsilon_\gamma-fg)$$

and the length

$$L=N_\alpha a+N_\gamma g$$

where $N_\alpha,\;N_\gamma$ are, respectively, the number of $\alpha$ and $\gamma$ molecules in the polymer, $N=N_\alpha+N_\gamma$, and $f$ is the force by which the fiber is pulled.
a: Find the entropy of a chain with $N_\alpha,\;N_\gamma$, monomers of each kind. Use Stirling's formula on the form

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

b: Find a formula for the length $L$ of the fiber as a function of $N,T,f$, by minimizing the Gibbs free energy.
c: Plot the length in reduced units $\frac{L}{Ng}$ vs. the effective force $\frac{fg}{k_BT}$ if $\epsilon_\alpha-\epsilon_\gamma=k_BT$, $a=2g$.
Problem 2:

A cylinder of cross sectional area $A$ and height $L$ is sealed at both ends. It contains an airtight piston (mass $m$) and $N$ monatomic ideal gas molecules on each side of the piston (see figure). The temperature is $T$. Find the equilibrium height of the piston by minimizing the free energy of the gas $+$ piston. (Since the entropy of the piston doesn't change as it moves up and down, you can take the free energy if the piston to be just $mgz$). Does this result agree with what you would expect by considering mechanical equilibrium?
Problem 3:
Do problem 5.33 of text.