PHYSICS 313
Problem set 4 2001
Given October 15
Due Oct 24

Problem 1:
In the model of a polymer considered in class the entropy was found to be

$$\frac{S}{k_B}=N\ln N - N_R\ln N_R-(N-N_R)\ln(N-N_R)$$

where $N$ is the total number of links between monomers and $N_R$ the number of links in the direction of an applied force. The stretched length $L$ is

$$L=l(2N_R-N)$$

a:
Express the entropy as a function of the thermodynamic variables $N$ and $L$ and derive a formula for the force required to stretch the polymer to length $L$ using

$$F=-T\left.\frac{\partial S}{\partial L}\right\vert _N$$

b:
A mass M is hanging by a rubber band from a fixed support. If the simple model above is correct, will the mass move up or down if the temperature is incresed?
c: Find the chemical potential per monomer of a polymer chain of $N$ links and stretched length $L$.
Problem 2:
a:
A monatomic ideal gas is at constant temperature and in a gravitational field $g$. The internal energy per particle at height $z$ is

$$u=\frac{3}{2}k_BT+mgz$$

Use the Sackur Tetrode equation for entropy and

$$\mu=\left.-T\frac{\partial S}{\partial N}\right\vert _{U,V}$$

to finde a formula for $\mu$.
b:
Show that for the chemical potential to stay constant the pressure decreases with height according to the barometric formula

$$P(z)=P(z=0)\exp{\frac{-mgz}{k_BT}}$$

c:
For the chemical potential of a monatomic ideal gas to stay constant while the temperature decreases will the density $N/V$ (and hence the pressure) have to increase or decrease?
d:
In the atmosphere the temperature tends to decrease with altitude. Does this mean that the barometric formula over-estimates or under-estimates the pressure at high altitudes?