PC2352 Examples 7

Questions 14&15 are on new material since the last sheet. Questions 16 to 18 are exam questions on material from earlier in the course, for Easter revision.

14.

(a)

A single particle system has 3 non-degenerate energy levels 0, 0.02 and 0.05 eV. It is in thermal equilibrium with a heat reservoir at $20^\circ$C. Calculate the partition function, the probability that the particle is in each level and the average energy. What is the average energy in the limit of very large temperatures ($kT\gg0.05$ eV)?

(b)

Now consider a similar system with the 0.02 eV level being two-fold degenerate and the 0.05 eV being three-fold degenerate, and repeat the question. Explain the differing values you get for the large-temperature limit of the energies.

In this question, we are just interested in the partition function as a normalisation constant. For the high-temperature limit, ask yourself what value the probabilities tend to as $T\to\infty$. See here.

15.

A single particle system has 3 non-degenerate energy levels, $-\varepsilon$, 0 and $\varepsilon$. Write down the probability that the particle is in each level, and hence show that the average energy is given by

$$\left\langle E \right\rangle = -\frac{2 \varepsilon \sinh (\... ...le B}T)}{1 + 2 \cosh (\varepsilon /k_{\scriptscriptstyle B}T)}$$

and the heat capacity by

$$\left\langle C_{\scriptscriptstyle V} \right\rangle = 2k_{\s... ... + 2 \cosh (\varepsilon /k_{\scriptscriptstyle B}T)\right)^2}.$$

Plot both these functions, find their low and high temperature limits, and explain your findings.

In this question, we use the partition function to caluculate other properties. The calculation is very like the two-state system, see here and, for details, here. Hazy on hyperbolic trig functions and their limits? See here

16.

PC225, 1994. Define the Gibbs free energy for a hydrostatic system and derive the Maxwell relation

$$\left({\partial S\over \partial P}\right)_T =-\left({\partial V\over \partial T}\right)_P.$$

By considering entropy as a function of pressure and temperature, show that for any substance a change of entropy ${\rm d}S$ is related to changes of temperature ${\rm d}T$ and pressure ${\rm d}P$ by

$$T{\rm d}S=C_{\scriptscriptstyle P}{\rm d}T-TV\alpha {\rm d}P$$

where $C_{\scriptscriptstyle P}$ is the heat capacity at constant pressure of the substance, $V$ is the volume and $\alpha$ is the coefficient of thermal expansion, $\alpha= V^{-1}({\partial V\over \partial T})_{\lower0.4ex\hbox{${\scriptscriptstyle P}$} }$.

Consider $10^{-6}$m$^{3}$ of mercury at 273K and atmospheric pressure for which the heat capacity $C_{\scriptscriptstyle P}=1.9$JK$^{-1}$ and $\alpha=1.78\times10^{-4}$K$^{-1}$. You may assume to a good approximation that $C_{\scriptscriptstyle P}$ and $\alpha$ are independent of temperature and pressure and that mercury is incompressible. The pressure on the mercury is now increased to 100atm. What is the change in temperature if the pressure change occurs reversibly and adiabatically?

See here for a somewhat similar example of writing ${\rm d}S$ in terms of other variables, and also the question 8 on examples 2. In the last part, the conditions are such that the entropy is constant, ${\rm d}S=0$, so we have ${\rm d}T/T=(V\alpha/C_{\scriptscriptstyle P}) {\rm d}P$ and hence $\ln(T_f/T_i)=(V\alpha/C_{\scriptscriptstyle P})(P_f-P_i)$. (The question tells us we can treat $V$, $\alpha$ and $C_{\scriptscriptstyle P}$ constant.) The temperature change is 0.26K.

17.

PC235, 1999.

a)

Write down an expression for the work done in stretching a wire of length $L$ under tension $\Gamma$. State the fundamental thermodynamic relation for a stretched wire.

b)

Over a certain temperature range the tension of a stretched plastic rod is related to its length by the expression $\Gamma=aT^2(L-L_0)$, where $a$ and $L_0$ are positive constants, $L_0$ being the unstretched length of the rod. The entropy $S(T,L)$ of the rod is a function of $T$ and $L$. Derive the Maxwell relation that involves $({\partial S\over \partial L})_{\lower0.4ex\hbox{${\scriptscriptstyle T}$} }$, and hence show that

$$\left({\partial S\over \partial L}\right)_T=-2aT(L-L_0).$$

c)

At $L=L_0$ the heat capacity of the rod (measured at constant length) is given by the relation $C_L=bT$, where $b$ is a constant. Given the value $S_0$ of the entropy at $T=T_0$ and $L=L_0$, find $S(L,T)$ at any other temperature and length. (Hint: it is most convenient first to calculate the change in entropy with temperature at the length $L_0$ at which the heat capacity is known.)

See qu. 19 for the fundamental thermodynamic relation in this case. We still have for the Helmholtz free energy $F=E-TS$, and the desired Maxwell relation is derived from ${\rm d}F$ just as the one for $({\partial S\over \partial V})_{\lower0.4ex\hbox{${\scriptscriptstyle T}$} }$ is in a hydrostatic system. Part c) is similarto, but a bit more involved thant, the example here. You will need to integrate $({\partial S\over \partial T})_{{\lower0.4ex\hbox{$\scriptscriptstyle L=L_0$}}} =C_{\scriptscriptstyle L}(L=L_0)/ T$ to get $S(T,L_0)$, and then integrate $({\partial S\over \partial L})_{\lower0.4ex\hbox{${\scriptscriptstyle T}$} }$ to get the final result, which is $S(T,L)=S_0+b(T-T_0)-aT(L-L_0)^2$.

18.

PC302, 1996. The fundamental thermodynamic relation for a rubber band of length $L$ is ${\rm d}E=T{\rm d}S+\Gamma {\rm d}L$ where $\Gamma$ is the tension in the band. Under what circumstances may $T{\rm d}S$ be identified as heat? What is the corresponding interpretation of $\Gamma {\rm d}L$?

A simple model of a rubber band in one dimension is a chain of $N$ links each of length $a$. Each link may point to the left or to the right without any difference in energy, as illustrated above. Hence if there are $n_+$ ($n_-$) links to the right (left), we have

$$L=a(n_+-n_-)=a(2n_+-N)$$

and the energy is independent of $n_+$. Obtain an expression for the entropy $S$ in terms of $n_+$ on the assumption that $n_+$, $n_-\gg 1$.

Find an expression for the tension $\Gamma$ as a function of the length of the band, and show that it reduces to Hooke's Law $\Gamma=C(T)L$ in the limit $L\ll Na$. Find expressions for the coefficient $C(T)$ and for the coefficient of thermal expansion at fixed tension in the same limit.

The coefficient of thermal expansion is negative, showing that the band shrinks when it is warmed. Explain why one might have expected this result without calculation.

[You may use Stirling's approximation $\ln n!=n\ln n-n$.]

The expression for $S$ is obtained just like that for the two-state paramagnet in a magnetic field which was question 13 of examples 6. The tension plays a similar role for a band to the pressure for a gas, so the relation we want is analogous to $P/T=({\partial S\over \partial V})_{\lower0.4ex\hbox{${\scriptscriptstyle E}$} }$ (see here); rearranging the fundamental thermodynamic relation to get ${\rm d}S$ on the left hand side, we see that $-\Gamma/T=({\partial S\over \partial L})_{\lower0.4ex\hbox{${\scriptscriptstyle E}$} }$. This can then be calculated in terms of ${{\rm d}S\;\over {\rm d}n_+}$ and ${ {\rm d}n_+ \over{\rm d}L\;}$; again see the paramagnet example for guidence with algebra. You will need $\ln (1+x)\approx x$ for small $x$. The answers are $C(T)=k_{\scriptscriptstyle B}T/Na^2$ and $L^{-1}({\partial L\over \partial T})_{\lower0.4ex\hbox{$\scriptscriptstyle \Gamma$}}=-1/T$.