Question 1

A system with constant volume <math>V</math> and temperature <math>T</math> contains two non-interacting particles. The solutions to the single particle Schr��dinger equation indicate that there are two single particle quantum states <math>1</math> and <math>2</math> with energy <math>\epsilon_1</math> and <math>\epsilon_2</math> respectively.

<p>
</p>

a) Assume that the two particles in the system are distinguishable. How many microstates are available to the system. Write down the energy for each distinct microstate?

<p>
</p>

<center>

<table cellpadding=10>

<tr>

<td>
<center>
<math>\mbox

</math>
</center>
</td>

<td>
<center>
<math>\epsilon_1</math>
</center>
</td>

<td>
<center>
<math>\epsilon_2</math>
</center>
</td>

<td>
<center>
<math>\epsilon_1</math>
</center>
</td>

<td>
<center>
<math>\epsilon_2</math>
</center>
</td>

</tr>
<tr>

<td>
<center>
<math>\mbox

</math>
</center>
</td>

<td>
<center>
<math>\epsilon_1</math>
</center>
</td>

<td>
<center>
<math>\epsilon_2</math>
</center>
</td>

<td>
<center>
<math>\epsilon_1</math>
</center>
</td>

<td>
<center>
<math>\epsilon_2</math>
</center>
</td>

</tr>

<tr>

<td>
<center>
<math>\mbox

</math>
</center>
</td>

<td>
<center>
<math>2 \epsilon_1</math>
</center>
</td>

<td>
<center>
<math>\epsilon_1 + \epsilon_2</math>
</center>
</td>

<td>
<center>
<math>2 \epsilon_2</math>
</center>
</td>

</tr>

</table>

</center>

d) If the particles are fermions, how many microstates are available to the system. Write down the energy for each distinct microstate.

<p>
</p>

Particles are still indistinguishable and are now fermions (no more than one can be in the same particle state). Now there is only one possible microstate.

<center>

<table cellpadding=10>

<tr>

<td>
<center>
<math>\mbox

</math>
</center>
</td>

<td>
<center>
<math>\epsilon_1</math>
</center>
</td>

</tr>
<tr>

<td>
<center>
<math>\mbox

</math>
</center>
</td>

<td>
<center>
<math>\epsilon_2</math>
</center>
</td>

</tr>

<tr>

<td>
<center>
<math>\mbox

</math>
</center>
</td>

<td>
<center>
<math>\epsilon_1 + \epsilon_2</math>
</center>
</td>

</tr>

</table>

</center>

Question 2

A magnetic solid containing <math>N</math> atoms is placed in a magnetic field <math>H</math> at constant temperature <math>T</math> and pressure <math>P</math>. Each atom in the solid has a magnetic moment <math>m_o</math> which can independently align itself parallel or anti-parallel with the magnetic field <math>H</math>.

<p>
</p>

a) Write an expression for the characteristic potential of this system in terms of appropriate Legendre transforms of the internal energy.

<p>
</p>

Thermodynamic boundary conditions are constant <math>N</math>, <math>T</math>, <math>P</math>, and <math>H</math>. Below is a characteristic potential.

<center>

<br>

<math>\wedge (T, P, N, H) = E(S, V, N, M) - TS + PV - MH</math>

<br>

<math>d \wedge = - SdT + VdP + \mu d N - M dH</math>

<br>

</center>

b) Write down a formal expression for the partition function of this system and indicate which microstates you are summing over.

<p>
</p>

Find out what goes in the box by Legendre Transforming energy. Microstates being summed over correspond to all possible energies, volumes, and magnetizations.

<center>

<br>

<math>-\beta \wedge (T, P, N, H) = \frac

- \beta E - \beta PV + \beta HM</math>

<br>

<math>\Gamma = \sum_

- \beta PV_

</math>

<br>

</center>

c) Derive a general expression for the fluctuations of the magnetization <math>M</math> of the solid (i.e. <math>\overline

- \overline M^2</math>

<br>

<math>\overline M = \frac{ \sum_

e^{-\beta E_

+ \beta H M_

M_

- \beta PV_

) }</math>

<br>

<math>\mbox

</math>

<br>

<math>\frac

\Gamma + \overline M \frac

\right )</math>

<br>

</center>

Question 3

a) A solid is a ferromagnet at low temperature, but undergoes a second order phase transition to the paramagnetic state at <math>T_c</math>. Sketch how you expect the heat capacity to behave around <math>T_c</math> and explain why.

<p>
</p>

At the critical point, fluctuations of the energy (which are related to the heat capacity) become very large and C_V diverges.

<center>

Unable to render embedded object: File (Heat_capacity_versus_T.PNG) not found.

</center>

b) Which of the following thermodynamic quantities are mechanical variables.

<p>
</p>

<center>
Entropy <math>S</math> Volume <math>V</math> Chemical Potential <math>\mu</math> Temperature <math>T</math> Pressure <math>P</math>
</center>

<p>
</p>

The following thermodynamic quantities are mechanical variables:

• Volume
• Chemical potential
• Pressure

c) In a crystalline solid, which phonon modes require more energy to excite, long wavelength phonon modes <math>( k \right 0 )</math> or short wave length phonon modes (e.g. <math>k \right \frac

</math> for a 1-dimensional crystal with lattice parameter a)?

<p>
</p>

<center>
______long wave-length phonon modes
</center>

<p>
</p>

<center>
_____short wave length phonon modes
</center>

<p>
</p>

Short wave length phonon modes require more energy.

<p>
</p>

d) The degeneracy of the quantum mechanical energy spectrum of a particular system is found to be well characterized with <math>\Omega(E) = A \exp (B(E-E_o))</math> , where <math>E_o</math> is the energy of the ground state and <math>A</math> and <math>B</math> are constants. Write an expression for the entropy of the system at <math>0 K</math>.

<center>

<br>

<math>S = k \ln \Omega</math>

<br>

<math>S = k \ln \left ( A e^{B(E - E_o) \right )</math>

<br>

<math>S = k \left ( \ln A + B(E - E_o ) \right )</math>

<br>

<math>S(T=0) = k \ln A</math>

<br>

</center>

Question 4

<math>M</math> atoms of <math>Fe</math> are crystallized on the fcc lattice. As a first approximation, the vibrational degrees of freedom of the <math>Fe</math> atoms can be treated with the Einstein model where each iron atom has an Einstein vibrational frequency of <math>\omega_

</math>. The fcc crystal of <math>M Fe</math> atoms has <math>M</math> interstitial sites which can be filled with carbon atoms. The binding energy of carbon in the interstitial sites of fcc <math>Fe</math> with respect to pure carbon (in the graphite reference phase) is <math>\epsilon</math>. Carbon in the interstitial sites has an Einstein frequency of <math>\omega_C</math>.

<p>
</p>

The canonical ensemble is applied to systems with fixed <math>T</math>, <math>V</math>, and <math>M</math>. Below is the energy spectrum of an Einstein harmonic oscillator.

<center>

<br>

<math>\epsilon_n = \left (n+\frac

\right ) \hbar \omega_c + \epsilon</math>

<br>

<math>q_c = e^{\beta \epsilon} \frac{e^{\Theta _c / 2T}}{1 - e^{-\Theta _c / T}</math>

<br>

<math>\Theta_c = \frac

</math>

<br>

<math>F = -kT \log W</math>

<br>

<math>F = -3 M kT \log q_

- 3NkT \log q_c - kT \left [M \log M - N \log N - (M-N ) \log (M-N) \right]</math>

<br>

</center>

c) Use the result from b) to calculate the chemical potential for <math>Fe</math> and for <math>C</math>. Try to express your result in terms of the carbon concentration <math>x=N/M</math>.

<center>

<br>

<math>\mu_

= -3kT \log q_

= - 3kT \log q_

= -3kT \log q_

+ kT \log (1-x)</math>

<br>

<math>\mu_c = -3kT \log q_c - kT \left [-\log N - 1 + \log (M-N) + 1 \right]</math>

<br>

<math>\mu_c = -3kT \log q_c + kT \left [\log \left ( \frac

\right ) \right]</math>

<br>

<math>\mu_c = - 3kT \log q_c + kT \log \left ( \frac

\right )</math>

<br>

</center>

Question 5

Some researchers have stated that when <math>Li_xCoO_2</math> is annealed at high temperature it is susceptible to Li loss due to the high vapor pressure of <math>Li</math> in the material. In this problem you will estimate the vapor pressure of <math>Li</math> in the material, using commonly available data.

<p>
</p>

DATA for elemental Li:

• boiling point: <math>1597 K</math>
• entropy of vaporization: <math>92.5 J/mol-K</math>
• melting point: <math>454 K</math>
• entropy of melting: <math>6.45 J/mol-K</math>

<p>
</p>

<center>

Unable to render embedded object: File (Voltage_versus_lithium_concentration.PNG) not found.

</center>

a) Using some or all of the data below estimate the vapor pressure of <math>Li</math> above liquid <math>Li</math> at <math>1473K</math>.

<p>
</p>

Use Clausisu Clapeyron to exttrapoloate from the boiling point.

<center>

<br>

<math>\frac

{d\left (\frac

- \frac

\right ) \right |]</math>

<br>

<math>p(1473) = 0.39 atm</math>

<br>

</center>

b) The data below shows the potential of a <math>Li_xCoO_2</math> electrode (referenced against a <math>Li</math> metal) as function of <math>x</math> when the two are in contact through a <math>Li+</math> ion conductor. The data is taken at <math>300 K</math>. Note that a positive potential indicates that the activity of <math>Li</math> in <math>Li_xCoO_2</math> is lower than in <math>Li</math> metal. Estimate the vapor pressure above <math>Li_

CoO_2</math> at <math>1100 C</math> against hypothetical <math>Li</math> solid is the same as at <math>300K</math>.

<center>

<br>

<math>- \left ( \mu_

^

- F \Delta \phi</math>

<br>

<math>-F \Delta \phi = RT \ln a_i</math>

<br>

<math>a_i = \exp \left ( \frac{-F \Delta \phi}{RT_{300}}</math>

<br>

<math>\mu_

^

- \mu_

^

- \mu_

= 0.16 \times a_i</math>

<br>

<math>P_

^

- \mu_

^

- \mu_

^

- \mu_

^{Li(s) = -1.63 MJ</math>

<br>

<math>a_i = 3.4 \times 10^{-65}</math>

<br>

<math>p_i = 5.5 \times 10^{-66}</math>

<br>

</center>