Differentials, Controlling Variables, Legendre Transform to find Partition Function
It may be helpful to see how problems similar to question 2 of exam 1, 2002 http://www.onethread.org/wiki/index.php?title=3.20_Exam_Questions/Exam1/2002#Question_2
and question 1 of exam 2, 2002 http://www.onethread.org/wiki/index.php?title=3.20_Exam_Questions/Exam2/2001#Question_1 are generally solved.
Defining Thermodynamic Potentials
How is it known whether combinations of extensive and intensive quantities, such as force and length or applied field and magnetization, are added or subtracted in defining thermodynamic potentials? Examples are question 2 of exam 1, 2001 http://www.onethread.org/wiki/index.php?title=3.20_Exam_Questions/Exam1/2001#Question_2 and question 1 of exam 1, 2003 http://www.onethread.org/wiki/index.php?title=3.20_Exam_Questions/Exam1/2003#Question_1 Guidelines in how to derive potentials may be helpful
Heat Capacity
Thin Film on Thick Substrate
It may be helpful to see why the following logic is true in a solution of problem 2 of exam 1, 2005 http://www.onethread.org/wiki/index.php?title=3.20_Exam_Questions/Exam1/2005#Question_2.
<p>
</p>
We are interested in a heat capacity under a strange constraint. Start with <tex>dS</tex>.
<center>
<br>
<tex>dS = \left ( \frac
\right )_V dT + \left ( \frac
\right )_T dV</tex>
<br>
</center>
The relation below follows.
<center>
<br>
<tex>T \left ( \frac
\right )_
= T \left ( \frac
\right )_
+ T \left ( \frac
\right )_
\left ( \frac
\right )_
</tex>
<br>
</center>
Bosons
Could it be explained what it means physically in a situation wherein bosons are not conserved and why <tex>\mu</tex> is zero in an equation below? I believe this is related to question 4 of exam 2, 2001 http://www.onethread.org/wiki/index.php?title=3.20_Exam_Questions/Exam2/2001#Question_4.
<center>
<br>
<tex>\overline n_k = \frac
{e^
- 1}</tex>
<br>
</center>
Entropy
AB alloy
Could a solution of <tex>S_
</tex> found in part c of question 5 in exam 2, 2001 be explained?http://www.onethread.org/wiki/index.php?title=3.20_Exam_Questions/Exam2/2001#Question_5 A solution is below.
<center>
<br>
<tex>S_
= -Nk \left [\ln 0.01 + 0.99 \ln 0.99 \right ]</tex>
<br>
</center>
Intercept rule
To get the partial molar volume of B or A start with the definition of
partial molar quantities
<center>
<br>
<tex>V = \overline
_A N_A + \overline
_B N_B</tex>
<br>
</center>
where <tex>N_A</tex> and <tex>N_B</tex> is the number of moles of <tex>A</tex> and <tex>B</tex> respectively. This holds because <tex>V</tex> is an extensive quantity, so it is a homogeneous function of degree one. Now divide by the total number of moles:
<center>
<br>
<tex>\underline
= \overline
_A x_A + \overline
_B x_B (1)</tex>
<br>
</center>
<tex>\underline
</tex> means "molar <tex>X</tex>" and <tex>\overline
_i</tex> is the partial molar <tex>X</tex> of component "<tex>i</tex>". using the constraint that <tex>x_A + x_B = 1</tex> and taking the derivative we have
<center>
<br>
<tex>d\underline
= (\overline
_B - \overline
_A) dx_B</tex>
<br>
</center>
so
<center>
<br>
<tex>\overline
_B = \overline
_A + d\underline
/dx_B</tex>
<br>
</center>
substitute in (1) for <tex>\overline
_A</tex> and solve for <tex>\overline
_B</tex> you get
<center>
<br>
<tex>\overline
_B = \underline
+ (1-x_B) d\underline
/dx_B</tex>
<br>
</center>
which is the intercept rule.
Now, to solve the problem from 2002, E2, Q4, you just have to cast the
r.h.s. in terms of the lattice parameter
<center>
<br>
<tex>\underline
= a^3/4</tex>
<br>
</center>
and
<center>
<br>
<tex>d\underline
/dx_B = d\underline
/da * da/dx_B</tex>
<br>
</center>