Physics 112, B.C. Regan, Problem Set #7. KK refers to Kittel and Kroemer’s Thermal Physics, YF refers to Young and Freedman’s University Physics, 15th edition.

1. KK 6.5 Integration of the thermodynamic identity for an ideal gas

2. KK 6.6 Entropy of mixing

3. KK 6.8 Time for a large fluctuation

4. In class we worked out the temperature and chemical potential of a N=2 (distinguishable)

particle system with total energy E = 2 ε. Generalizing this problem gives us a chance to internalize how the entropy function completely characterizes a thermodynamic system.

a. Make a table with N ∈ [1,5] columns and m∈ [0,4] rows. Fill in all the elements of the table with the multiplicity ��������� ��� ������, ������ by hand. By the time you are done with the ��� ��� 4 column you should recognize a pattern that you can use to do the last ������ ��� 5��� column.

b. Write down an analytic expression for ���������, ������.

c. Calculate an analytic expression for the temperature ���������, ������. (There’s an ambiguity in

how you choose to define the discrete derivative. Do you choose the +1 definition, the ‐ 1 definition, or the average? The average is more complicated, and the ‐1 definition messes with the beginning of your tables if you are putting these calculations into a spreadsheet – which you are strongly encouraged to do [but don’t turn these in]. So let’s choose the +1 definition.)

d. Calculate an analytic expression for the chemical potential ���������, ������. Since ��� ≃ ��� ������ and

��� ≃ ������ ������, does it follow that ��� ≃ ������ ������? (Hint: review lecture 3.) ������ ������

e. Make a table with N ∈ [1,5] columns and g∈ [1,21] rows. Fill in all the elements of the table with the energy ���������, ��� ��� ��������� by hand. Not every element has an integer value – you can just put a dash “‐“ in those spots. Again, you will notice some patterns.

f. Invert ���������, ��� ��� 2��� to determine an exact analytic expression for ��� ������, ��� ��� 2���.

g. Invert ���������, ��� ��� 3��� to determine an exact analytic expression for ��� ������, ��� ��� 3���.

h. Because of the Abel‐Ruffini theorem, it seems unlikely that ���������, ������ can be determined

for all ���. For a given number of particles N, how does U scale with g for large g? Is the energy a rapidly increasing function of the multiplicity? Is the energy a rapidly increasing function of the entropy?

KK6.1‐4 are all worth a look. 6.1 and 6.2 should seem very plausible based on your knowledge of the shape of the Fermi‐Dirac distribution. 6.4 you should be able to do in your head (the approach KK have in mind is longer because they need to derive a result we already have).

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