Please print this sheet, sign the statement at the bottom, and use it as cover page when you submit your homework. Remember to always include a complete explanation of your reasoning, and to show all calculations.

• 1: Cold Fermionic Gas
  Using the expression for the Fermi energy of a free-fermion gas derived in class, estimate its value for copper and state whether the conduction electrons in copper at room temperature should be considered as a cold or a hot gas (justify your statement).

• 2: Non-Ideal Gas
  Consider a non-ideal gas of N particles at temperature T in a box of volume V, with interparticle potential given by \(v(r) = A\,\exp\{-r^2/r_0^2\}\), where \(A\) and \(r_0\) are constants. Calculate the second virial coefficient of this gas, treated classically, and identify any assumptions you made on the values of \(A\) and \(r_0\), and any approximations.

• 3: Hydrogen Gas [adapted from Kennett Problem 4.19]
  Molecular hydrogen gas (H\(_2\)) can exist in two slightly different forms, ortho and para hydrogen, depending on the state of the spins of the hydrogen nuclei. Treating the H\(_2\) molecule as a linear 3D rotor, the rotational energy is

\[ \epsilon_J^{~} = {\hbar^2\,J(J+1)\over2I}\;. \]

For ortho hydrogen the spins are in a triplet state, for which the degeneracy is 3 and J can only take odd integer values 1, 3, 5, ..., while for para hydrogen the spins are in a singlet state, for which the degeneracy is 1 and J can only take even integer values 0, 2, 4, .... Noting that the proton-proton bond distance in hydrogen is 0.747 Å, find the ratio of ortho to para hydrogen at (i) a temperature of 25 K and (ii) high temperatures. You may ignore non-rotational degrees of freedom since these are the same for both species.

Aside from discussions I may have had with other students in the class about how to solve these problems, the solutions to this homework set I am submitting are entirely my own. While I was writing these solutions I did not look at anyone else's solutions, and I was not told by anyone what to write.

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