1. The fundamental hypothesis of statistical mechanics (S=k_B log G) is formulated for the entropy of a system at fixed energy U, volume V, and number of particles N.  The variables S,U,V,N are all "extensive."  Both S=S(U,V,N) and U=U(S,V,N) can be regarded as "thermodynamic potentials."  An important property of a thermodynamic potential is that all thermodynamic information can be derived from it.  When we want to switch to other variables, like T in place of S, p in place of V, or µ in place of N, we introduce new potentials using the Legendre transformation.  Why?  The following may help.

a. For the monatomic ideal gas, derive the formula for S(U,V,N) using the microcanonical ensemble.  You may introduce a factor 1/N! in the phase space volume, and use the fact that the area of a hypersphere of radius R in d dimensions is approximately R^d-1/(d/2)!. Taking appropriate partial derivatives, derive the properties of the ideal gas (that is, derive equations for the intensive variables T, p, and µ.)
b. Solve S=S(U,V,N) for U=U(S,V,N).  Taking appropriate partial derivatives of U(S,V,N), again derive equations for the intensive variables.
c. Use an appropriate equation for T to convert U(S,V,N) into U(T,V,N).  Explain why the result helps to confirm that the Helmholtz free energy F(T,V,N)=U-TS, and NOT U(T,V,N), is  the appropriate thermodynamic potential for systems with fixed T,V, and N.

2.  The picture shows the 2-d triangular lattice with N sites indicated by "O".  There are 2N interstitial sites, labeled by "X" (forming a honeycomb lattice.)  Suppose there are N atoms to distribute, and that the interstitial sites cost energy D compared to the triangular "O" sites which cost energy 0. Except for the multiplicity 2 for interstitials relative to regular sites, this problem is the same as the one solved in class.  Your problem is again, to find the number N_X of occupied interstitial sites in equilibrium at temperature T.  You should separate the problem into two problems: the regular lattice of N sites and the hexagonal lattice of 2N sites, each in thermal and chemical equilibrium with a reservoir of particles at temperature T and chemical potential m.  Find the grand partition functions Z for each problem.  Use the grand partition function to find the equilibrium averages of N_O and N_X.  Use the fact that N_O + 2N_X = N to fix m.  Solve for N_X.  Make the approximation that k_BT is small compared to D, to simplify the answer.  Compare with the answer found in class for the similar problem with equal numbers of regular and interstitial sites, N_X = N exp(-D/2k_BT).

3. Here is a problem that fills a missing link in Kittel's discussion in section 16 of chemical equilibrium.  We have a system which is an ideal gas of r components.  The ith component has N_i molecules of mass M_i.  Your job is to prove that the total pressure p=Nk_BT/V is the sum of the partial pressures p_i=N_ik_BT/V of the species, and that the chemical potential µ_i=k_BT log(N_i l_i³/V), where the thermal DeBroglie wavelengths l_i have the usual formulas and thus depend on the mass M_i in the usual way.  Do this in two stages:

a. Starting from the total entropy written as a sum of system and reservoir, motivate the generalized grand partition function as the sum overall states with N₁,N₂,...,N_rparticles, also summed over the particle numbers N₁, N₂, ... ,N_r, with separate chemical potentials µ_i and the usual exponent (numbers times chemical potentials minus the energy of the state, all over k_BT).
b. Making the appropriate choice for the energy of the multicomponent ideal gas, show that the generalized grand partition function factorizes, so the grand potential is just the sum of grand potentials for r ideal gasses, each with its own mass M_i and chemical potential µ_i.
c.Derive the desired formulas for partial pressures and chemical potentials.

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