Homework #5 Due Tuesday Oct. 11

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 R2. 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^{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, andNOTU(T,V,N), is the appropriate thermodynamic potential for systems with fixed T,V, and N.

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 *i*th component
has *N _{i }*molecules of mass

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 withNparticles, also summed over the particle numbers_{1},N_{2},...,N_{r}Nwith separate chemical potentials_{1}, N_{2}, ... ,N_{r},µand the usual exponent (numbers times chemical potentials minus the energy of the state, all over_{i }k)._{B}T

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 forrideal gasses, each with its own massMand chemical potential_{i}µ._{i}

c.Derive the desired formulas for partial pressures and chemical potentials.

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