3. Kittel, Ch2-3. Quantum harmonic oscillator. (a) Find the entropy of a set of N oscillators of frequency @ as a function of the total quantum number n. Use the (N + n − 1)! multiplicity function (1.55), i.e., g(N,n) = and make the Sterling n! (N − 1)! approximation log N!≈ N log N- N. Replace N-1 by N. (b) Let U denote the total energy nhw of the oscillators. Express the entropy o(U,N). Show that the total energy at temperature Tis U = Nhw exp(ħw/T)-1 " This is the Planck result; it is derived again in Chapter 4 by a powerful method that does not require us to find the multiplicity function.

3. Kittel, Ch2-3. Quantum harmonic oscillator. (a) Find the entropy of a set of N oscillators of frequency @ as a function of the total quantum number n. Use the (N + n − 1)! multiplicity function (1.55), i.e., g(N,n) = and make the Sterling n! (N − 1)! approximation log N!≈ N log N- N. Replace N-1 by N. (b) Let U denote the total energy nhw of the oscillators. Express the entropy o(U,N). Show that the total energy at temperature Tis U = Nhw exp(ħw/T)-1 " This is the Planck result; it is derived again in Chapter 4 by a powerful method that does not require us to find the multiplicity function.

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9.1. Consider the results for ideal gases derived from quantum mechanics. Write an expression for the function s(T, v) that includes the parameter A(T). (a) Show that the behavior of the entropy as T0 is unrealistic in a quantum sense. What approximation in the derivation is responsible for this behavior? (b) The Sackur-Tetrode equation is valid when the thermal de Broglie wave- length is much less than the average molecular separation distance, A(T) < (V/N)13, which is generally the case at high temperatures. Calculate the temperature at which this inequality is first violated for an ideal gas at atmospheric pressure and with a mass of 6.6 × 10-26 kg/molecule (which is typical of argon). x
Problem 1: Consider a classical ideal gas in three dimensions, with N indistinguishable atoms confined in a box of volume N³. Assume the atoms have zero spin and neglect any internal degrees of freedom. Starting from the energy levels of a single atom in a box, find: (a) The Helmholtz free energy F' Hint: ſ. -ax² d.x e Va (b) The entropy o (c) The pressure p