(a) Fun fact about factorials: (N - 1)! = N! / N, since dividing by N cancels the final factor in N! and leaves just the first N-1 factors. Use this to show that the multiplicity of an Einstein solid can be expressed as: (g + N)! q! N! N q+ N Then apply Stirling's Approximation to each of the factorials, to express the multiplicity as approximately (q + N)ª+N qª NN 2nq(q+ N)' (b) When N and q are both large, we can set the entire square-root in the above multiplicity expression to 1, leaving just: (q + N)N+q N(N,q) = Using this formula, find an expression for the total entropy of the Einstein solid. (c) Use your result from part (b) to find the solid's temperature as a function of its energy. (d) Invert your answer from part (c) to find the energy as a function of temperature, then use it to find a formula for the solid's heat capacity C.

(a) Fun fact about factorials: (N - 1)! = N! / N, since dividing by N cancels the final factor in N! and leaves just the first N-1 factors. Use this to show that the multiplicity of an Einstein solid can be expressed as: (g + N)! q! N! N q+ N Then apply Stirling's Approximation to each of the factorials, to express the multiplicity as approximately (q + N)ª+N qª NN 2nq(q+ N)' (b) When N and q are both large, we can set the entire square-root in the above multiplicity expression to 1, leaving just: (q + N)N+q N(N,q) = Using this formula, find an expression for the total entropy of the Einstein solid. (c) Use your result from part (b) to find the solid's temperature as a function of its energy. (d) Invert your answer from part (c) to find the energy as a function of temperature, then use it to find a formula for the solid's heat capacity C.