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: Thermodynamics of a plastic rod
  In a certain temperature range, the tension force F applied to a stretched plastic rod is related to its length by F = aT ² (L–L₀), where L₀ is the rod's unstretched length and a a positive constant. When L = L₀, the heat capacity of the rod at constant length is C_L = bT, where b is a constant.
  (a) Write down the fundamental thermodynamic identity for this system, relating dE, dS and dL.
  (b) The entropy S of the rod is a function of T and L. Compute (∂S/∂L)_T.
  (c) Assuming you know S(T₀, L₀), find S(T, L) at any other values of T and L. (Hint: Calculate first the change of entropy with temperature at the length L₀, where the heat capacity is known.)
  (d) If one starts at T = T_i and L = L_i, and stretches the thermally insulated rod quasistatically until it attains a length L_f, what is the final temperature T_f? Is T_f larger or smaller than T_i?

• 2: Probability distributions [adapted from Kennett, Problem 1.1]
  (a) We are told that \(P(x) = {\cal N}\,x\,{\rm e}^{-x}\) is a probability distribution on \(0 \le x < \infty\). Find the value of \(\cal N\) so that \(P(x)\) is properly normalized and then calculate \(\langle x\rangle\) and \(\sigma_x^2\). (b) Now consider the following probability distribution that depends on two variables, x and y: $$ P(x,y) = \cases{A(x^2+y^2), &$0 < x < 1$, $0 < y < 1$,\cr 0,&elsewhere.} $$ Calculate the value of the normalization constant A and then find \(\langle x\rangle\), \(\langle y\rangle\) and \(\langle x^2+y^2\rangle\).

• 3: Microstates for spins [Kennett, Problem 1.5]
  Consider a collection of 70 non-interacting spins that may point either up or down. How many microstates are there in this system? How can we label the macrostates in this system? If each microstate is equally probable, what is the most likely macrostate to find the system in, and what is the probability that we will find the system in this configuration?

Aside from oral discussions I may have had with other students in the class, the solutions to this homework set I am submitting are entirely my own.

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