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• 1: Relativistic ideal gas in a canonical state
  Consider an extreme relativistic gas consisting of N monatomic particles with energy-momentum relationship ε = pc, c being the speed of light. (a) Show that the partition function Z_N is given by

Z_N(V, T) = (1/N!) {8πV (k_BT/hc)³}^N.

(b) Find the energy and the pressure equations of state for this system.

• 2: Microcanonical thermodynamics of a rubber band
  Polymers, like rubber, are made of very long molecules, usually tangled up in a configuration that has lots of entropy. As a very crude model of a rubber band, consider a chain of N links, each of length \(\ell\). Imagine that each link has only two possible states, pointing either left or right. The total length L of the rubber band is the net displacement from the beginning of the first link to the end of the last link.
   
  (a) Find an expression for the entropy of this system in terms of N and \(N_{\rm R}\), the number of links pointing to the right. (b) Write down a formula for L in terms of N and \(N_{\rm R}\); (c) For a one-dimensional system such as this, the length L is analogous to the volume V of a three-dimensional system. Similarly, the pressure p is replaced by the tension force f. Taking f to be positive when the rubber band is pulling inward, write down and explain the appropriate thermodynamic identity for this system. (d) Using the thermodynamic identity, you can now express the tension force f in terms of a partial derivative of the entropy. From this expression, compute the tension in terms of L, T, N, and \(\ell\). (e) Show that when \(L \ll N\ell\) the tension force is directly proportional to L (Hooke's law). (f) Discuss the dependence of the tension force on temperature. If you increase the temperature of a rubber band, does it tend to expand or contract? Does this behavior make sense?

• 3: Equipartition Principle [Kennett, Problem 4.7]
  Use the equipartition principle to estimate the typical angular velocity of a nitrogen molecule at room temperature. (The N-N bond length is 1.0975 Å and the mass of an N atom is 14.0067 amu.)

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, and I did not use anyone else's worked out solution to any of these problems when writing mine.

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