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Consider a system of N identical particles. Each particle has two energy levels: a ground
state with energy 0, and an upper level with energy . The upper level is four-fold degenerate
(i.e., there are four excited states with the same energy ).
(a) Write down the partition function for a single particle.
(b) Find an expression for the internal energy of the system of N particles.
(c) Calculate the heat capacity at constant volume of this system, and sketch a graph to
show its temperature dependence.
(d) Find an expression for the Helmholtz free energy of the system.
(e) Find an expression for the entropy of the system, as a function of temperature. Verify
that the entropy goes to zero in the limit T ! 0. What is the entropy in the limit
T ! 1? How many microstates are accessible in the high-temperature limit?

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