PHY 418 Midterm Exam Spring 2004

1) [40 points total]

Consider a classical gas of N indistinguishable, non-interacting, particles with ultrarelativistic energies, i.e. their energy - momentum relation is given by e(p) = |p|c, with c the speed of light and p the particle's momentum.

(a) Compute the canonical partition function for this system. [10 pts]

(b) Show that this system obeys the usual ideal gas law, pV = NkBT. [10 pts]

(c) Show that the total average energy is, E = 3NkBT. [10 pts]

(d) Show that the ratio of specific heats is, Cp/CV = 4/3. [10 pts]

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2) [60 points total]

Consider a classical, non-relativistic, gas of indistinguishable non-interacting particles (i.e. the classical ideal gas).

a) Write down the N particle canonical partition function that describes the gas. You should evaluate any integrals and define all symbols. [12 pts]

b) Suppose now that you wish to describe this gas by an ensemble in which the pressure p is fixed, and the volume V is allowed to fluctuate. Compute the corresponding partition function Z(T, p, N). [12 pts]

c) If you defined Z properly in part (b), then the Gibbs free energy should be given by

G(T, p, N) = -kBT ln Z(T, p, N)

To demonstrate this, with G as defined above, show that [12 pts]

d) Using G(T, p, N) from part (c), compute the total specific heat at constant pressure, Cp [12 ps]

e) Derive a relationship between the isothermal compressibility,

and fluctuations in the system volume. [12 pts]