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* = *Nk** _{B}T.* [10
pts]

(c) Show that the total average energy is, *E* = 3*Nk** _{B}T.* [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*) =
-*k _{B}T* ln

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]