Problems and Solutions on Thermodynamics and Statistical Mechanics

Statistical Physics 295

2118
Consider the energy and fluctuation in energy of an arbitrary system
in contact with a heat reservoir at absolute temperature T = l/k,f?.

(a) Show that the average energy 3 of the system is

where z = c exp(-PE,) sums over all states of the system.
n

• (b) Obtain an expression for E2 in terms of the derivatives of lnz.

(c) Calculate the dispersion of the energy, (AE)2 = E2 - E.

(d) Show that the standard deviation

-- -2

= ((AE)2)1'2 can be ex-
pressed in terms of the heat capacity of the system and the absolute tem-
perature.
N_
(e) Use this result to derive an expression for AE/E for an ideal

(UC, Berkeley)

monatomic gas.

(e) For an ideal monatomic gas,

• 3 3
  E = -NkT,
  2 2
  C, = -Nk

and thus

E