1000 Solved Problems in Modern Physics

256 4 Thermodynamics and Statistical Physics

4.54 The probability for occupying the Fermi levelPF= 1 /2. If the probability for
occupying a levelΔEaboveEFisP+and that for a levelΔEbelowEFisP−,
then show that forΔkTE1,PFis the mean ofP+andP−

4.55 Find the number of ways in which two particles can be distributed in six states
if
(a) the particles are distinguishable
(b) the particles are indistinguishable and obey Bose-Einstein statistics
(c) the particles are indistinguishable and only one particle can occupy any
one state.

4.56 From observations on the intensities of lines in the optical spectrum of nitro-
gen in a flame the population of various vibrationally excited molecules rela-
tive to the ground state is found as follows:

v 0123

Nv/N 0 1.000 0.210 0.043 0.009

Show that the gas is in thermodynamic equilibrium in the flame and calcu-

late the temperature of the gas (θv= 3 ,350 K)

4.57 How much heat (in eV) must be added to a system at 27◦C for the number of
accessible states to increase by a factor of 10^8?

4.58 The counting rate of Alpha particles from a certain radioactive source shows
a normal distribution with a mean value of 10^4 per second and a standard
deviation of 100 per second. What percentage of counts will have values
(a) between 9,900 and 10,100
(b) between 9,800 and 10,200
(c) between 9,700 and 10,300

4.59 A system has non-degenerate energy levels with energyE =

(

n+^12

)

ω,

whereω= 8. 625 × 10 −^5 eV, andn= 0 , 1 , 2 , 3 ...Calculate the probability

that the system is in then=10 state if it is in contact with a heat bath at room

temperature (T=300 K). What will be the probability for the limiting cases

of very low temperature and very high temperature?

4.60 Derive Boltzmann’s formula for the probability of atoms in thermal equilib-
rium occupying a stateEat absolute temperatureT.

4.2.4 Blackbody Radiation...............................

4.61 A wire of length 1 m and radius 1 mm is heated via an electric current to pro-
duce 1 kW of radiant power. Treating the wire as a perfect blackbody and
ignoring any end effects, calculate the temperature of the wire.
[University of London]