586
10 Quantum Statistics
2, or 3 bosons. Compute the probabilities for states lying at energies 10 −6 eV,
10 −3 eV, 0.01 eV, 0.1 eV, and 1 eV above μ.
11. In Chap. 4 we obtained an expression for the chemical potential for a system
that obeys MB statistics. Suppose that you knew the distribution function,
n MB = e −β((−μ) , but didn’t know the expression for μ. You could still determine μ by requiring that the total number of particles, summed over all states
be N . Carry out this calculation to rederive the formula μ = −k B T ln(z/N ).
This is the actual procedure employed to determine μ in quantum statistics.
12. How large must − μ be before the FD, BE, and MB distributions are within
1% of one another? Is it possible for this condition to be violated for gases at
temperatures normally accessible on Earth?
13. Obtain an expression for N for an ideal quantum gas subject to a gravitational
field with acceleration g. (Hint: see Example 10.2.)
14. Use Eq. (4.2.10) giving the internal energy U(T , V , λ) together with
Eq. (10.2.8) for the natural logarithm of the grand partition function to show
that
U(T , V , λ) =
k
k λe −ββ k
1 ± λe −ββ k
and, by utilizing the continuum approximation to the summation over the
energy states for an ideal quantum gas in the presence of a gravitational field,
show that U(T , V , λ) is given by
U(T , V , λ) = k B T
V
3 (T )
5
2η
g
±
5
2
(α) − g
±
5
2
(α − η)
− g
±
3
2
(α − η)
.
15. Employ the results obtained in Example 10.2 and Problems 13 and 14 to
show that the entropy S/(Nk B ) associated with an ideal quantum gas in a
gravitational field is given by
S
Nk B
=
7
2η
g
±
5
2
(α) − g
±
5
2
(α − η)
−
α
η
g
±
3
2
(α) − g
±
3
2
(α − η)
− g
±
3
2
(α − η) .
16. Obtain Eq. (10.2.28) for the entropy S(T , V , μ) from Eq. (4.2.30b).
17. Temperatures at the cores of stars typified by our sun are of the order of 10 7 K,
so that all electrons have been stripped from the atomic nuclei to form an
electron gas with a number density N/V of the order of 10 32 m −3 . Would
it be a valid approximation to treat these electrons as a degenerate Fermi gas?
18. The mass density, ρ, of platinum metal is ρ(Pt) = 21.5 g cm 3 and that of
europium is ρ(Eu) = 5.24 g cm 3 , while the corresponding molar masses are
M(Pt) = 0.19508 kg mol
−1 and M(Eu) = 0.15196 kg mol
−1 . If each atom in
a metallic sample is assumed to contribute a single electron to the conduction
band, what are the values for their Fermi temperatures? Would it be justifiable
10 Quantum Statistics
2, or 3 bosons. Compute the probabilities for states lying at energies 10 −6 eV,
10 −3 eV, 0.01 eV, 0.1 eV, and 1 eV above μ.
11. In Chap. 4 we obtained an expression for the chemical potential for a system
that obeys MB statistics. Suppose that you knew the distribution function,
n MB = e −β((−μ) , but didn’t know the expression for μ. You could still determine μ by requiring that the total number of particles, summed over all states
be N . Carry out this calculation to rederive the formula μ = −k B T ln(z/N ).
This is the actual procedure employed to determine μ in quantum statistics.
12. How large must − μ be before the FD, BE, and MB distributions are within
1% of one another? Is it possible for this condition to be violated for gases at
temperatures normally accessible on Earth?
13. Obtain an expression for N for an ideal quantum gas subject to a gravitational
field with acceleration g. (Hint: see Example 10.2.)
14. Use Eq. (4.2.10) giving the internal energy U(T , V , λ) together with
Eq. (10.2.8) for the natural logarithm of the grand partition function to show
that
U(T , V , λ) =
k
k λe −ββ k
1 ± λe −ββ k
and, by utilizing the continuum approximation to the summation over the
energy states for an ideal quantum gas in the presence of a gravitational field,
show that U(T , V , λ) is given by
U(T , V , λ) = k B T
V
3 (T )
5
2η
g
±
5
2
(α) − g
±
5
2
(α − η)
− g
±
3
2
(α − η)
.
15. Employ the results obtained in Example 10.2 and Problems 13 and 14 to
show that the entropy S/(Nk B ) associated with an ideal quantum gas in a
gravitational field is given by
S
Nk B
=
7
2η
g
±
5
2
(α) − g
±
5
2
(α − η)
−
α
η
g
±
3
2
(α) − g
±
3
2
(α − η)
− g
±
3
2
(α − η) .
16. Obtain Eq. (10.2.28) for the entropy S(T , V , μ) from Eq. (4.2.30b).
17. Temperatures at the cores of stars typified by our sun are of the order of 10 7 K,
so that all electrons have been stripped from the atomic nuclei to form an
electron gas with a number density N/V of the order of 10 32 m −3 . Would
it be a valid approximation to treat these electrons as a degenerate Fermi gas?
18. The mass density, ρ, of platinum metal is ρ(Pt) = 21.5 g cm 3 and that of
europium is ρ(Eu) = 5.24 g cm 3 , while the corresponding molar masses are
M(Pt) = 0.19508 kg mol
−1 and M(Eu) = 0.15196 kg mol
−1 . If each atom in
a metallic sample is assumed to contribute a single electron to the conduction
band, what are the values for their Fermi temperatures? Would it be justifiable