4.5 Problems for this Chapter
209
24. Consider the expression obtained in Problem 23(a) for the partial derivative
of the chemical potential μ with respect to the specific volume v at constant
temperature T . Employ the definition v ≡ V /N to rewrite it in terms of the
partial derivative of μ with respect to N (with V held fixed); similarly, rewrite
the partial derivative of P with respect to v (with T held fixed) in terms of the
partial derivative of P with respect to V (with N held fixed). Employ these
results to show that
∂N
∂μ
T ,v
=
κ T N 2
V
,
in which κ T the isothermal compressibility of Eq. (2.5.2).
25. Determine the values for the ratio n/N of Schottky defects to be expected for
samples of crystalline NaF, NaCl, NaBr, and NaI at temperatures T = 300 K,
500 K, and 800 K if the E f values for these substances are E f = 2.888, 2., 2.452,
and 2.050 eV. What can you say about the relative abilities of these halides to
form Schottky defects?
26. If we assume that the number of interstices M in a crystal is roughly equal to
the number of lattice sites N in the crystal, we may compare the equilibrium
number n of Frenkel defects with the equilibrium number
n
N
e
−
1
2 βE f ,
of Schottky defects obtained in Problem 25. Use the values for E f and w given
in the table below to calculate the equilibrium fraction of defects in KCl, KBr,
AgCl, and AgBr crystals at room temperature (use T = 300 K) to see which
type of defect predominates in the various crystals.
Crystal E f /eV w /eV Crystal E f /eV w /eV
KCl
2.530
3.461 AgCl
1.713
1.318
KBr
2.331
3.158 AgBr
1.499
0.945
27. Show that the term
N,r N
ln dp N,r N appearing in Eq. (4.2.13) vanishes and,
by a similar argument, show that the term
i,r i
V i dp i,r i vanishes in expression
(4.3.8) for the internal energy in the isothermal–isobaric ensemble.
28. Use Eq. (4.2.5) for the grand partition function (λ, T , V ) to show that the
ensemble average number, N, of particles is given by N = ln , and show that
the ideal gas equation of state follows from this expression, with N playing
the role of the thermodynamic N. Show further, starting from Eqs. (4.2.30c)
giving the pressure P and the internal energy U in terms of the grand partition
function, that both P (N, T , V ) and U(N, T , V ) have the same forms as those
obtained from the canonical partition function of Eq. (3.2.27), but with N
replaced by N .
209
24. Consider the expression obtained in Problem 23(a) for the partial derivative
of the chemical potential μ with respect to the specific volume v at constant
temperature T . Employ the definition v ≡ V /N to rewrite it in terms of the
partial derivative of μ with respect to N (with V held fixed); similarly, rewrite
the partial derivative of P with respect to v (with T held fixed) in terms of the
partial derivative of P with respect to V (with N held fixed). Employ these
results to show that
∂N
∂μ
T ,v
=
κ T N 2
V
,
in which κ T the isothermal compressibility of Eq. (2.5.2).
25. Determine the values for the ratio n/N of Schottky defects to be expected for
samples of crystalline NaF, NaCl, NaBr, and NaI at temperatures T = 300 K,
500 K, and 800 K if the E f values for these substances are E f = 2.888, 2., 2.452,
and 2.050 eV. What can you say about the relative abilities of these halides to
form Schottky defects?
26. If we assume that the number of interstices M in a crystal is roughly equal to
the number of lattice sites N in the crystal, we may compare the equilibrium
number n of Frenkel defects with the equilibrium number
n
N
e
−
1
2 βE f ,
of Schottky defects obtained in Problem 25. Use the values for E f and w given
in the table below to calculate the equilibrium fraction of defects in KCl, KBr,
AgCl, and AgBr crystals at room temperature (use T = 300 K) to see which
type of defect predominates in the various crystals.
Crystal E f /eV w /eV Crystal E f /eV w /eV
KCl
2.530
3.461 AgCl
1.713
1.318
KBr
2.331
3.158 AgBr
1.499
0.945
27. Show that the term
N,r N
ln dp N,r N appearing in Eq. (4.2.13) vanishes and,
by a similar argument, show that the term
i,r i
V i dp i,r i vanishes in expression
(4.3.8) for the internal energy in the isothermal–isobaric ensemble.
28. Use Eq. (4.2.5) for the grand partition function (λ, T , V ) to show that the
ensemble average number, N, of particles is given by N = ln , and show that
the ideal gas equation of state follows from this expression, with N playing
the role of the thermodynamic N. Show further, starting from Eqs. (4.2.30c)
giving the pressure P and the internal energy U in terms of the grand partition
function, that both P (N, T , V ) and U(N, T , V ) have the same forms as those
obtained from the canonical partition function of Eq. (3.2.27), but with N
replaced by N .