4.5 Problems for this Chapter
207
system members. For a 2-state system, in which the system members have
internal energies 0 and 1 = 0 + , express your results in terms of the ratio
of the system temperature T to the characteristic temperature int , defined as
int ≡ B . Construct a plot of U int (T )/(Nk B int ) vs. T // int for T / int
varying between 0 and 10.
14. Obtain an expression for the entropy S int (T ) associated with a 2-state system
and determine expression for the behaviour of S int (T ) for T int and T
int , with int defined as in Problem 13. Plot the ratio S int (T )/(Nk B ) as a
function of T // int and comment upon the manner in which S int approaches its
limiting high- and low-temperature values.
15. Consider a sample of a molecular solid containing N molecules in thermal
equilibrium at temperature T . If each individual molecule in this sample may
be in only one of two energy eigenstates, so that its energy is either + 0 or − 0 ,
and if the solid sample at equilibrium contains N + excited molecules having
energy 0 , obtain an expression for the configurational entropy S(N, N + ) (i.e.,
the entropy associated with the number of ways in which the excitation can be
distributed amongst the N molecules in the sample).
16. For a molecular crystal of the type considered in Problem 15, in which
individual molecules can be found in one of two internal states having energies
− 0 or + 0 , obtain expressions for the probabilities p + and p − that an
individual molecule may be found in the states at energies 0 , − 0 , and obtain
expressions for the internal state contribution, U int (T ), to the thermodynamic
internal energy U(T ) and for the contribution C int
V (T ) to the heat capacity
C V (T ) for such a solid. Express the heat capacity contribution in terms of the
separation between the molecular energy states.
17. Consider N atoms arranged in a regular three-dimensional lattice so as to form
a perfect crystal. The displacement of n of these atoms (1 n N) from
their lattice sites to interstices of the crystal lattice creates an imperfect crystal,
with n crystal defects said to be of the Frenkel type. The energy of such an
imperfect crystal is higher than that of the corresponding perfect crystal by
an amount U(n) ≡ E(n) = nw, with w the energy required to move one
atom from a crystal site to an interstitial site. By assuming that the number
M of interstitial sites into which an atom can enter is similar in magnitude
to N and that the concentration of such defects is sufficiently small that the
creation of any one defect can be treated as an independent event, argue that the
number of configurations associated with n Frenkel defects in a crystal having
N lattice sites and M defect sites is given by the number of ways M, n) of
removing n atoms from the N lattice sites and then independently redistributing
them among the M interstitial sites. From your expression for N, n),
obtain an expression for the configurational entropy S(n) associated with n
Frenkel defects. Note that S depends parametrically upon the fixed values N
and M.
18. Employ your expression for S(n) from Problem 17 to obtain an expression
for the configurational contribution A(n) to the Helmholtz free energy of an
imperfect crystal caused by Frenkel defects. As the configurational contribution
207
system members. For a 2-state system, in which the system members have
internal energies 0 and 1 = 0 + , express your results in terms of the ratio
of the system temperature T to the characteristic temperature int , defined as
int ≡ B . Construct a plot of U int (T )/(Nk B int ) vs. T // int for T / int
varying between 0 and 10.
14. Obtain an expression for the entropy S int (T ) associated with a 2-state system
and determine expression for the behaviour of S int (T ) for T int and T
int , with int defined as in Problem 13. Plot the ratio S int (T )/(Nk B ) as a
function of T // int and comment upon the manner in which S int approaches its
limiting high- and low-temperature values.
15. Consider a sample of a molecular solid containing N molecules in thermal
equilibrium at temperature T . If each individual molecule in this sample may
be in only one of two energy eigenstates, so that its energy is either + 0 or − 0 ,
and if the solid sample at equilibrium contains N + excited molecules having
energy 0 , obtain an expression for the configurational entropy S(N, N + ) (i.e.,
the entropy associated with the number of ways in which the excitation can be
distributed amongst the N molecules in the sample).
16. For a molecular crystal of the type considered in Problem 15, in which
individual molecules can be found in one of two internal states having energies
− 0 or + 0 , obtain expressions for the probabilities p + and p − that an
individual molecule may be found in the states at energies 0 , − 0 , and obtain
expressions for the internal state contribution, U int (T ), to the thermodynamic
internal energy U(T ) and for the contribution C int
V (T ) to the heat capacity
C V (T ) for such a solid. Express the heat capacity contribution in terms of the
separation between the molecular energy states.
17. Consider N atoms arranged in a regular three-dimensional lattice so as to form
a perfect crystal. The displacement of n of these atoms (1 n N) from
their lattice sites to interstices of the crystal lattice creates an imperfect crystal,
with n crystal defects said to be of the Frenkel type. The energy of such an
imperfect crystal is higher than that of the corresponding perfect crystal by
an amount U(n) ≡ E(n) = nw, with w the energy required to move one
atom from a crystal site to an interstitial site. By assuming that the number
M of interstitial sites into which an atom can enter is similar in magnitude
to N and that the concentration of such defects is sufficiently small that the
creation of any one defect can be treated as an independent event, argue that the
number of configurations associated with n Frenkel defects in a crystal having
N lattice sites and M defect sites is given by the number of ways M, n) of
removing n atoms from the N lattice sites and then independently redistributing
them among the M interstitial sites. From your expression for N, n),
obtain an expression for the configurational entropy S(n) associated with n
Frenkel defects. Note that S depends parametrically upon the fixed values N
and M.
18. Employ your expression for S(n) from Problem 17 to obtain an expression
for the configurational contribution A(n) to the Helmholtz free energy of an
imperfect crystal caused by Frenkel defects. As the configurational contribution