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5 Atomic Systems
18. Obtain an expression for a modified vibrational partition function z vib (T ),
defined as z vib (T ) ≡
∞
v=0
e
−vv vib /T , in the classical limit that the discrete
summation can be replaced by integration over the vibrational quantum number
v. When is this result appropriate?
19. Show that the difference between the (classical) limiting value Nk B of C V (T )
and C V (T ) integrated over the temperature for an ensemble of N quantum
mechanical simple harmonic oscillators (SHOs) precisely represents the heat
capacity contribution associated with the excluded zero-point energy of the N
quantum mechanical SHOs.
20. An approximate partition function for a gas of hard spheres can be deduced
from the monatomic gas expression simply by replacing the system volume V
by the volume available to a gas, each of whose atoms occupies a finite volume
typically denoted in thermodynamics by b, namely, replacing V by V − Nb
in the expressions for the partition functions for the gas. In this way, obtain
expressions for the internal energy U and the pressure P for a system of N
hard spheres.
21. If N atoms (or molecules) of a gas adsorbed onto M adsorption sites on
a surface are bound sufficiently strongly that they cannot readily leave the
surface, but not so strongly that they cannot move from an occupied site to an
unoccupied site, then such an adsorbed gas can at low coverage (i.e., N M)
of the surface be treated as a two-dimensional gas for which the canonical
partition function is
Z(T , A, N) =
1
N!
2πmk B T
h 2
N
A
N ,
in which m is the mass of an atom (molecule) and A is the area of the surface
upon which the atoms (molecules) are adsorbed. Obtain an expression for the
internal energy, U , and the heat capacity, C A , for such a two-dimensional gas.
22. Obtain an approximation to the molar Einstein heat capacity C V (T ) that may
be employed at temperatures T such that T E ; similarly, obtain an
approximation to C V (T ) that applies at temperatures T such that T E .
23. Show that in the limit that u goes to zero, the Debye function D(u) defined by
Eq. (5.6.19) goes to unity.
24. Show that for temperature T such that T D , expression (5.6.21) for the
molar Debye heat capacity C V (T ) can be approximated as
C V (T )
12Rπ 4
5
T
D
3
,
while for temperature T such that T D , C V (T ) has the value C V (T )
3R.
5 Atomic Systems
18. Obtain an expression for a modified vibrational partition function z vib (T ),
defined as z vib (T ) ≡
∞
v=0
e
−vv vib /T , in the classical limit that the discrete
summation can be replaced by integration over the vibrational quantum number
v. When is this result appropriate?
19. Show that the difference between the (classical) limiting value Nk B of C V (T )
and C V (T ) integrated over the temperature for an ensemble of N quantum
mechanical simple harmonic oscillators (SHOs) precisely represents the heat
capacity contribution associated with the excluded zero-point energy of the N
quantum mechanical SHOs.
20. An approximate partition function for a gas of hard spheres can be deduced
from the monatomic gas expression simply by replacing the system volume V
by the volume available to a gas, each of whose atoms occupies a finite volume
typically denoted in thermodynamics by b, namely, replacing V by V − Nb
in the expressions for the partition functions for the gas. In this way, obtain
expressions for the internal energy U and the pressure P for a system of N
hard spheres.
21. If N atoms (or molecules) of a gas adsorbed onto M adsorption sites on
a surface are bound sufficiently strongly that they cannot readily leave the
surface, but not so strongly that they cannot move from an occupied site to an
unoccupied site, then such an adsorbed gas can at low coverage (i.e., N M)
of the surface be treated as a two-dimensional gas for which the canonical
partition function is
Z(T , A, N) =
1
N!
2πmk B T
h 2
N
A
N ,
in which m is the mass of an atom (molecule) and A is the area of the surface
upon which the atoms (molecules) are adsorbed. Obtain an expression for the
internal energy, U , and the heat capacity, C A , for such a two-dimensional gas.
22. Obtain an approximation to the molar Einstein heat capacity C V (T ) that may
be employed at temperatures T such that T E ; similarly, obtain an
approximation to C V (T ) that applies at temperatures T such that T E .
23. Show that in the limit that u goes to zero, the Debye function D(u) defined by
Eq. (5.6.19) goes to unity.
24. Show that for temperature T such that T D , expression (5.6.21) for the
molar Debye heat capacity C V (T ) can be approximated as
C V (T )
12Rπ 4
5
T
D
3
,
while for temperature T such that T D , C V (T ) has the value C V (T )
3R.