Gaseous systems composed of molecules with internal motion'

1
Lecture 7

• Gaseous systems composed of molecules with
  internal motion.
• Monatomic molecules.
• Diatomic molecules.
• Fermi gas.
• Electron gas.
• Heat capacity of electron gas.

2
Gaseous systems composed of molecules with
internal motion
In most of our studies so far we have consider
only the translation part of the molecular
motion.
Though this aspect of motion is invariably
present in a gaseous system, other aspects, which
are essentially concerned with the internal
motion of the molecules, also exist. It is only
natural that in the calculation of the physical
properties of such a system, contributions
arising from these motions are also taken into
account.
In doing so, we shall assume here that a) the
effects of the intermolecular interactions are
negligible and
3
b) the nondegeneracy criterion
(7.1)
is fulfilled effectively, this makes our system
an ideal, Boltzmannian gas.
Under these assumptions, which hold sufficiently
well in a large number of practical applications,
the partition function of the system is given by
(7.2)
where
(7.3)
4
The factor in brackets is the transitional
partition function of a molecule, while the
factor j(T) is supposed to be the partition
function corresponding to the internal motions.
The latter may be written as
(7.4)
where ?i is the molecular energy associated with
an internal state of motion (which characterized
by the quantum numbers i), while gi represents
the degeneracy of that state.
The contributions made by the internal motions of
the molecules to the various thermodynamic
quantities of the system follow straightforwardly
from the function j(T). We obtain
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(7.5)
Fint - N kT lnj
(7.6)
?int - kT lnj
(7.7)
(7.8)
EintNkT2
(7.9)
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Thus the central problem in this study consists
of deriving an explicit expression for the
function j(T) from a knowledge of the internal
states of the molecules. For this purpose, we
note that the internal state of a molecule is
determined by

• electronic state,
• state of nuclei,
• vibrational state and
• rotational state.

Rigorously speaking, these four modes of
excitation mutually interact in many cases,
however, they can be treated independently of one
another. We then write
j(T)jelec(T) jnuc(T) jvib(T) jrot(T)
(7.10)
with the result that the net contribution made by
the internal motions to the various thermodynamic
quantities of the system is given by a simple sum
of the four respective contributions.
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Monatomic molecules
At the very outset we should note that we cannot
consider a monatomic gas except at temperatures
such that the thermal energy kT is small in
comparison with the ionization energy Eion for
different atoms, this amounts to the condition
TltltEion/k?104-105 oK.
At these temperatures the number of ionized atoms
in the gas would be quite insignificant. The same
would be true for atoms in excited states, for
the reason that separation of any of the excited
states from the ground state of the atom is
generally comparable to the ionization energy
itself. Thus, we may regard all the atoms of the
gas to be in their (electronic) ground state.
8
Now, there is a well-known class of atoms, namely
He, Ne, A,..., which, in their ground state,
possess neither orbital angular momentum nor spin
(LS0).
Their (electronic) ground state is clearly a
singlet ge1.
The nucleus, however, possesses a degeneracy,
which arises from the possibility of different
orientations of the nuclear spin. (As is well
known, the presence of the nuclear spin gives
rise to the so-called hyperfine structure in the
electronic state.
However the intervals of this structure are such
that for practically all temperatures of interest
they are small in comparison with kT.) If the
value of this spin is Sn, the corresponding
degeneracy factor gn2Sn1. Moreover, a monatomic
molecule is incapable of having any vibrational
or rotational states
9
The internal partition function (7.10) of such a
molecule is therefore given by
j(T)ggr.st.ge? gn2Sn1
(7.11)
Equations (7.4-7.9) then tell us that the
internal motions in this case contribute only
towards properties such as the chemical potential
and the entropy of the gas they do not make
contribution towards properties such as the
internal energy and the specific heat.
In other cases, the ground state of the atom may
possess both orbital angular momentum and spin
(L?0,S?0- as, for example, in the case of alkali
atoms), the ground state would then possess a
definite fine structure.
10
The intervals of this structure are in general,
comparable with kT hence, in the evaluation of
the partition function, the energies of the
various components of the fine structure must be
taken into account. Since these components
differ from one another in the value of the total
angular momentum J, the relevant partition
function may be written as
(7.12)
The forgoing expressions simplifies considerably
in the following limiting cases
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kTltlt all ?J then
(7.14)
where J0 is the total angular momentum, and ?0
the energy of the atom in the lowest state. In
their case, the electronic motion makes no
contribution towards the specific heat of the
gas. And, in view of the fact that both at high
temperatures the specific heat tends to be equal
to the translational value 3/2 Nk, it must be
passing through a maximum at a temperature
comparable to the separation of the fine levels.
Needless to say, the multiplicity (2Sn1)
introduced by the nuclear spin must be taken into
account in each case.
12
Diatomic molecules
Now, just as we could not consider a monatomic
gas except at temperatures for which kT is small
compared with the energy of ionization, for
similar reasons one may not consider a diatomic
gas except at temperatures for which kT is small
compared with the energy of dissociation for
different molecules this amounts once again to
the condition TltltEdiss/k?104-105 oK.
At this temperatures the number of dissociated
molecules in the gas would be quite
insignificant.
13
At the same time, in most cases, there would be
practically no molecules in the excited states as
well, for the separation of any of these states
from the ground state of the molecule is in
general comparable to the dissociation energy
itself.
The heat capacitance of the diatomic gas is
consist from three parts
Cv(Cv)elec(Cv)vib(Cv)rot
(7.15)
Let us consider them consequently.
In the case of electron contribution the
electronic partition function can be written as
follows
(7.16)
14
where g0 and g1 are degeneracy factors of the two
components while ? is their separation energy.
The contribution made by (7.16) towards the
various thermodynamic properties of the gas can
be readily calculated with the help of the
formula (7.4-7.9).
In particular we obtain for the contribution
towards specific heat
(7.17)
We note that this contribution vanishes both for
Tltlt?/k and for Tgtgt?/k and has a maximum value for
a certain temperature ??/k cf. the corresponding
situation in the case of monatomic atom.
15
Let us now consider the effect of vibrational
states of the molecules on the thermodynamic
properties of the gas. To have an idea of the
temperature range, over which this effect would
be significant, we note that the magnitude of the
corresponding quantum of energy, namely , for
different diatomic gases is of order of 103 oK.
Thus we would obtain full contributions
(consistent with the dictates of the
equipartition theorem) at temperatures of the
order of 104 oK or more, and practically no
contribution at temperatures of the order of 102
oK or less.
We assume, however, that the temperature is not
high enough to excite vibrational states of large
energy the oscillations of the nuclei are then
small in amplitude and hence harmonic.
16
The energy levels for a mode of frequency ? are
then given by the well-known expression (n1/2)
h?/2?. The evaluation of the vibrational
partition function jvib(T) is quite elementary.
In view of the rapid convergence of the series
involved, the summation may formally be extended
to n?. The corresponding contributions towards
the various thermodynamic properties of the
system are given by eqn.(4.64 -4.69). In
particular, we have
(7.18)
We note that for Tgtgt?v the vibrational specific
heat is very nearly equal to the equipatition
value Nk otherwise, it is always less than Nk.
In particular, for Tltlt?v , the specific heat
tends to zero (see Figure 7.1) the vibrational
degrees of freedom are then said to be "frozen".
17
Figure 7.1 The vibrational specific heat of a
gas of diatomic molecules. At T?v the specific
heat is already about 93 of the equipartition
value.
18

• Finally, we consider the effect of
• the states of the nuclei and
• the rotational states of the molecule
• wherever necessary, we shall take into account
  the mutual interaction of these modes.

This interaction is on no relevance in the case
of the heternuclear molecules, such as AB it is,
however, important in the case of homonuclear
molecules, such as AA.
In the case of heternuclear molecules the states
of the nuclei may be treated separately from the
rotational states of the molecule. Proceeding in
the same manner as for the monatomic molecules we
conclude that the effect of the nuclear states is
adequately taken care of through degeneracy
factor gn. Denoting the spins of the two nuclei
by SA and SB, this factor is given by
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gn (2SA1)(2SB1)
(7.19)
As before, we obtain a finite contribution
towards the chemical potential and the entropy of
the gas but none towards the internal energy and
specific heat.
Now, the rotational levels of a linear "rigid"
with two degrees of freedom (for the axis of
rotation) and the principle moments of inertia
(I, I, 0), are given by
(7.20)
here IM(r0)2 , where Mm1m2/(m1m2) is the
reduced mass of the nuclei and r0 the equilibrium
distance between them. The rotational partition
function of the molecule is then given by
20
(7.21)
For Tgtgt?r the spectrum of the rotational states
may be approximated by a continuum.
The summation (7.21) is the replaced by
integration
(7.22)
21
The rotational specific heat is the given by
(CV)rotNk
(7.23)
which is indeed consistent with equipartition
theorem.
A better evaluation of the sum (7.21) can be made
with the help of the Euler-Maclaurin formula
22
which is the so-called Mulholland's formula as
expected, the main term of this formula is
identical with the classical partition function
(7.22). The corresponding result for the specific
heat is
(7.25)
which shows that at high temperatures the
rotational specific heat decreases with
temperatures and ultimately tends to the
classical value Nk.
23
Fig.7.2. The rotational specific heat of a gas of
heteronuclear diatomic molecules.
Thus, at high (but finite) temperatures the
rotational specific heat of diatomic gas is
greater than the classical value. On the other
hand, it must got to zero as T? 0. We, therefore,
conclude that it must pass through at least one
maximum. (See Figure 7.2)
24
In the opposite limiting case, namely for Tltlt?r ,
one may retain only the first few terms of the
sum (7.21) then
whence one obtains, in the lowest approximation
(7.27)
Thus, as T? 0, the specific heat drops
exponentially to zero (Fig. 7.2). Now we can
conclude that at low temperatures the rotational
degrees of freedom of the molecules are also
"frozen".
25
Fermi gas
Let us consider the perfect gas composed of
fermions. Let us consider in this case the
behavior of the Fermi function given by equation
(5.46)
for the case when the assembly is at the absolute
zero of temperature and the Fermi energy is ?F(0).
When T0 the quantity ?-?F(0)/kT has two
possible values
for ? gt?F(0), ?-?F(0)/kT? while for ?
lt?F(0), ?-?F(0)/kT-?.
26
There are therefore two possible values of the
Fermi function
Figure 7.3. Fermi-Dirac distribution function
plotted at absolute zero and at a low temperature
kTltlt?. The Fermi level ?o at T0 is shown.
27
Equation (7.28) implies that, at the absolute
zero of temperature, the probability that a state
with energy ? lt?F(0) is occupied is
unity, i.e such states are all occupied.
Conversely, all states with energies ? gt?F(0)
will be empty. The form of f(?) at T0 is shown
as a function of energy in Figure 7.3.
This behavior may be explained as following. At
the absolute zero of temperature, the fermions
will necessarily occupy the lowest available
energy states.
Thus with only one fermion allowed per state, all
the lowest states will be occupied until the
fermions are all accommodated. The Fermi level,
in this case, is simply the highest occupied
state and above this energy level the states are
unoccupied.
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Figure 7.3. Fermi-Dirac distribution function
plotted at absolute zero and at a low temperature
kTltlt?. The Fermi level ?o at T0 is shown.
For the temperatures TltltTF??/k??F/k the
Fermi-Dirac distribution behavior is shown in the
Fig. 7.3 by bold line. The Fermi temperature TF
and the Fermi energy ?F are defined by the
indicated identities. The Fermi energy ?F is
defined as the value of the chemical potential at
the absolute temperature ?F ??(0).
29
We note that f1/2 when ??. The distribution for
T0 cuts off abruptly at ??, but at a finite
temperature the distribution fuzzes out over a
width of the order of several kT. At high
energies ?-?gtgtkT the distribution has a classical
form.
The value of the chemical potential is a function
of temperature, although at low temperature for
an ideal Fermi gas the temperature dependence of
? may often be neglected.
The determination of ?(?) is often the most
tedious stage of a statistical problem,
particularly in ionization problems. We note that
? is essentially a normalization parameter and
that the value must be chosen to make the total
number of particles come out properly.
30
An important analytic property of f at low
temperatures is that -df/d? is approximately a
delta function. We recall the central property of
the Dirac delta function ?(x-a)
(7.29)
Now consider the integral
At low temperatures -df/d? is very large for ???
and is small elsewhere. Unless F(?) is rapidly
varying in this neighborhood we may replace it by
F(?) and the integral becomes
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(7.30)
But at low temperatures f(0)?1, so that
(7.31)
a result similar to (7.29).
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Electron gas
The conducting electrons in a metal may be
considered as nearly free, moving in a constant
potential field like the particles of an ideal
gas. Electrons have half-integral spin, and hence
the Fermi-Dirac statistics are applicable to an
ideal gas of electrons.
We use k to specify the state of the electron,
and ?k its energy. Let ? be the chemical
potential of the electron. Each state can
accommodate at most one electron. Let fk be the
FD average population of state k.
fkf(?k-?)
(7.32)
(7.33)
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The energy distribution of the states is an
important property. Let
(7.34)
???(T0)
(7.35)
The function g(?) is the energy distribution of
the states per unit volume, which we simply call
density of states, and ? is the energy with
respect to ?0. The calculation of g(?) gives
(7.36)
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The thermodynamic properties of this model can be
largely expressed in terms of f and g, e.g. the
density N/V of the electrons and the energy
density E/V are
(7.37)
(7.38)
If T0, then all the low energy states are filled
up to the Fermi surface. Above this surface all
the states are empty. The energy at the Fermi
surface is ?0, i.e the chemical potential when
T0, and is always denoted by ?F
(7.39)
?F??(T0)?0
The Fermi surface can be thought of as spherical
surface in the momentum space of the electrons.
The radius of the sphere pF is called Fermi
momentum
(7.40)
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There are N states with energy less than ?F
(7.41)
i.e (volume of sphere in momentum space) ?
(volume) ? (spin state (2)) ? (h)3, with h/2?
1. Hence
pF(3?2n)1/3
(7.42)
where n?N/V. Let ah/(2? pF) is approximately
the average distance between the electrons, hence
(7.43)
is approximately the zero-point energy of each
electron. This zero-point energy is a result of
the wave nature of the electron or a necessary
result of the uncertainty principle. To fix an
electron to within a space of size a, its
momentum would have to be of order h/ (2? a).
36
In most metals, the distance between electrons is
about 10-8 cm and ?F 1 eV 104 oK (See Table
7.1). Therefore, at ordinary temperatures,
Tltlt?F, i.e. the temperature is very low, only
electrons very close to the Fermi surface can be
excited and most of the electrons remain inside
the sphere experiencing no changes.
(7.44)
(7.45)
(7.46)
(7.47)
37
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38
Heat Capacity
Only a very small portion of the electrons is
influenced by temperature. Hence the concept of
holes appears naturally.
The states below the Fermi surface are nearly
filled, and empty states are rare. We shall call
an empty state a hole.
Now this model becomes a new mixed gas of holes
together with electrons above Fermi surface. (We
shall call these outer electrons.)
The momentum of a hole is less than pF, while
that of the outer electrons is larger than pF.
The lower the temperature T , the more dilute the
gas is. At T0, this gas disappears.
39
At T0 the total energy is zero, i.e. there are
no holes or outer electrons. The holes are also
fermions because each state has at most one hole.
Hence, a state of energy -?' can produce a hole
of energy ?'. The average population of the hole
is (for each state)
1-f(-?'-?) f(?'?)
(7.48)
Now the origin of the energy is shifted to the
energy at the Fermi surface, i.e. ?0. The energy
of a hole is the energy required taking an
electron from inside the Fermi surface to the
outside.
As Tltlt?F, the energy of the holes or the outer
electrons cannot exceed T by too much. In this
interval of energy, g(?) is essentially
unchanged, i.e.
40
g(?)?g(0)mpF/?2
(7.49)
Hence the energy distribution is the same for the
holes or the outer electrons. Therefore, ??0. All
the calculations can now be considerably
simplified, e.g. the total energy is
(7.50)
where 2g(0) is the density of states of the holes
plus the outer electrons.
The energy of a hole cannot exceed ?F, but ?FgtgtT
and so the upper limit of the integral in (7.50)
can be taken to be ?. This integration is easy
41
(7.51)
Substituting in (7.50), and differentiating once,
we get heat capacity
(7.52)
This result is completely different from that of
the ideal gas in which Cv3/2 N.
In that case each gas molecule contribute a heat
capacity of 3/2. Now only a small portion of the
electrons is involved in motion and the number of
active electrons is about NT/?F
42
(7.53)
Each active electron contribute approximately 1
to heat capacity C.
Hence CvN(T/?F). From (7.52) we get
(7.54)
43
Figure 7.4 Electron (hole) energy versus
momentum.