Transcript Document

Lecture 7
Gaseous systems composed of molecules with
internal motion.
•
Monatomic molecules.
•
Diatomic molecules.
•
Fermi gas.
•
Electron gas.
•
Heat capacity of electron gas.
1
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
2
b) the nondegeneracy criterion
nh 3
3

n

 1, here n=N/V
3/ 2
( 2 MkT )
  h /2MkT 

1
2

(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
1
Z1 N
N!
(7.2)
V 
Z1   3  j (T )
 
(7.3)
Z
where
3
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
j (T )   gi e i / kT
(7.4)
i
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
4
Fint= - N kT lnj
(7.5)
int= - kT lnj
(7.6)



 int  Nk  ln j  T
ln j 
T


Eint
=NkT2

CV int  Nk
T
(7.7)

ln j
T
 2 
ln
T
 T
(7.8)

j

(7.9)
5
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.
6
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: T<<Eion/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.
7
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 (L=S=0).
Their (electronic) ground state is clearly a singlet: ge=1.
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 gn=2Sn+1. Moreover, a
monatomic molecule is incapable of having any vibrational
or rotational states
8
The internal partition function (7.10) of such a molecule is
therefore given by
Fint= - N kT lnj
j(T)=ggr.st.=ge gn=2Sn+1
int= - kT lnj
(7.11)
Equations (7.4-7.9) then tell us that the internal motions in





Nk
ln
j

T
ln
j
 towards properties

int
this case contribute
only
such as the

T


chemical potential and the entropy of the gas; they do not
 properties such as the internal
2
make contribution
towards
ln j
Eint=NkT
 Theat.
energy and the specific
In other cases, the ground
may possess
 state
 the atom
 2 of

CV int  Nk T
ln j 
both orbital angular momentum
 T  and
T spin
 (L0,S0- as, for
example, in the case of alkali atoms), the ground state
would then possess a definite fine structure.
9
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
jelect (T )   (2 J  1)e
 J /kT
(7.12)
J
The forgoing expressions simplifies considerably in the
following limiting cases:
kT >> all J ; then
jelect (T )   (2 J  1)  (2L  1)( 2S  1)
J
(7.13)
10
kT<< all J ; then
jelect (T )  (2 J 0  1)e 0 /kT
(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 (2Sn+1) introduced by the
nuclear spin must be taken into account in each case.
11
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: T<<Ediss/k104-105 oK.
At this temperatures the number of dissociated molecules
in the gas would be quite insignificant.
12
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
jelec(T )  g0  g1e
  / kT
(7.16)
13
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
  2 

Cv int  Nk
ln j 
T
T T

( / kT ) 2
(Cv )elec  Nk
[1  ( g0 / g1 )e  / kT ][1  ( g1 / g0 )e / kT ]
(7.17)
We note that this contribution vanishes both for T<</k
and for T>>/k and has a maximum value for a certain
temperature /k; cf. the corresponding situation in the
case of monatomic atom.
14
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.
15
The energy levels for a mode of frequency  are then
given by the well-known expression (n+1/2) 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
2
e v /T

 v 
(CV )vib  Nk 
; v 


/
T
2
v
k
 T  (e
 1)
(7.18)
We note that for T>>v the vibrational specific heat is very
nearly equal to the equipatition value Nk; otherwise, it is
always less than Nk. In particular, for T<<v , the specific
heat tends to zero (see Figure 7.1); the vibrational degrees
16
of freedom are then said to be "frozen".
Cvib/Nk
1.0
0.5
0
0.5
1.0
T/ V

1.5
2.0
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.
17
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
18
gn= (2SA+1)(2SB+1)
(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
 rot  l (l  1) 2 /2I
(7.20)
here I=M(r0)2 , where M=m1m2/(m1+m2) 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
19


2

j rot (T )   (2l  1) exp l (l  1)
2IkT

l 0




=  (2l  1) exp l (l  1) r
T

l 0





2

;  r 
2Ik

(7.21)
For T>>r the spectrum of the rotational states may be
approximated by a continuum.
The summation (7.21) is the replaced by integration:

r

j rot (T )   (2l  1) exp l (l  1)
T

0
T

dl 
r

(7.22)
20
The rotational specific heat is the given by
(7.23)
(CV)rot=Nk
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


 f (n)   f ( x)dx 
n 0
Putting
1
2
1
1
f (0)  121 f ' (0)  720
f ' ' ' (0)  30240
f V (0)  ....
0
f ( x)  (2 x  1) exp x( x  1)r / T 
one can obtain
2
T 1 1 r
4  r 
jrot (T ) 
 


  ....
 r 3 15 T
315  T 
(7.24)
21
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
2
3




 r 
 r 
16
1
(CV ) rot  Nk 1  45 


....




945
T
T








(7.25)
which shows that at high temperatures the rotational
specific heat decreases with temperatures and ultimately
tends to the classical value Nk.
22
Cvib/Nk
1.0
0.5
0
0.5
1.0
T/ V
1.5
2.0

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)
23
In the opposite limiting case, namely for T<<r , one may
retain only the first few terms of the sum (7.21); then
jrot (T )  1  3e 2r / T  5e 6r / T  ....
(7.26)
whence one obtains, in the lowest approximation
2
 r   2r / T
(CV ) rot  12Nk 
 e
 T 
(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".
24
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)
1
f ( )   (   )
e
1
for the case when the assembly is at the absolute zero of
temperature and the Fermi energy is F(0).
When T=0 the quantity {-F(0)}/kT has two possible
values:
for  >F(0), {-F(0)}/kT= while
for  <F(0), {-F(0)}/kT=-.
25
There are therefore two possible values of the Fermi
function:
1
f ( )  
1
e 1
for  <F(0),
(7.28)
1
for  >F(0),
f ( )  
0
e 1
f()
1
kT
T=0
0.5
Classical tail
0
0

Figure 7.3. Fermi-Dirac
distribution function plotted
at absolute zero and at a low
temperature kT<<. The
Fermi level o at T=0 is
shown.
26
Equation (7.28) implies that, at the absolute zero of
temperature, the probability that a state with energy
 <F(0) is occupied is unity, i.e such states are all occupied.
Conversely, all states with energies  >F(0) will be empty.
The form of f() at T=0 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.
f()
1
kT
T=0
0.5
Classical tail
0
0

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.
27
f()
1
kT
T=0
0.5
Classical tail
0
0

Figure 7.3. Fermi-Dirac distribution function plotted at absolute zero and
at a low temperature kT<<. The Fermi level o at T=0 is shown.
For the temperatures T<<TF/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
28
absolute temperature F (0).
We note that f=1/2 when =. The distribution for T=0
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 ->>kT 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.
29
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):

 F ( x ) ( x  a )  F (a )
(7.29)

Now consider the integral

  f
 F ( )  
0

d

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
30

  f
F ( )   

0

0
d


F
(

)[
f
(

)]

  F (  ) f ( 0)

(7.30)
But at low temperatures f(0)1, so that

  f
F
(

)


 
0

d  F (  )

(7.31)
a result similar to (7.29).
31
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.
fk=f(k-)
(7.32)
1
(7.33)
f ( )   / T
e
1
32
The energy distribution of the states is an important
property. Let
1
g ( )    [ ( k  0 )]
V k
(T=0)
(7.34)
(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
d3p
p2
m
1/ 2
g ( )  2 

(




)

[
2
m
(



)]
0
0
3
2
2
m
( 2 )

(7.36)
33
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)
N /V  g ( ) f (   )d

E /V   g ( ) f (   )(  0 )d
(7.38)
If T=0, 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 T=0, and is
always denoted by F:
F(T=0)=0
(7.39)
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
pF2
 F
(7.40)
2m
34
There are N states with energy less than F:
4 3
2
pF  V 
N
3
3
( 2 )
(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 a=h/(2 pF) is approximately the average distance
between the electrons, hence:
F ~
2
2
(7.43)
ma
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).
35
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,
T<<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.
2
F 
(3 2 N / V ) 2 / 3
2m
k F  (3 N / V )
2
1/ 3
pF


(7.44)
(7.45)
pF
vF 

(7.46)

TF  F
k
(7.47)
36
Table 7.1 Properties of the electron gas at the Fermi surface.
N/V cm-3
kF cm-1
vF cm/sec
F eV
TF=F/k
Li
4.61022
1.1108
1.3108
4.7
5.5104
Na
2.5
0.9
1.1
3.1
3.7
K
1.34
0.73
0.85
2.1
2.4
Rb
1.08
0.68
0.79
1.8
2.1
Cs
0.86
0.63
0.73
1.5
1.8
Cu
8.50
1.35
1.56
7.0
8.2
Ag
5.76
1.19
1.38
5.5
6.4
Au
5.90
1.2
1.39
5.5
6.4
37
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 T=0, this
gas disappears.
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At T=0 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 T<<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.
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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

1
[ E (T )  E (0)]   2 g (0) f ( ) d
V
0
(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 F>>T and so
the upper limit of the integral in (7.50) can be taken to be
. This integration is easy:
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

0
f ( ) d  T
2

x
 ex 1
0
dx 
2
12
T2
(7.51)
Substituting in (7.50), and differentiating once, we get
heat capacity
1
1  E 2
C

g (0)T
V
VT
3
(7.52)
This result is completely different from that of the ideal gas
in which Cv=3/2 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
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
Vg (0)  f ( )d ~
0
NT
(7.53)
F
Each active electron contribute approximately 1 to heat
capacity C.
Hence Cv~N(T/F). From (7.52) we get
2  T

Cv  N
2  F

2
  N
2

T
k 
 TF



(7.54)
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energy of a hole
F
energy of an
outer elctron
pF
-F
Figure 7.4 Electron (hole) energy versus momentum.
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