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A Brief Introduction to
Statistical Thermodynamics
A Web-Book by Rituraj Kalita, Dept. of
Chemistry, Cotton College,
Guwahati-781001 (Assam, India)
Preface
Ch. 1
Ch. 2 Ch. 3
Ch. 4
Ch. 5
Ch. 6
Ch.
7
Bibliog.
Background topics/ vocabulary
General topics
Advanced
(avoidable) topics
© 2009.
Copyright reserved. The book or any portion of it
can't be
reproduced/
re-published/
circulated.
Ch. 5: Statistical
Thermodynamics of Vibrations in Solids
and Some Non-Boltzmann Systems
5.0 The Normal Modes of Vibrations in a Polyatomic Molecule:
In a diatomic molecule such as HCl,
there arises only one mode of vibration, namely the stretching mode in which the
lone internuclear distance of the molecule changes. But in a polyatomic molecule
such as H2O, the vibrational motion of the nuclei of the molecule is
more complicated. The complicated vibrational motion therein may be explained as
a superimposed combination of several simple kinds of vibrational motions, each
of which is termed a vibrational normal mode. The normal modes in general have
different fundamental frequencies of vibrations, but sometimes two modes might
have the same frequency. For example, in the bent
tri-atomic molecule H2O, there arises three normal modes of
vibrations, namely symmetric stretching, asymmetric stretching, and bending
(Figure 5.1). The second one is associated with the largest inter-level
energy differences while the third one (bending) with the smallest differences, with their fundamental
vibrational frequencies in wavenumbers being 3657, 3756 and 1595 cm–1
respectively. The actual vibration in any H2O molecule may be
expressed as a superimposed combination of these three basic forms of
vibrations.
Figure 5.1: The three normal modes of vibration
(sym. stretching, asym. stretching & bending) in
H2O
For a polyatomic molecule with atomicity A > 2, the number of
normal modes of vibrations is either (3A–5) or (3A–6), depending
respectively on whether the molecule is linear or non-linear. This pair of
values are based on the fact that the total (nuclear) independent degrees of
freedom for the molecules is 3A (three each from each nucleus), out of which
three are translational ones, another two or three are rotational ones depending
on whether the molecule is linear or non-linear respectively, while the rest are
vibrational ones.
5.1 The Normal Modes of Vibrations in an Elementary Solid:
A piece of an elementary (i.e., an element, in contrast to a compound) solid crystal (e.g., lead or graphite) composed of N atoms may be compared to a single extremely large molecule, and thus may considered to have 3N–6 (~3N) normal modes of vibration. The translational and rotational motions of the piece of solid obviously do not contribute to their system internal energy U, which means that the internal energy of a solid is wholly vibrational, unlike the case for gases discussed earlier. As the number of atoms N in the solid piece is very large (say, of the order of 1020 or higher), 3N–6 is practically identical with 3N. Thus the elementary solid is assumed as composed of 3N number of linear harmonic oscillators (LHOs), the vibrations of which impart its internal energy. It is the vibrational frequencies and the number of oscillators vibrating with a given frequency that differs for the two major theories about vibrations in solids (discussed below), namely in the simpler Einstein theory and the comparatively evolved Debye theory, but the total number of oscillators is same (i.e., 3N) according to both. In Einstein theory all the oscillators are assumed to vibrate with the same frequency (for a given solid) nE implying that all the normal modes have the same vibrational frequency, an obviously too crude approximation regarding the actual normal modes (recall the H2O example above). As per the Debye theory, however, the frequencies of different oscillators are considered to be continuously differing up to a maximum value nD.
This concept of 3N number of
vibrational normal modes or oscillators is consistent with the well-known Dulong
and Petit law (an experimental law discovered much earlier, in 1819) about
molar heat capacity at constant volume (Cv,mol)
of elementary solids, this law stating that Cv,mol
of any elementary solid at sufficiently high temperatures approximately equal
24.9 J K–1
mol–1, i.e. 3R. This is because at a 'sufficiently
high temperature' (its necessary value dependent upon the solid considered*), each
vibrational mode gets fully expressed and contributes a share kT to the internal
energy, so that the internal energy of one mole of
solid (containing NA
number of atoms) becomes 3NAkT
= 3RT, implying that the molar heat capacity Cv,mol
= (∂Umol/∂T)V
equals 3R. However the lower value of molar heat
capacity observed at lower temperatures requires
further explanations, which are provided by the Einstein and the Debye theories
detailed below.
* As examples, for lead at room temperature (say, at 290 K) the molar heat
capacity practically equals 3R, whereas for
diamond even 1500 K is not a 'sufficiently
high' temperature!
5.2 The Einstein Theory of Vibrations in Solids:
In this early theory explaining the heat capacity of solids, propounded by the renowned physicist Albert Einstein in 1907, Einstein postulated that all the 3N oscillators within a piece of elementary solid vibrates (as linear harmonic oscillators) with a same frequency nE. The value of nE is dependent on the nature of the solid, or in other words, for a given solid (say, diamond) nE has a definite value (say, 3.022 x 1013 Hz for diamond).
The energy e
of a linear harmonic oscillator, as per quantum mechanics, is:
e = (v + ½) h nE
[where v is a non-negative integer, say 0, 1, 2, 3 etc.]
This means that the partition function q for these oscillators is:
q = S exp{–e/(kT)}
= Sv
exp{–(v + ½) h nE/(kT)}
= exp{– hnE/(2kT)}.
Sv
exp{–
vhnE/(kT)}
= exp{– hnE/(2kT)}.
Sv
[exp{– hnE/(kT)}]v
= exp{– hnE/(2kT)}/
[1 – exp{–
hnE/(kT)}]
Recalling the derivation of the vibrational p.f. for gases in Section 3.5.
Substituting hnE/k
as qE,
where qE,
the Einstein temperature, is a temperature-dimension (do check its
dimension) physical constant for the given solid (for diamond, qE
= 1450 K -- check it using the corresponding nE
value mentioned above), we get the following simple relation for q:
q = exp{–
qE/(2T)}/
{1 – exp(–qE
/ T
)}
The system internal energy U, as per the relation in Section 2.5.1, equals
(3N).kT2 (∂ lnq/∂T)V,
as the number of particles here is nothing other than the number of oscillators
3N meaning that here N must be replaced with (3N). We now have:
ln q = ln [exp{–
qE/(2T)}]
–
ln{1 – exp(–qE
/ T )} = –qE/(2T)
–
ln{1 – exp(–qE
/ T )}
Thus we get: U = (3N).kT2 (∂
lnq/∂T)V
= 3N.kT2 (–qE/2)(–1/
T2) –
[1/{1 – exp(–qE
/ T )}].{–exp(–qE
/ T ).(–qE).(1
/ T2 )}
= 3N.k.qE
[½ + exp(–qE
/ T )/{1 – exp(–qE
/ T )}]
Multiplying both numerator and denominator of the ratio exp(–qE
/ T )/{1 – exp(–qE
/ T )} in the last expression for U, we
finally get:
U = 3N k qE
[½ + 1/{exp(qE
/ T ) –
1}]
This is the expression for U of an elementary solid as per the Einstein theory of
solids. However, an expression for the heat capacity Cv
would be particularly meaningful, as the experimental value for Cv
may be directly measured and compared with this theoretical expression.
As Cv
= (∂U/∂T)V,
we get:
Cv
= 3N k qE.
d/dT [½ + 1/{exp(qE
/ T ) –
1}]
= 3N k qE.
[0 + (–1)/{exp(qE
/ T ) –
1}2. exp(qE
/ T ). qE.
(–1) / T2]
or, Cv
= 3N k. (qE
/ T )2.
exp(qE
/ T ) / {exp(qE
/ T ) –
1}2
and, Cv,m
= 3R. (qE
/ T )2.
exp(qE
/ T ) / {exp(qE
/ T ) –
1}2
That the above expression for Cv
satisfies the aforesaid experimentally known (Dulong and Petit) value at
sufficiently high temperatures is rather obvious. [At sufficiently high
T,
T >> qE
so that (qE
/ T ) << 1, meaning that {exp(qE
/ T) –
1} ~ (qE
/ T ), remembering the expansion (1 + y + y2/2
+ ...) for exp(y). This means that the denominator in the above
expression equals (qE
/ T )2,
so that Cv
~ 3N k.{1 + (qE
/ T ) + ...} ~ 3Nk. Thus the molar heat
capacity Cv,mol
equals 3NAk,
i.e., 3R.] However, at lesser temperatures the above Einstein-theory expression
of Cv
does not give values properly consistent with the experimentally measured
values of Cv.
Thus for lead in the temperature range 0–50K,
i.e. at temperatures T comparable to and less than its Einstein temperature qE,
the deviation of experimental values of Cv
from its aforesaid Einstein-theory expression
is quite significant (Figure 5.2, adapted from: Elements of Statistical Thermodynamics, 2nd
edition, by L.K. Nash, Addison-Wesley, 1972.). Such
a deviation arises mainly due to the assumption of equal frequency for all the
3N vibrational modes, and so gets greatly corrected by the Debye theory (see
next section) that
improves upon that drastic assumption.
Note: As the frequency of a vibrational mode bears a
proportionality to the square-root of the force constant but a inverse
proportionality to the square-root of the atom masses [recalling the well-known
relation n
= (2p)–1.(K/m)0.5
for frequency of an LHO], it becomes immediately obvious why hard and
light-atomic crystals such as diamond have much larger characteristic
frequencies (as well as Einstein temperatures) compared to soft and
heavy-atomic crystals such as lead!
Problem: Calculate, as
per Einstein theory, the molar heat capacity of an element at thrice its
Einstein temperature, at its Einstein temperature, and at one-tenth of its
Einstein temperature.
Ans: We have, respectively, (T
/ qE)
= 3, 1 & 1/10. So, (qE
/ T) = 1/3, 1 & 10. So we
get:
(i) at (qE
/ T) = 1/3,
Cv,m
= 3R (1/3)2
exp(1/3)/{exp(1/3)–1}2
= 24.712 J K–1
mol–1 (almost 3R, isn't it?)
(ii) at (qE
/ T) = 1, Cv,m
= 3R e/{e –1}2
= 22.963 J K–1
mol–1
(iii) at (qE
/ T) = 10,
Cv,m
= 3R (10)2 exp(10) /{exp(10)–1}2
= 0.113 J K–1
mol–1 (so smaller than
3R)
Figure 5.2: Discrepancy seen between
Figure 5.3: Frequency distribution
experiment and Einstein theory at low T
for the 3N oscillators as per Debye
5.3 The Debye Theory of Vibrations in Solids:
As per this improved theory of
elementary solids, the oscillators within the piece of solid are assumed to have
continuously varying frequencies of vibration n,
starting from n = 0 to n =
nD,
a maximum limit dependent on the nature of the solid (as an example, for lead nD
= 1.834 x 1012 Hz). The number of oscillators dN associated with
frequencies between n and n+dn
is given as: dN = g(n).dn
where g(n) is proportional to n2
(i.e., g(n) = Cn2)
as is shown in Figure 5.3 above. It is obvious that the proportionality constant C
here is (9N/nD3),
i.e., g(n) = (9N/nD3).n2,
as 
g(n).dn
(= CnD3/3)
must equal 3N, the total number of oscillators within the solid (the integral is
shown in figure as the shaded area under the curve).
Note: Instead of specification of nD,
the given solid is more commonly characterized by specifying its Debye
temperature qD
defined as qD
= hnD
/ k. The larger is nD,
higher is the value of qD.
[The Debye temperature for lead is 88 K (do cross-check using its nD
value above), whereas for diamond it is 1860 K.]
For the dN linear harmonic oscillators within the said
infinitesimal frequency range n to n+dn,
the energy per oscillator e
is obviously (v+½)hn, so the corresponding partition function is:
q = exp{–e/(kT)}
= Sv
exp{–(v + ½) hn/(kT)}
= exp{–
q/(2T)}/
{1 – exp(–q
/ T )}
where q
= hn/ k,
and the above result may be obtained similarly to the corresponding one in the above section.
Now, the energy component dU corresponding to these dN oscillators is
given in terms of q as: dU = (dN)kT2.(∂ lnq/∂T)V
= dN.kq
[½ + 1/{exp(q
/ T ) –
1}]
Where we use the obvious similarities in calculations from the above
section. Using the
expressions dN = g(n).dn
= (9N/nD3)
n2 dn
& q
= hn/ k,
we would finally get:
dU = (9N/nD3).n2
dn. kq
[½ + 1/{exp(q
/ T ) –
1}].
= (9Nh/nD3).n.[½ +
1/{exp(q
/ T ) –
1}].n2 dn
The total internal energy U is
obviously an integral over all such dU values, and so we have:
U =
(9Nh/nD3).
[½ + 1/{exp(q
/ T ) –
1}].n3 dn
= ½.(9Nh/nD3)
n3
dn + (9Nh/nD3).
[n3 /{exp(q
/ T ) –
1}].dn
= ½.(9Nh/nD3).(nD4/4)
+ (9Nh/nD3).{n3
/(ez –
1)} dn
where we substitute z
= q
/ T [= hn/(kT)].
May note that z
≤ qD
/ T, as n
≤ nD.
This gives U = (9/8).NhnD
+ (9Nh/nD3).
{n3 /(ez –
1)} dn
The expression for the heat capacity Cv
is [as Cv
= (∂U/∂T)V]:
Cv
= 0 + (9Nh/nD3).(d/dT)
[
{n3 /(ez –
1)} dn]
The differentiation with respect to T may now be performed within the integral
expression. To do this, we note that only z
= hn/(kT),
found within the denominator (ez –
1) of the integrand n3/(ez –
1), is dependent on the variable T. So we finally
get:
Cv
= (9Nh/nD3).
n3 (d/dT){1/(ez –
1)}.dn
= (9Nh/nD3).
n3 {(–1)/(ez –
1)2.ez.(hn/k).(–1)/
T2}.dn
= (9Nh2/nD3).{1/(kT2)}.{n4 ez
/ (ez –
1)2} dn
Using the substitution n
= (kT/h).z
and dn =
(kT/h).dz
we get:
Cv
= (9Nh2/nD3).{1/(kT2)}.(kT/h)4.(kT/h).{z4 ez
/ (ez –
1)2} dz
= 9Nk{k3T3/(h3nD3)}.{z4 ez
/ (ez –
1)2}
dz
= 9Nk (T/qD)3.
{z4 ez
/(ez –
1)2}dz
[substituting new limits of integration]
Thus, as per Debye theory,
Cv
= 9Nk (T/qD)3.
{z4 ez
/ (ez –
1)2}dz, where z
= hn/(kT)
At very low temperatures, this relation gives a significantly better fit to the experimental
heat capacity data compared to the Einstein model.
Now for high enough temperatures, T
>> qD
or qD/
T << 1 so that any z
<< 1. This gives
ez
~ 1 and (ez –
1) ~ z (similar to above
section),
implying that
z4
ez
/ (ez –
1)2 ~ z2.
Thus
{z4ez
/(ez –1)2}
dz equals z2
dz
= (qD/
T)3/3, meaning that Cv
= 3Nk, consistent with the Dulong and Petit law, as happened in the case of
Einstein theory. On the other hand, at very low temperatures (so that T is about
0.1qD
or lower), the upper limit
(qD/
T) of the above integral may be replaced with
infinity. So the above integral then equals
o∫a
{z4 ez
/(ez –
1)2}
dz which is 4p4/15
(from knowledge of a standard integral in mathematics). Thus Cv
equals (12p4/5).Nk. (T/qD)3.
This leads to the Debye's T3 law, which states that at very low
temperatures (i.e., T ≤ 0.1qD),
molar heat capacity at constant volume (Cv,mol)
of an elemental solid is proportional to the 3rd power of its absolute
temperature T. Obviously, we find here that the proportionality constant is (12p4/5).(R/qD3),
i.e., Cv,mol
~ (12p4/5).(R/qD3).T3
Problem: Calculate the proportionality constant in Debye's T3
law for lead. Ans: 0.002852 J K–4
mol–1