1

The Heat Capacity of a Diatomic Gas

5.1 INTRODUCTION

We have seen how statistical thermodynamics provides deep insight into the classical description of a monatomic ideal gas. We might have reason to hope, therefore, that the statistical model can resolve a thorny problem we encountered in the application of classical thermodynamics to a diatomic gas. The principle of equipartition of energy fails to give the observed value of the specific heat capacity. The explanation of this discrepancy was considered by Maxwell to be the most important challenge facing the statistical theory. In this chapter we shall see how the problem is solved.

5.2 THE QUANTIZED LINEAR OSCILLATOR

Until now we have confined our attention to systems of particles that have translational degrees of freedom only. To deal with particles such as diatomic molecules, we need to investigate so-called internal degrees of freedom such as vibrations, rotations, and electronic excitations. We consider an assembly of

N one-dimensional harmonic oscillators. We assume that the oscillators are loosely coupled in that the energy exchange among them is small. This means that each particle can oscillate nearly independently of the others. We further assume that each oscillator is free to vibrate in one dimension only with some natural frequency

" . From quantum mechanics, the single particle energy levels are given by

" j = j +1

2( )h#, j = 0,1,2,... (5.1)

Note that the energies are equally spaced and that the ground state has zero-point energy equal to

h" 2 . The states are nondegenerate in that

g j =1 for all

j . For the internal degrees of freedom, Boltzmann statistics applies. The assumption may seem questionable since Boltzmann statistics characterizes distinguishable particles (localized in a crystal lattice, for example) and would therefore appear to be inappropriate for the treatment of an assembly of diatomic molecules. However, in the dilute gas approximation, the number

2

of translational quantum states is so much larger than the number of particles (

g j >> N j ) that the great majority of states are unoccupied, a few are occupied by a single particle, and virtually none have a population greater than one. Thus, in our treatment of the internal degrees of freedom of diatomic molecules, we can regard the particles as differentiated by the translational quantum states that they occupy. We begin by evaluating the partition function

Z = g je"# j kT

j= 0

$

% = e" j+1

2( )h& kT

j= 0

$

% (5.2)

The temperature at which

kT = h" is called the characteristic temperature

" :

" =h#

k

Using this in Equation (5.2), we have

Z = e" j+ 1

2( )# T

j= 0

$

% = e"# 2T e" j# Tj= 0

$

% = e"# 2T 1+ e"# T + e"2# T + ...( )

The sum in this expression is just an infinite geometric series of the form

1+ y + y2

+ ...=1

1" y.

Then

Z =e"# 2T

1" e"# T

(5.3)

We shall also be concerned with the occupation numbers, or with

N j N , the fraction of the total number of particles with energy

" j . The Boltzmann distribution for

g j =1 is

N j

N=e"# j kT

Z= e

"# j kTe$ 2T

1" e"$ T( ) = 1" e"$ T( )e"# j kT +$ 2T . (5.4)

The exponent of the term outside the parentheses can be written

3

"# j

kT+$

2T= " j + 1

2( )h%

kT+h%

2kT= " j

h%

kT= " j

$

T (5.5)

Thus Equation (5.4) becomes

N j

N= e" j# T 1" e"# T( ) (5.6)

A sketch of Equation (5.6) for two temperatures shows that the lower the temperature, the more rapidly the occupation numbers decrease with

j (Figure 5.1). At higher temperatures, more particles populate the higher energy levels.

Figure 5.1 Fractional occupation numbers for quantized linear oscillators with (a)

T =" 2 , and (b)

T = 2" .

Next we compute the internal energy of the assembly of oscillators. The expression we obtained for

U in terms of the derivative of

lnw (Equation (3.7)) applies here because the Boltzmann distribution is the same as the Maxwell-Boltzmann distribution. We have

U = NkT2 "lnZ

"T

#

$ %

&

' ( V

(5.7)

From eqn (5.3)

lnZ = "#

2T" ln 1" e

"# T( )

4

"

"TlnZ =

#

2T2$e$# T % $# T 2( )1$ e$# T( )

=#

T2

1

2+

e$# T

1$ e$# T& ' (

) * +

so

U = Nk"1

2+

1

e" T #1

$

% &

'

( ) (5.8)

LIMIT:

T" 0, U = Nk# 2 = Nh$ 2 , the zero-point energy. LIMIT: As

T"# such that

T " >>1 or

" T

5

CV

= Nk"#

#Te" T

$1( )$1 ,

or

CV

= Nk"

T

#

$ %

&

' (

2

e" T

e" T )1( )

2 (5.9)

LIMIT: As

T"# , at very high temperatures

T " >>1 or

" T >1, we have

e" T

>>1 and

e" T

e" T#1( )

2$

1

e" T

= e#" T so that

CV" Nk

#

T

$

% &

'

( )

2

e*# T

The rate at which the exponential factor approaches zero as

T" 0 is greater than the rate of growth of

" T( )2 that

CV" 0 as T" 0,

consistent with the third law. The variation of

CV

with

T is shown schematically in Figure 5.3. At an extremely low temperature

T

6

At the opposite extreme of temperature for which

T >>" or

kT >> h" , we reach a classical limit. Here the expressions for

U and

CV

do not involve

h" . Plancks constant, the scale of the energy level separation, is irrelevant. Oscillators of any frequency have the same average energy

kT . The dependence of

U on

kT is associated with the classical law of energy equipartition: each degree of freedom contributes

NkT 2 to the internal energy. But the question left unanswered by the classical law is: when is a degree of freedom excited and when is it frozen out? The total internal energy of a diatomic molecule is made up of four contributions that can be treated separately:

(1) the kinetic energy associated with the translational motion of the center of the mass of the molecule (this is

3

2kT , the same as that for a monatomic molecule);

(2) the rotational energy due to the rotation of the two atoms about the center of mass of the molecule;

(3) the vibrational motion of the two atoms along the axis joining them; and (4) the energy of excitation of the atomic electrons.

The last three are internal modes of possessing energy. Because the four contributions can vary independently, it follows that the partition function is a product of the corresponding factors. This can be seen by noting that the internal energy is a function of the logarithm of

Z . Having considered the vibrational motion in Section 5.3, we turn our attention to the rotational contribution.

5.4 ROTATIONAL MODES OF DIATOMIC MOLECULES

Figure 5.3 Variation with temperature of the heat capacity of an assembly

of linear oscillators.

7

The rotation of a diatomic molecule is modeled as the motion of a quantum mechanical rigid rotator. The rotation takes place about an axis through the center of mass of the molecule and perpendicular to the line joining the two atoms. We take

I as the moment of inertia of the molecule about this axis;

I = r0

2 where

= m1m2(m

1+ m

2) is the reduced mass of the two

atoms and

r0 is the equilibrium value of the distance between the nuclei.

Quantum mechanics states that the allowed values of the square of the angular momentum are

l(l +1)h2 , where

h is Plancks constant divided by

2" and

l = 0,1,2,3,.... Recall from classical mechanics that the rotational energy is

1

2I" 2 , where

" is the angular velocity. The angular momentum

L is

I" so the energy is

L22I . The quantized energy levels are therefore

"l= l(l +1)

h2

2I (5.10)

We define a characteristic temperature for rotation:

"rot

=h2

2Ik (5.11)

so that

"l

= l(l +1)k#rot

(5.12)

Experimental values of

"rot

are given in Table 5.1 for several gases; they are found from infrared spectroscopy, in which the energies required to excite the molecules to higher rotational states are measured. The energy levels of equation (5.12) are degenerate; quantum mechanics gives

gl = (2l +1) states for level

l corresponding to different possible directions of the angular momentum vector. Given these results, we can write down the partition function for rotation:

Z = gle"# l kT

l

$ = (2l +1)e" l( l+1)% rot Tl

$ (5.13)

8

The important quantity in this expression is the argument of the exponential. For

T > "rot

. (5.16)

Thus, at significantly high temperatures, the rigid rotator exhibits the equipartition of energy between two degrees of freedom. At very low temperatures such that

"rot

>> T , the partition function, Equation (5.13), can be expanded as

Z = (2l +1)e" l( l+1)#

rotT

l= 0

$

% =1+ 3e"2# rot T + 5e"6# rot T + ....

9

Retaining only the first two terms, we can write

lnZ " ln 1+ 3e#2$ rot T( )

Since the exponential term is small compared with unity, we can use the approximate relation

ln(1+ ") # " for

"

10

5.6 THE TOTAL HEAT CAPACITY

We are finally in a position to account fully for the behavior of

CV

as a function of

T for a diatomic gas at temperatures in the usual range of interest. Assuming that electronic excitations are negligible, we have

CV

= CV ,tr

+ CV ,rot

+ CV ,vib

Thus, at ordinary temperatures,

CV

=3

2Nk + Nk + Nk

"vib

T

#

$ %

&

' ( 2

e" vib T

e" vib T )1( )

2= Nk

5

2+

"vib

T

#

$ %

&

' ( 2

e" vib T

e" vib T )1( )

2

*

+

, ,

-

.

/ / (5.21)

Table 5.1 gives values for

"rot

and

"vib

for several diatomic gases. At the lowest temperatures at which the system exists in the gaseous phase, the molecular motion is solely translational, contributing

3Nk 2 = 3nR 2 to the heat capacity. As the temperature increases and approaches

"rot

, rotational quantum states are excited. Eventually, for

T >>"rot

, the rotational contribution becomes equal to

nR, corresponding to two rotational degrees of freedom. Therefore, the heat capacity steps up to

5

3nR in the

region

"rot

11

TABLE 5.1 Characteristic temperatures of rotation and vibration of diatomic molecules. Substance

!

"rot

(K)

!

"vib

(K)

!

H2 85.4 6140

!

O2 2.1 2239

!

N2 2.9 3352

!

HCl 4.2 440

!

CO 2.8 3080

!

NO 2.4 2690

!

Cl2 0.36 810

At elevated temperatures, the higher vibrational states are excited and the heat capacity exhibits two additional degrees of freedom, rising asymptotically to a value of

!

7nR 2 for

!

T >>"vib

. The characteristic temperature of vibration depends on the bond strength between the two atoms of the molecule and on their masses. In some cases, the diatomic molecules disassociate as the vibrational energy overcomes the bonding energy.

Experimental values of the heat capacity at constant volume are close to predicted values over a wide range of temperatures. Values of

!

CVnR are shown for

hydrogen on a logarithmic scale in Fig 5.5. An understanding of the temperature variation of the heat capacity of diatomic gases is surely an outstanding triumph of quantum statistical theory. The exact theory is

apparently so firmly established that heat capacities of gases can be computed theoretically from optical measurements, more accurately than they can be measured experimentally (using calorimetry).

Figure 5.5 Values of

!

CVnR for hydrogen (

!

H2) as a function of

temperature. The temperature scale is logarithmic.