The Molecular Partition Function

The Boltzmann distribution can be written as

pi = exp(-i) / q

where pi is the probability of a molecule being found in a state iwith energy i. q is called the molecular partition function,

q = i exp(-i)

The summation is over all possible states (not the energy levels).

Molecular Partition Function

Independent Molecules

Consider a system which is composed of N identical molecules.We may generalize the molecular partition function q to thepartition function of the system Q

Q = i exp(-Ei)

where Ei is the energy of a state i of the system, and summationis over all the states. Ei can be expressed as assuming there isno interaction among molecules,

Ei = i(1) + i(2) +i(3) + … + i(N)

where i(j) is the energy of molecule j in a molecular state i

The partition function Q

Q = i exp[-i(1) - i(2) - i(3) - … -i(N)] = {i exp[-i(1)]}{i exp[-i(2)]} … {i exp[-i(N)]} = {i exp(-i)}

N

= qN

where q i exp(-i) is the molecular partition function. Thesecond equality is satisfied because the molecules areindependent of each other. The above equation applies only tomolecules that are distinguishable, for instance, localizedmolecules. However, if the molecules are identical and free tomove through space, we cannot distinguish them, and theabove equation is to be modified!

Translational Partition Function of a molecule qT

Although usually a molecule moves in a three-dimensionalspace, we consider first one-dimensional case. Imagine amolecule of mass m. It is free to move along the x directionbetween x = 0 and x = X, but confined in the y- and z-direction.We are to calculate its partition function qx.

The energy levels are given by the following expression,En = n2h2 / (8mX2) n = 1, 2, …

Setting the lowest energy to zero, the relative energies can thenbe expressed as,

n = (n2-1) with = h2 / (8mX2)

qx = n exp [ -(n2-1) ] is very small, then

qx = 1 dn exp [ -(n2-1) ] = 1 dn exp [ -(n2-1) ] = 0 dn exp [ -n2 ] = (2m/h22)1/2 X

Now consider a molecule of mass m free to move in a container of volume V=XYZ. Its partition function qT may be expressed as qT = qx qy qz

= (2m/h22)1/2 X (2m/h22)1/2 Y (2m/h22)1/2 Z = (2m/h22)3/2 XYZ = (2m/h22)3/2 V = V/3 where, = h(/2m)1/2, the thermal wavelength. The thermal wavelength is small compared with the linear dimension of the container. Noted that qT as T . qT 2 x 1028 for an O2 in a vessel of volume 100 cm3, = 71 x 10-12 m @ T=300 K

Partition function of a perfect gas,

Q = (qT) N / N! = V N / [3N N!]

EnergyE = - (lnQ/)V = 3/2 nRT

where n is the number of moles, and R is the gas constant

Heat CapacityCv = (E/T)V = 3/2 nR

Diatomic Gas

Consider a diatomic gas with N identical molecules. A molecule is made of two atoms A and B. A and B may be the same or different. When A and B are he same, the molecule is a homonuclear diatomic molecule; when A and B are different, the molecule is a heteronuclear diatomic molecule. The mass of a diatomic molecule is M. These molecules are indistinguishable. Thus, the partition function of the gas Q may be expressed in terms of the molecular partition function q,

The molecular partition q

where, i is the energy of a molecular state I, β=1/kT, and ì is the

summation over all the molecular states.

!/ NqQ N

i

iq )exp(

Factorization of Molecular Partition Function

The energy of a molecule j is the sum of contributions from its different modes of motion:

where T denotes translation, R rotation, V vibration, and E the electronic contribution. Translation is decoupled from other modes. The separation of the electronic and vibrational motions is justified by different time scales of electronic and atomic dynamics. The separation of the vibrational and rotational modes is valid to the extent that the molecule can be treated as a rigid rotor.

)()()()()( jjjjj EVRT

i

Ei

Vi

Ri

Tii iq )](exp[)exp(

i i i i

Ei

Vi

Ri

Ti )]exp()][exp()][exp()][exp([ EVRT qqqq

The translational partition function of a molecule

ì sums over all the translational states of a molecule.

The rotational partition function of a molecule

ì sums over all the rotational states of a molecule.

The vibrational partition function of a molecule

ì sums over all the vibrational states of a molecule.

The electronic partition function of a molecule

ì sums over all the electronic states of a molecule.

i

Ti

Tq )exp(

i

Ri

Rq )exp(

i

Vi

Vq )exp(

i

Ei

Eq )exp(

RVT qqqq 1/ Eqw

3/ VqT2/1)2/( Mh

kT/1where

Vibrational Partition Function

Two atoms vibrate along an axis connecting the two atoms. The vibrational energy levels:

If we set the ground state energy to zero or measure energy from the ground state energy level, the relative energy levels can be expressed as

5--------------5hv4--------------4hv3--------------3hv2--------------2hv1--------------hv0--------------0

hvnVn )2/1(

nhvVn

n= 0, 1, 2, …….

kT

hv

Then the molecular partition function can be evaluated

Therefore,

nn n

v hvhvnq )]exp(1/[1)exp()exp(

...1 32 eeeqv

1....32 vv qeeeqe

hvv

eeq

1

1)1/(1

Consider the high temperature situation where kT >>hv, i.e.,

hvkThvhv v //1q ,1

Vibrational temperature v

High temperature means that T>>v

hvk v

he hv 1

F2 HCl H2

v/K 309 1280 4300 6330v/cm-1 215 892 2990 4400m

kvwhere

Rotational Partition Function

If we may treat a heteronulcear diatomic molecule as a rigid rod, besides its vibration the two atoms rotates. The rotational energy

where B is the rotational constant. J =0, 1, 2, 3,…

where gJ is the degeneracy of rotational energy level εJR

Usually hcB is much less than kT,

=kT/hcB

)1( JhcBJRJ

states rotational all

]exp[ RJ

Rq

levelsenergy rotational all

]exp[ RJJg

J

)]1(exp[)12( JhcBJJ

0

)]1(exp[)12( dJJhcBJJqR

dJ /)]}1({exp[)/1(0

dJJhcBJdhcB

0]}1(){exp[/1( lJhcBJhcB

h/8cI2

c: speed of light I: moment of Inertia

i

i rmI i

2

hcB<<1

Note: kT>>hcB

For a homonuclear diatomic molecule

Generally, the rotational contribution to the molecular partition function,

Where is the symmetry number.

Rotational temperature R

hcBkTqR 2/

hcBkTqR /

12

CH

3

NH

2432OH

hcBk R

Electronic Partition Function

=g0

=gE

where, gE = g0 is the degeneracy of the electronic ground state, and the

ground state energy 0E is set to zero.

If there is only one electronic ground state qE = 1, the partition function of a diatomic gas,

]exp[]exp[states electronic all energies electronic all

Ejj

Ej

E gq

]exp[ 00Eg

NhvNN ehcBkTVNQ )1()/()/)(!/1( 3

At room temperature, the molecule is always in its ground state

Mean Energy and Heat Capacity

The internal energy of a diatomic gas (with N molecules)

(T>>1)

Contribution of a molecular to the total energyTranslational contribution(1/2)kT x 3 = (3/2)kTRotational contribution(1/2)kT x 2 = kTVibrational contribution(1/2)kT + (1/2)kT = kT kinetic potentialthe total contribution is (7/2)kT

venNNnNUU hvvv ]/)1(1[)/ln()/1(3)0(

)1/(/1/1)2/3( hveNhvNN )1/()2/5( hveNhvNkT

kTN )2/7(

qV = kT/hvqR = kT/hcB

The rule: at high temperature, the contribution of one degree of freedom to the kinetic energy of a molecule (1/2)kT

the constant-volume heat capacity

(T>>1)Contribution of a molecular to the heat capacityTranslational contribution

(1/2) k x 3 = (3/2) kRotational contribution

(1/2) k x 3 = kVibrational contribution

(1/2) k + (1/2) k = kkinetic potential

Thus, the total contribution of a molecule to the heat capacity is (7/2) k

vv TUC )/( 22 )1/(hv)(K N k )2/5( hvhv eeN

k )2/7( N

Partition Function

q = i exp(-i) = j gjexp(-j) Q = i exp(-Ei)

Energy E= N i pi i = U - U(0) = - (lnQ/)V

Entropy S = k lnW = - Nk i pi ln pi = k lnQ + E / T

A= A(0) - kT lnQ

H = H(0) - (lnQ/)V + kTV (lnQ/V)T

Q = qN or (1/N!)qN

EVRT qqqqq