The document discusses partition functions in the context of statistical mechanics, particularly focusing on the Boltzmann distribution law and molecular partition functions. It explains how partition functions relate microscopic properties of molecules, such as energy states, to macroscopic properties like entropy and heat capacities, along with specific formulations for ideal gases and diatomic molecules. Key characteristics and calculations for the translational, rotational, and vibrational partition functions are also detailed, highlighting dependencies on temperature and volume.

1. Presented by:
Dr. Sharayu M. Thorat
Associate Professor
Shri Shivaji College of Arts, Commerce &
Science, Akola
PARTITION FUNCTIONS

2. According to Boltzmann distribution law, the fraction of molecules which is in the
most probable state at temperature T, having energy εi is given by
Where ni is number of molecules having energy εi at temperature T,
N is total number of molecules
gi is the degeneracy (or statistical weight) of the energy level εi
The denominator of above equation which gives the sum of the terms
for all energy levels is called partition function, represented by q.
Thus partition function q =

3. Partition function indicates how the particles /molecules are
partitioned ( distributed) in various energy levels.
• Introduced by Fowler, it may be defined as the sum of
probability factors or the way in which energy of an
assembly is partitioned among molecules of an assembly.
Molecular partition function is also called “ sum -over-states”
• It’s value depends on molar mass, molar volume,
temperature, intermolecular distances, molecular motion
and the intermolecular forces.

4. Characteristics of Molecular Partition Function ‘q’
Partition function is
a pure number,
hence it is a
dimensional
quantity. It can
never be zero and
cannot have
negative values.
As the temperature
is raised, value of q
becomes very large
since the particles
occupy higher
energy states.
The lowest value of
q is one at absolute
zero i.e. no= N. At
absolute zero all
the particles
occupy lowest
ground energy
state.

5. Characteristics of Molecular Partition Function ‘q’
Partition function
is a measure of the
tendency of the
molecules to
escape from the
ground state.
When T 0,
β=1/kT =∞,hence
q=0 except for first
term with zero
energy, for which
q=q0
q links the
microscopic
properties of
individual
molecules such as
discrete energy
levels,moment of
inertia etc with
macroscopic
properties like
entropy, heat
capacities etc.

6. Molecular Partition Functions for an Ideal gas:
Molecules are associated with energy of different types. All these forms of energy
must be taken into account while mentioning partition function.
The molecular energy levels needed for the evaluation of the molecular partition
functions are obtained from the solution of the Schrodinger equation.
According to Born-Oppenheimer approximation, the total energy of a molecule
is composed of contributions from the translational, rotational, vibrational and
electronic modes of motion.
ε = εtran +εrot + εvib +εelectr
The equation holds good if there is no coupling between the different modes of motion.
Electronic energy ε electr can be obtained for the simple atoms.
q=
The molecular partition function q is given by ………..(1)
The total partition function of a molecule is the product of the translational,
rotational, vibrational and electronic contribution.

7. i.e. q = qtransx qrot x qvib xqelec …………(2)
The Translational Partition Function
For a particle of mass m, moving in an infinite 3-dimensional box of sides a, b and c,
assuming that the potential is zero within the box, the energy levels obtained by the
solution of the Schrodinger equation are given by the expression
εnx ny nz = εtran= ………….(3)
where each of the quantum numbers nx ,ny ,nz vary from one to infinity.
Using equations (1) & (3) the translational partition function ,neglecting degeneracy
is given by
qtran = …………(4)
Where the triple summation is taken over all integral values of nx,ny & nz from
one to infinity. The motion of the particles in the three x, y & z directions being
independent, we can replace the triple summation as a product of three
summations.

8. Thus, qtran = x x ………..(5)
The spacing between the energy levels of a particle in a three dimensional box is very
small compared with the thermal energy ,kt. Hence we can replace the summation
by integration.
Accordingly,
qtran = dnx dny
x x dnz
………..(6)
dx =
From calculus, the standard integral ……….(7)
Using this result, the three integrals in equation 6 which are identical except for
the constants a,b and c can be calculated giving
qtran = a/h(2𝝿mkt)1/2 x b/h(2𝝿mkt)1/2 x c/h(2𝝿mkt)1/2
=
3/2
x abc = x V ………..(8)

9. Equation 8⟹ Translational partition function depends upon volume & temperature
V= volume of the box in which the molecule moves
Equation 8 can be written as qtrans = q0
trans x V ……….(9)
q0
trans = partition function per unit volume
If M is the mass per mole then m = M/NA , NA is Avogadro’s number, k = R/NA
Hence x
qtrans =
NA
3
……….(10)
In the case of a perfect monoatomic gas ,the molar partition function is given by
Z = ……………(11)

10. Partition function for diatomic molecules: Rotational partition function
The partition function for rotational energy of a diatomic molecule is given by
………..(1)
From quantum mechanical principles, the rotational energy 𝟄r for a diatomic
molecule at the Jth quantum level is given by
……….(2) Where I= moment of inertia = 𝞵r2
As axis of rotation is defined by two co-ordinates ,which means that there are two
rotational degrees of freedom. Each quantum level of rotation will bring in two possible
modes of distribution of rotational energy.
Thus statistical weight factor J is given by 2J+1
Hence equation 1 becomes,
fr = Σ( 2J+1)e – {J(J+1)h2}/ 8𝝿2IkT
……….(3)
Since the levels are closely spaced, the summation can be replaced by integration

11. f’r = ∫( 2J+1)e – {J(J+1)h2}/ 8𝝿2IkT dJ
0
∞
∴
∫( 2J+1)e – J(J+1)𝛽 dJ
=
0
∞
……….(4) where 𝛽= h2/8𝝿2IkT
Suppose G= J(J+1), On differentiating it we get, dG=(2J+1) dJ
Hence equation 2 becomes,
fr = ∫e-G𝛽 dG = 1/𝛽 = 8𝝿2IkT/h2
0
∞
..…….(5)
The value of fr is valid for hetero- nuclear molecules like NO, HCl etc whereas in the
case of homonuclear molecules like O2, N2 etc where the molecule when reversed,
becomes indistinguishable from initial state, the partition function is to be divided
by the number of symmetry viz 2.
∴ fr =
In general , when the symmetry number is δ,partition function is
fr = ………..(6)

12. Equation 6 holds good for diatomic molecules, other than hydrogen & deuterium.
fr = For homonuclear diatomic molecules like H2, N2,O2 etc σ =2
For heteronuclear diatomic molecules like CO, NO ,HCl etc σ =2
fr = Where B is rotational constant =

13. Vibrational Partition function
The partition function for vibrational energy of a diatomic molecule is given by
fv = Σgv e-𝜀v/kT
As the statistical weight of each vibrational level is unity, we have,
Σ e-𝜀v/kT
fv =
At the nth quantum level, the vibrational energy of a diatomic molecule is given by,
𝜀v = 〔n+1/2〕 hν Where ν = fundamental frequency of vibration
n = an integer 0,1,2,3……etc
………(1)
……..(2)
∴ from equation 2 we have, fv =
=
= ……..(3)
For a diatomic molecule vibrating as a simple harmonic oscillator, the vibrational
energy levels are obtained by the solution of the Schrödinger's wave equation .

14. =
∵
The quantity hν/ kT is very small & as a first approximation,
fv = ………(4)
The value of ν is equal to cw, where c is velocity of light & wcm-1 is the vibration
frequency in wave numbers of the given oscillator. Hence
fv =
………(5)
Equation 5 may be used for the vibrational partition function of a diatomic molecule
at all temperatures ; the only approximation involved is that the oscillations are
supposed to be harmonic in nature.