Partition Functions for Independent Particles
Independent Particles
• We now consider the partition function for independent particles,
i.e., particles that do not in any way interact or associate with other
molecules
• We consider two cases: distinguishable and indistinguishable
Distinguishable Particles
• Particles that can be differentiated from each other
• The particles could be in some way labeled (e.g. red vs. blue) or
kept at a fixed position (e.g. particles in a crystal lattice)
Indistinguishable Particles
• Particles that cannot be differentiated from each other
• These particles can interchange locations, so you cannot tell which
particle is which (e.g. gas particles)
Partition Functions for Independent Particles
Model System
• Consider a system with energy levels Ej
• The system consists of two independent subsystems with energy
levels ei and em
Distinguishable Particles
• Label the two systems A and B
• The system energy is
E j  e iA  e mB
• Where, i = 1,2,…,a and m = 1,2,…,b
• The partition function of each subsystem is
a
qA   e
i 1
e iA
b
kT
and
qB   e
m1
e mB kT
Partition Functions for Independent Particles
Distinguishable Particles
• The partition function for the entire system is
t
Q  e
i 1
 E j kT
a
b
  e

 e iA e mB
 kT
i 1 m1
a
b
  e
e iA kT e mB kT
e
i 1 m1
• Because the subsystems are independent and distinguishable by
their labels, the sum over i levels of A has nothing to do with the
sum over m levels of B
• Therefore, the partition function can be factored
a
Q  e
i 1
e iA
b
kT
e
e mB kT
m1
 q A qB
• More generally, when we have N independent and distinguishable
particles the partition function simplifies to
Q  qN
Partition Functions for Independent Particles
Indistinguishable Particles
• There are no labels A or B the particles from each other
• The system energy is
E j  ei  e m
• Where, i = 1,2,…,t1 and m = 1,2,…,t2
• The system partition function is
t
Q  e
i 1
 E j kT
t1
t2
  e e i e m  kT
i 1 m1
• The summation can no longer be separated
• As a result of performing the full summation, you overcount by a
factor of 2!
• Therefore, for N indistinguishable particles, the partition function
evaluates to
qN
Q
N!
Ideal Gases
Molecular Definition
• For an ideal gas there are no intermolecular interactions, i.e., F = 0
• The molecules are unaware of each other’s existence and behave
independently – the state molecule j is in is completely unaffected
by the state of the other molecules in the gas
Molecular Energies
• We write the total energy of each molecule as the sum of the
translational (kinetic) and internal energies
e  e trans  eint
• The internal energies include rotational, vibrational, electronic, and
nuclear energies
eint  e rot  e vib  e elect  e nucl
Ideal Gases
Molecular Energies
• The schematic below gives an indication of the relative spacing
between energy levels for the various energies
electronic
vibrational
rotational
translational
Energy Levels
Spectroscopy
• Spectroscopy measures the
frequency n of
electromagnetic radiation
that is absorbed by an atom,
a molecule, or a material
• Adsorption of radiation by
matter leads to an increase
in its energy by an amount
e  hn
• This change is the difference
from one energy level to
another on an energy ladder
Rotational (l) and vibrational
(n) energy levels for HBr
Ideal Monatomic Gas
Monatomic Gas
• A monatomic gas has translational, electronic, and nuclear degrees
of freedom
• The translational Hamiltonian is separable from the electronic and
nuclear degrees of freedom
• The electronic and nuclear Hamiltonians are separable to a very
good approximation
• The molecular partition function can be written as
qV , T   qtrans qelect qnucl
• Each factor is treated independently – we now look at the various
contributions
Energy Levels
Quantum Mechanics
• The basis for predicting quantum mechanical energy levels is the
Schrödinger equation
Hy  E jy
• The wavefunction y(x,y,z) is a function of spatial position
• The square of this function, y2, is the spatial probability
distribution of the particles for the problem of interest
• The Hamiltonian operator H describes the forces relevant for the
problem of interest
• In classical mechanics the Hamiltonian is given by the sum of
kinetic and potential V energies
• For example, for the one-dimensional translational motion of a
particle having mass m and momentum p
p2
H 
 V  x
2m
Energy Levels
Quantum Mechanics
• While classical mechanics regards p and V as functions of time
and spatial position, quantum mechanics regards p and V as
mathematical operators that create the right differential equation
for the problem at hand
• In one dimension, the translational momentum operator is
2
2


h
d
p2    2  2
 4  dx
• Schrödinger’s equation is now
 h 2  d 2y  x 
 2 
 V  x y  x   E jy  x 
2
 8 m  dx
• Only certain functions y(x) satisfy this equation
• The quantities Ej are called eigenvalues and represent the discrete
energy levels that we seek
• The index j for the eigenvalues is called the quantum number
Translational Motion
The Particle-in-a-Box Model
• The particle-in-a-box is a model for the freedom
of a particle to move within a confined space
• A particle is free to move along the x-axis over
the range 0 < x < L
• At the walls (x = 0 and x = L) the potential is
infinite (V(0) = V(L) = ∞)
• Inside the box the molecule has free motion
(V(x) = 0 for 0 < x < L)
• Schrödinger’s equation is
d 2y
2

K
y 0
2
dx
with
2
8π
me
2
K 
h2
• The solution to this differential equation is
y  x   A sin  Kx   B cos  Kx 
Translational Motion
The Particle-in-a-Box Model
• The boundary conditions are
y  0  0
and
• Applying the boundary conditions and
normalizing the probability distribution
gives the following expressions for the
wavefunction yn and energy en for a
given quantum number n
2
n x 
y n  x     sin 

12
 L
n2 h2
en 
8mL2
 L 
n  1, 2,3,
y  L  0
Ideal Monatomic Gas
Translational Partition Function
• Quantum mechanics (particle in a box) tells us that the translational
energy levels for a particle in a three-dimensional box are given by
h2
2
2
2

n

n

n
y
z 
2 x
8mL
en n n
x y z
nx , ny , nz  1, 2,
• The translational molecular partition function is now given by
qtrans 


e
 e nx n y nz
nx ,n y ,nz 1

 e
h 2 nx2 

8 mL2
e
nx 1

n y 1

 n t T 
  e

 n1

2
3

h 2n 2y
8 mL2

e
nz 1

h 2nz2
8 mL2
h2
t 
8mL2 k
Example: For argon
in a 1 cm3 box,
t  6.0 1016 K
Ideal Monatomic Gas
Translational Partition Function
• At most temperatures of practical importance, the energy levels are
narrowly spaced in comparison to kT
• Therefore, we can replace the sum with an integral

qtrans    e
 0
  h2n2 8 mL2
dn 

3
• which evaluates to
 2 mkT 
qtrans V , T   
 V
2
 h

32
• Using the de Broglie wavelength L, the result is
qtrans
V
 3
L
with
12
 h

L

2

mkT


2
Ideal Monatomic Gas
Electronic Partition Function
• The electronic partition function is usually written as a sum over
energy levels rather than a sum over states
qelect  wei e ei
• Where wei and ei are the degeneracy and energy of the ith electronic
level respectively
• Generally, the first electronic level is set as the ground state (e1 = 0)
and the remaining energy levels are expressed in terms of their
spacing relative to that of the ground state
qelect  we1  we 2e e12 
• Where e1j is the energy of the jth electronic level relative to the
ground state
• At ordinary temperatures, usually only the ground state and perhaps
the first excited state need to be considered
Ideal Monatomic Gas
Electronic Partition Function
• Here are the electronic energies for a number of atoms
Ideal Monatomic Gas
Example
1. What fraction of helium atoms are in their first excited
state at T = 300 K and T = 3000 K
2. What fraction of fluorine atoms are in their first excited
state at T = 200, 400, and 1000 K
Ideal Monatomic Gas
Nuclear Partition Function
• The nuclear partition function has a form similar to that of the
electronic partition function
• Nuclear energy levels are separated by millions of electron volts,
which means temperatures on the order of 1010 K need to be
reached before the first excited state becomes populated to any
significant degree
• Therefore, at ordinary temperatures we can simply express the
nuclear partition function in terms of the degeneracy of the ground
state
qnucl  wn1
• With this simplification the nuclear partition function only
contributes to entropies and free energies
Ideal Diatomic Gas
Overview
• To a high degree of accuracy diatomic molecules can be described
using the rigid rotor – harmonic oscillator approximation
• With this approximation, in addition to the translational, electronic,
and nuclear energies, the molecule has two additional energies
• Rotational (rigid rotor): rotary motion about the center of mass
• Vibrational (harmonic oscillator): relative vibratory motion of
the two atoms
• Again, we assume that all energies are independent, which gives the
following expression for the molecular partition function
q  qtrans qrot qvib qelect qnucl
Ideal Diatomic Gas
Energy levels
• A quantum mechanical solution to the rigid rotor – harmonic
oscillator problem gives the following energy levels and
degeneracies for the rotational and vibrational motion
Rigid Rotor
h 2 J  J  1
eJ 
8
I
Harmonic Oscillator
J  0,1, 2,
I   re2
re → bond length
 → reduced mass
n  0,1,2,
wn  1
wJ  2 J  1
I → moment of inertia
e n  hv  n  12 
m1m2

m1  m2
2
B → rotational const. B  h 8 Ic
12
v → frequency
1 k
v
2   
k → force constant
w → wave number
w
v
c
Ideal Diatomic Gas
Ground State
• Before obtaining the partition function,
we need to set the ground state
• We take the rotational ground state
as the J = 0 state
De  Do  12 hn
• For the vibrational energy, we have
 Do  12 k v
two choices
 Take the lowest vibrational state
to be the ground state
 Take the bottom of the interatomic
potential well as zero energy
• We adopt the second choice
• We take the zero of the electronic energy to be the separated,
electronically unexcited atoms at rest
• The electronic partition function is now
qelect  we1e  De  we 2e e 2 
Ideal Diatomic Gas
Vibrational Partition Function
• With our choice of the ground state as the minimum in the
interatomic potential well, the vibrational energies are given as
e n   n  12  hv
n  0,1,2,
• The partition function is given by a sum over all energy levels
qvib T    e  e n
n

 e  hv 2  e   hvn
n 0
• This summation can be evaluated analytically

 nx
e
 
n 0
1
1  e x
Ideal Diatomic Gas
Vibrational Partition Function
• Performing the summation gives the following exact solution
e  hv 2
qvib T  
1  e  hv
• The vibrational temperature, v ≡ hv/k, is often used instead of the
vibrational frequencies
ev 2T
qvib T  
1  e v T
Ideal Diatomic Gas
Vibrational Partition Function
• Thermodynamic properties are found in the usual way
U vib
v 
  ln qvib 
 v
 NkT 
  Nk   v T 
1
 T 
 2 e
2
v T

U

e


 
  vib   k  v 
2
 T 
 T   ev T  1
2
Cv, vib
• Here is a plot of the vibrational contribution to the heat capacity as a
function of temperature
Ideal Diatomic Gas
Rotational Partition Function
• The rotational energy levels are
h 2 J  J  1
eJ 
8
I
wJ  2 J  1
J  0,1, 2,
• The partition function is given by a sum over all energy levels

qrot T     2 J  1 e   BJ  J 1
J 0
• This summation cannot be written in closed form
• At ordinary temperatures the energy levels are usually narrowly
spaced compared to kT and we can approximate the sum by an
integral
qrot T   

0
 2 J  1 e J  J 1 T dJ
r
• Where we have introduced the rotational temperature r  B k
Ideal Diatomic Gas
Rotational Partition Function
• The integral evaluates to
T
qrot T  
r
• This approximate solution is valid for temperatures that far exceed
the characteristic temperature for rotation, i.e., T >> r
• Replacing the rotational temperature with the moment of inertia
gives
8 2 IkT
qrot T  
h2
• As T approaches r the approximate solution is no longer valid and
the actual summation must be performed
Ideal Diatomic Gas
Rotational Partition Function
• The approximate solution gives the following values for the energy
and heat capacity
U rot  NkT
Cv, rot  k
Ideal Diatomic Gas
Molecular Constants
• Here are the molecular constants for several diatomic molecules
Ideal Diatomic Gas
Symmetry Number
• The symmetry number (s) is a classical correction factor introduced
to avoid over counting indistinguishable configurations
• Its origin is quantum mechanical in nature and the approach
presented here is appropriate at ordinary temperatures
• The symmetry number represents the number of different ways a
molecule can be rotated into a configuration indistinguishable from
the original
• By accounting for the symmetry of a molecule, the rotational
partition function becomes
T
qrot T  
s r
Ideal Diatomic Gas
Symmetry Number
• Here are the symmetry numbers for a number of common molecules
Formula
HCl
N2
s
1
2
Carbonyl Sulfide
Carbon Dioxide
Water
Ammonia
COS
CO2
H2O
NH3
1
2
2
3
Methane
Benzene
CH4
C6H6
12
12
Molecule
Hydrogen Chloride
Nitrogen
Ideal Polyatomic Gas
Overview
• We now extend the methods we have developed for diatomic
molecules to polyatomic systems
• We use an analogous approach to that adopted previously for the
translational, electronic, and nuclear energies
• The vibrational partition function is a straightforward extension of
the diatomic case
• The rotational partition function is a bit more complicated, however
a convenient solution exists as long as we can treat the molecule
classically
• A new feature that arises with polyatomic molecules is hindered (or
internal) rotation (e.g., rotation about the carbon-carbon bond in
ethane)
• The molecular partition function is
q  qtrans qrot qint rot qvib qelect qnucl
Ideal Polyatomic Gas
Translational Partition Function
• Once again, the translational partition function is expressed in terms
of the de Broglie wavelength
qtrans
V
 3
L
with
h2
L
2 mkT
• Where m is the molecular mass of the molecule
Electronic Partition Function
• Again, we choose as the zero of energy all n atoms completely
separated in their ground electronic states
• Therefore, the energy of the ground electronic state is –De
• It follows that the electronic partition function is
qelect  we1e  De  we 2e e 2 
Ideal Polyatomic Gas
Vibrational Partition Function
• The number of vibrational degrees of freedom a a molecule
possesses is the difference between its total number of degrees of
freedom (3n) and the sum of the translational (3) and rotational (2
for linear and 3 for nonlinear molecules) degrees of freedom
3n  5
a 
3n  6
linear molecule
nonlinear molecule
• A coordinate transformation can be completed such that a set of
coordinates (normal coordinates) are found in which the
Hamiltonian can be written in terms of a sum of a independent
harmonic oscillators
• The total vibrational energy is given by
a
e    n j  12  hv j
j 1
n j  0,1, 2,
Ideal Polyatomic Gas
Vibrational Partition Function
• Each of the vibrational modes can be treated independently and the
vibrational partition function becomes
a
qvib  
j 1
e
vj 2T
1  e
vj T

• It follows that the internal energy and heat capacity are given as
U vib

vj
vj
 Nk  
 vj T
e
1
j 1  2





Cv, vib
2

vj T



e
 k   vj 
 T
j 1  T 
e vj  1



2


a


a


Ideal Polyatomic Gas
Vibrational Partition Function
• Here’s a look at the vibrational modes for two common molecules
symmetric
stretch
asymmetric
stretch
bending mode
(doubly degenerate)
Ideal Polyatomic Gas
Rotational Partition Function (Linear Molecules)
• For linear molecules, the quantum states are the same as for a
diatomic molecule
h 2 J  J  1
eJ 
8
I
J  0,1, 2,
wJ  2 J  1
• Using the classical approximation the partition function is given by
8 2 IkT
T
qrot 

2
sh
sr
Ideal Polyatomic Gas
Rotational Partition Function (Nonlinear Molecules)
• We describe the geometry of a nonlinear molecule in terms of its
principal moments of inertia
• These values are found with respect to a molecules principal axes,
set such that all products of inertia are zero
I xy  I xz  I yz  0
• The principal moments of inertia are usually denoted as follows
I xx  I A
I yy  I B
I zz  I C
• The three principal moments of inertia are used to define the
corresponding rotational temperatures A, B, and C
Ideal Polyatomic Gas
Rotational Partition Function (Nonlinear Molecules)
• The quantum mechanical energy levels depend on the relative
values of the principal moments of inertia
• Three cases exist:
Spherical Top: IA = IB = IC
h 2 J  J  1
eJ 
8
I
IA = IB ≠ IC
Symmetric Top:
e JK
wJ  2 J  1
J  0,1, 2,
h 2  J  J  1
1 
2 1

 K   

8  I A
 IC I A 
Asymmetric Top:
J  0,1, 2,
K  J , J  1,
, J
IA ≠ I B ≠ IC
A closed form solution does not exist
wJK  2 J  1
Ideal Polyatomic Gas
Rotational Partition Function (Nonlinear Molecules)
• If T is close to A, B, or C, then the actual summations must be
performed to obtain the partition function (the asymmetric top case
requires a numerical solution)
• In the classical limit T >>  a general solution for the rotational
partition function can be obtained
12
12
12
  8 I A kT   8 I B kT   8 I C kT 
qrot 
s  h 2   h 2   h 2 
12
2
2
2
12

  T
qrot 
s   A B C 
12
3
• Simple relationships result for the internal energy and heat capacity
Cv, rot  32 k
U rot  32 NkT
Ideal Polyatomic Gas
Molecular Constants
• Here are the molecular constants for several polyatomic molecules
• For a polyatomic molecule Do and De are related by
a
Do  De   12 hv j
j 1
Ideal Polyatomic Gas
Hindered Rotation
• Consider the rotation about the carbon-carbon bond in ethane
• The potential energy as a function of the angle f is as follows
• The treatment of this problem depends upon the value of Vo relative
to that of kT
• Let’s take a look at three possibilities
Ideal Polyatomic Gas
Hindered Rotation
• kT >> Vo: in this case the internal rotation is essentially free and can
be treated by methods similar to that for the rigid rotor
• kT << Vo: in this case the molecule is trapped at the bottom of the
wells and the motion is that of a simple torsional vibration, which
can be treated by a method similar to that used for the simple
harmonic oscillator
• kT ≈ Vo: in this case neither of the above approximations is valid and
the full quantum mechanical problem must be solved – the solutions
have been tabulated for many common molecules
• Unfortunately, for many molecules of practical significance, the
latter case is found