1. Partition Functions for 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)

2. 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
Where, i = 1,2,…,a and m = 1,2,…,b
The partition function of each subsystem is
and

3. Partition Functions for Independent Particles
Distinguishable Particles
The partition function for the entire system is
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
More generally, when we have N independent and distinguishable particles the partition function simplifies to

4. Partition Functions for Independent Particles
Indistinguishable Particles
There are no labels A or B the particles from each other
The system energy is
Where, i = 1,2,…,t1 and m = 1,2,…,t2
The system partition function is
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

5. 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
The internal energies include rotational, vibrational, electronic, and nuclear energies

6. 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

7. Rotational (l) and vibrational (n) energy levels for HBr
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
This change is the difference from one energy level to another on an energy ladder
Rotational (l) and vibrational (n) energy levels for HBr

8. 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
Each factor is treated independently – we now look at the various contributions

9. Energy Levels Quantum Mechanics
The basis for predicting quantum mechanical energy levels is the Schrödinger equation
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

10. 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
Schrödinger’s equation is now
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

11. 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
The solution to this differential equation is
with

12. Translational Motion The Particle-in-a-Box Model
The boundary conditions are
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
and

13. 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
The translational molecular partition function is now given by
Example: For argon in a 1 cm3 box,

14. 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
which evaluates to
Using the de Broglie wavelength L, the result is
with

15. 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
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
Where De1j 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

16. Ideal Monatomic Gas Electronic Partition Function
Here are the electronic energies for a number of atoms

17. Ideal Monatomic Gas Example
What fraction of helium atoms are in their first excited state at T = 300 K and T = 3000 K
What fraction of fluorine atoms are in their first excited state at T = 200, 400, and 1000 K

18. 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
With this simplification the nuclear partition function only contributes to entropies and free energies

19. 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

20. 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
Harmonic Oscillator
I → moment of inertia
v → frequency
re → bond length
k → force constant
m → reduced mass
→ wave number
→ rotational const.

21. 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
For the vibrational energy, we have 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

22. 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
The partition function is given by a sum over all energy levels
This summation can be evaluated analytically

23. Ideal Diatomic Gas Vibrational Partition Function
Performing the summation gives the following exact solution
The vibrational temperature, Qv ≡ hv/k, is often used instead of the vibrational frequencies

24. Ideal Diatomic Gas Vibrational Partition Function
Thermodynamic properties are found in the usual way
Here is a plot of the vibrational contribution to the heat capacity as a function of temperature

25. Ideal Diatomic Gas Rotational Partition Function
The rotational energy levels are
The partition function is given by a sum over all energy levels
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
Where we have introduced the rotational temperature

26. Ideal Diatomic Gas Rotational Partition Function
The integral evaluates to
This approximate solution is valid for temperatures that far exceed the characteristic temperature for rotation, i.e., T >> Qr
Replacing the rotational temperature with the moment of inertia gives
As T approaches Qr the approximate solution is no longer valid and the actual summation must be performed

27. Ideal Diatomic Gas Rotational Partition Function
The approximate solution gives the following values for the energy and heat capacity

28. Ideal Diatomic Gas Molecular Constants
Here are the molecular constants for several diatomic molecules

29. 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

30. Ideal Diatomic Gas Symmetry Number
Here are the symmetry numbers for a number of common molecules
Molecule
Formula s
Hydrogen Chloride
HCl 1
Nitrogen N2 2
Carbonyl Sulfide
COS
Carbon Dioxide
CO2
Water
H2O
Ammonia
NH3 3
Methane
CH4 12
Benzene
C6H6

31. 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

32. Ideal Polyatomic Gas Translational Partition Function
Once again, the translational partition function is expressed in terms of the de Broglie wavelength
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
with

33. 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
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

34. Ideal Polyatomic Gas Vibrational Partition Function
Each of the vibrational modes can be treated independently and the vibrational partition function becomes
It follows that the internal energy and heat capacity are given as

35. 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)

36. Ideal Polyatomic Gas Rotational Partition Function (Linear Molecules)
For linear molecules, the quantum states are the same as for a diatomic molecule
Using the classical approximation the partition function is given by

37. 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
The principal moments of inertia are usually denoted as follows
The three principal moments of inertia are used to define the corresponding rotational temperatures QA, QB, and QC

38. 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
Symmetric Top: IA = IB ≠ IC
Asymmetric Top: IA ≠ IB ≠ IC
A closed form solution does not exist

39. Ideal Polyatomic Gas
Rotational Partition Function (Nonlinear Molecules)
If T is close to QA, QB, or QC, 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 >> Q a general solution for the rotational partition function can be obtained
Simple relationships result for the internal energy and heat capacity

40. Ideal Polyatomic Gas Molecular Constants
Here are the molecular constants for several polyatomic molecules
For a polyatomic molecule Do and De are related by

41. 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

42. 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