Partition Functions for Independent ParticlesIndependent 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 ParticlesModel System

• Consider a system with energy levels Ej

• The system consists of two independent subsystems with energy levels i and m

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

A Bj i mE

1

Ai

akT

Ai

q e

1

Bm

bkT

Bm

q e

and

Partition Functions for Independent ParticlesDistinguishable 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

1 1 1 1 1

A B A Bi mj i m

t a b a bkTE kT kT kT

i i m i m

Q e e e e

1 1

A Bi m

a bkT kT

A Bi m

Q e e q q

NQ q

Partition Functions for Independent ParticlesIndistinguishable 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

!

NqQ

N

j i mE

1 2

1 1 1

j i m

t ttE kT kT

i i m

Q e e

Ideal GasesMolecular Definition

• For an ideal gas there are no intermolecular interactions, i.e., = 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

Ideal GasesMolecular 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 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

h

Rotational (l) and vibrational () energy levels for HBr

Ideal Monatomic GasMonatomic 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

Energy Levels

Quantum Mechanics• The basis for predicting quantum mechanical energy levels is the

Schrödinger equation

• The wavefunction (x,y,z) is a function of spatial position• The square of this function, 2, 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

jE H

2

2p

xm

H V

Energy LevelsQuantum 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 (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

2 22

2 24h d

pdx

22

2 28j

d xhx x E x

m dx

V

Translational MotionThe 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

22

2 0d

Kdx

22

2

8π mK

hwith

sin cosx A Kx B Kx

Translational MotionThe 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 n and energy n for a given quantum number n

0 0 0L and

1 2

2sinn

n xx

L L

2 2

2 1,2,3,8n

n hn

mL

Ideal Monatomic GasTranslational 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

2

28t

hmL k

Example: For argon in a 1 cm3 box,

166.0 10 Kt

Ideal Monatomic GasTranslational 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 , the result is

with

Ideal Monatomic GasElectronic Partition Function

• The electronic partition function is usually written as a sum over energy levels rather than a sum over states

• Where ei and i are the degeneracy and energy of the ith electronic level respectively

• Generally, the first electronic level is set as the ground state (1 = 0) and the remaining energy levels are expressed in terms of their spacing relative to that of the ground state

• Where 1j 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 GasElectronic Partition Function

• Here are the electronic energies for a number of atoms

Ideal Monatomic GasExample

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 GasNuclear 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

Ideal Diatomic GasOverview

• 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

Ideal Diatomic GasEnergy 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

re → bond length

→ reduced mass

v → frequency

k → force constant

→ wave number → rotational const.

Ideal Diatomic GasGround 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

Ideal Diatomic GasVibrational 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

Ideal Diatomic GasVibrational Partition Function

• Performing the summation gives the following exact solution

• The vibrational temperature, v ≡ hv/k, is often used instead of the vibrational frequencies

Ideal Diatomic GasVibrational 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

Ideal Diatomic GasRotational 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

Ideal Diatomic GasRotational Partition Function

• The integral evaluates to

• 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

• As T approaches r the approximate solution is no longer valid and the actual summation must be performed

Ideal Diatomic GasRotational Partition Function

• The approximate solution gives the following values for the energy and heat capacity

Ideal Diatomic GasMolecular Constants

• Here are the molecular constants for several diatomic molecules

Ideal Diatomic GasSymmetry Number

• The symmetry number () 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

Ideal Diatomic GasSymmetry Number

• Here are the symmetry numbers for a number of common molecules

Molecule Formula Hydrogen Chloride HCl 1

Nitrogen N2 2

Carbonyl Sulfide COS 1

Carbon Dioxide CO2 2

Water H2O 2

Ammonia NH3 3

Methane CH4 12

Benzene C6H6 12

Ideal Polyatomic GasOverview

• 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

Ideal Polyatomic GasTranslational 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

Ideal Polyatomic GasVibrational Partition Function

• The number of vibrational degrees of freedom 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 independent harmonic oscillators

• The total vibrational energy is given by

Ideal Polyatomic GasVibrational 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

Ideal Polyatomic GasVibrational Partition Function

• Here’s a look at the vibrational modes for two common molecules

symmetricstretch

asymmetricstretch

bending mode(doubly degenerate)

Ideal Polyatomic GasRotational 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

Ideal Polyatomic GasRotational 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 A, B, and C

Ideal Polyatomic GasRotational 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

Ideal Polyatomic GasRotational 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

• Simple relationships result for the internal energy and heat capacity

Ideal Polyatomic GasMolecular Constants

• Here are the molecular constants for several polyatomic molecules

• For a polyatomic molecule Do and De are related by

Ideal Polyatomic GasHindered Rotation

• Consider the rotation about the carbon-carbon bond in ethane• The potential energy as a function of the angle 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 GasHindered 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