The Microscopic View

Properties like the temperature and pressure can be related to microscopic quantities. You have already seen, for example, that temperature can be expressed in terms of a molecular velocity distribution function, the Maxwell-Boltzman probability. Likewise, the pressure can be understood by picturing the miniature collisions of molecules with the walls of a container. Before we consider that applied that connects microscopic momentum transfer to macroscopic pressure, we consider the three major contributors to molecular energy levels in the thermal range. Note that electronic levels do not get as much consideration since the transition energies of optical transitions are large relative to k_BT.

Molecular Energy Levels

There are quantized energy levels for electronic states, vibration, rotation, and translation. For our purposes we are mostly concerned with translational states in thermodynamics. We need a means of calculating the statistical average of the molecular velocity in order to calculate energy, pressure, and other thermodynamic quantities. The Boltzman factor tells us that if a system has states with energies E₁, E₂, E₃, …, the probability p_J that the system will be in the state with energy E_J depends exponentially on the energy of that state.

Molecular Vibration

A diatomic molecule has one vibrational mode.

Polyatomic molecules have 3N – 6 modes where N is the number of atoms. Each vibrational mode of a molecule has a ladder of energy levels. If we have an ensemble of many molecules the Boltzman formula says that the population of the levels is determined statistically such that the population of each higher level decreases by a factor e^-E/kBT.

Molecular Rotation

There are 3 rotational degrees of freedom for polyatomic molecules (and 2 for linear molecules). Rotational motion is quantized also giving a ladder of unequally spaced energy levels. The energy E_J is proportional to J(J+1) where J is the rotational quantum number. The population of successive levels is given by e^-EJ/kBT. It is important to realize that translational motion is much more important than vibrational and rotational motion in statistical thermodynamics. The reason for this is that translational energy levels are very closely spaced (and there are many of them) while the energy spacing of rotational and vibrational levels are much larger.

Energy spacing: Vibrations > Rotations >> Translations

Energy can be measured in terms of thermal energy.

For one molecule this is k_BT. For one mole this is RT.

Thermal energy can populate the quantized levels.

We will derive the quantized energy levels in Chemistry 433.

Quantization of Energy Levels

The constant h (called Planck’s constant) gives the scale for quantized energy levels in atoms and molecules.

h is 6.626 x 10^-34 J-s or 4 x 10^-13 (kJ/mol)-s.

The solutions of the Schrödinger equation (HY = EY) cannot only determine electronic state energies, but also the quantization of molecular motions.

These solutions predict energy levels that have quantum numbers to describe their discrete nature.

Solutions of the Schrödinger Equation

Motion	#	Solution	Set up of problem
Vibration	v	(v +1/2)hn	Harmonic oscillator
Rotation	J	{h^2/8p^2I}J(J+1)	Rigid rotator
Translation	n	{h^2/8ma^2}n^2	Particle in a box

# is the quantum number. The magnitude of the quantum number depends on how much energy is in the system.

n is the frequency O (10¹³ s^-1). (O () means “of the order”)

I is the moment of inertia O (10^-46 kg m²)

a is the length of the box (depends on context).

m is the mass of the particle O (10^-26 kg).

Thermal Population of Energy Levels

At room temperature the thermal energy of one mole is

RT = (8.314 kJ/mole-K)(300 K) = 2.4 kJ/mole.

Motion	#	Formula	kJ/mole
Vibration	v	(v +1/2)hn	1 – 20
Rotation	J	{h^2/8p^2I}J(J+1)	< 1
Translation	n	{h^2/8ma^2}n^2	10^-11

The Boltzman factor gives us a means of calculating how many levels will be occupied.

The spacing of quantized energy levels according to quantum mechanics is NOT the average energy!!!

The diagram shows the ground state (lowest energy level) and higher levels for vibrations, rotations, and translations. The vibrational energy level spacing is larger relative to RT while the rotational level spacing is small. There are so many translational levels that they are not distinguishable.

The Partition Function

The sum of all of the probabilities must equal 1. This is called normalization. The normalization constant of the probability is 1/Q where

Q is called the partition function.

The population of a particular state J with energy E_J is given by

This is known as the Boltzman distribution.

The derivation of the Boltzman distribution relies on the concept of an ensemble of system all in thermal equilibrium. Consider a huge number of systems in thermal contact (an ensemble). Then we ask what is the relative number of systems that will be found in any one state E_J. The proof for the Boltzman distribution is found in McQuarrie and Simon 17-1.