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 Boltzmann factor tells us that if a system has states with energies E1, E2, E3, …., the probability pJ that the system will be in the state with energy EJ 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 Boltzmann 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/kT.
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 EJ is proportional to J(J+1) where J is the rotational quantum number. The population of successive levels is given by e-E
J/kT.
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 kT. For one mole this is RT.
Thermal energy can populate the quantized levels.
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 (H
Y = EY) can not only determine electronic state energies, but also the quantization of molecular motions.Solutions of the Schrödinger Equation
|
Motion |
# |
Solution |
Set up of problem |
|
Vibration |
v |
( v +1/2)hn |
Harmonic oscillator |
|
Rotation |
J |
{h2/8 p2I}J(J+1) |
Rigid rotator |
|
Translation |
n |
{h2/8ma2} n2 |
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 |
{h2/8 p2I}J(J+1) |
< 1 |
|
Translation |
n |
{h2/8ma2} n2 |
10-11 |
The Boltzmann distribution tells us what the population of the levels will be at a given temperature.
The diagram below 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 level of thermal energy is shown by the red line. Although this is a qualitative picture, it shows that there will be many translation levels populated at room temperature, there will be tens or hundreds of rotational levels occupied and from one to ten vibrational levels.