ENGR-1100 Introduction to Engineering Analysis

1
Lecture 9 Energy Levels

• Translations, rotations, harmonic oscillator
• Independence of energy storage
• Many particle systems
• Independent distinguishable particles
• Independent indistinguishable particles

2
Energy levels from quantum mechanics
Need to calculate the partition functions to have
any use of statistical mechanics For a
single particle independent modes of motion will
contribute to energy and thus the partition
function. For independent particles the energy
is the sum of the individual particles energies
All this will provide a rather simple way of
calculating Q for many physical systems
3
Translation
Consider a single diatomic molecule is the
box Translation M - total mass of the
molecule a,b,c - dimensions of the box containing
the molecule h - Planck constant
4
Rotation and vibrations
Rotation I ml2 - moment of inertia, m
mass of one atom, l atom to atom
distance Vibrations - harmonic oscillator ? -
frequency
5
Single gas diatomic molecule particle partition
function
When energy is additive the partition function is
a product of partition function
Therefore free energy is the sum
6
Many independent particle
Total energy is the sum of single particle
energies So the partition function is the
product of single particle functions For N
independent, identical, distinguishable particles
7
Indistinguishable particles
States such as a) first particle in state one
and second particle in state 2 and b) second
particle in state one and first particle in
state 2 are the same total state since
particle are not distinguishable This leads to
over counting of the partition function That
has to be corrected by
8
Bosons and Fermions
1/N! correction is not quite right. If particles
1 and 2 are in the same state there was no over
counting. This is possible for bosons. But
typically number of states is much larger than
number of particles so this situation is very
rare and does not contribute in the thermodynamic
limit to the Q, such as to affect intensive
properties. For fermions two particles can not
be in the same state - we over counted even with
N! correction. Again for classical, not quantum
systems, we are fine.
9
Problem 10.6
N distinguishable particles, each having two
energy levels e11/2kT and e23/2kT. Calculate
energy, entropy and heat capacity at constant
volume. Particles are bosons.