1. Classical Statistical Mechanics: 1 dimensional Simple Harmonic Oscillator

2. Recall the Equipartition Theorem: In Ch 6,
Valid in Classical Stat. Mech. ONLY!!!
“Each degree of freedom in a system of particles contributes (½)kBT to the thermal average energy of the system.”
In Ch 6,
we proved this & discussed it in detail!

3. In the Classical Cannonical
Ensemble it is easy to show that
The average energy of a particle per independent degree of freedom is (½ )kBT.
Proof: See Ch. 6 Lectures!

4. The Boltzmann Distribution
Canonical Probability Function P(E):
This is defined so that P(E) dE  the probability
to find a particular molecule between E and E + dE Z
Define: The Energy Distribution Function
(Number Density) nV(E):
This is defined so that nV(E) dE  the number of
molecules per unit volume with energy between E
& E + dE

5. Examples: Equipartition of Energy in Classical Statistical Mechanics
Free Particle Z

6. Other Examples of the Equipartion Theorem
LC Circuit
Harmonic
Oscillator
Free Particle
in 3 D
Rotating
Rigid Body

7. 1 D Simple Harmonic Oscillator

8. Einstein Model for the Vibrational Heat Capacity of Solids

9. Vibrational Heat Capacity of Solids R is the Ideal Gas Constant!
Einstein Model:
Vibrational Heat Capacity of Solids
Starting as early as the late 1600’s, people started
measuring the heat capacities Cp (or Cv) of
various solids, liquids & gases at various temperatures.
1819: Dulong & Petit observed that the molar heat
capacity for many solids at temperatures near room
temperature (300 K) is approximately independent
of the material & of the temperature & is
approximately
R is the Ideal Gas Constant!
This result is called the “Dulong-Petit Law”

10. Cv = AT3 (A is a constant) Early 1900’s: A theory explanation for this
Late 1800’s & early 1900’s: experimental capabilities
became more sophisticated, & it became possible to
achieve temperatures well below room temperature.
Then, measurements clearly showed that the heat
capacities of solids depend on what the material is &
are also strongly dependent on the temperature. In fact,
at low enough temperatures, the heat capacities of many
solids were observed to depend on temperature as
Cv = AT3
(A is a constant)
Early 1900’s: A theory explanation for this
temperature dependence of Cv was one of the
major unsolved problems in thermodynamics.

11. The Molar Heat Capacity
Assume that the heat supplied to a solid is
transformed into kinetic & potential
energies of each vibrating atom.
To explain the classical Dulong
Petit Law, use a simple classical
model of the vibrating solid as in
the figure. Assume that each
vibrating atom is an independent
simple harmonic oscillator & use the
Equipartition Theorem to calculate the Thermal
Energy E & the Heat Capacity Cv

12. The Thermal Average Energy for each degree of freedom is (½) kT
Molar Thermal Energy of a Solid
Dulong-Petit Law: Can be explained using the
Equipartition Theorem of Classical Stat Mech:
The Thermal Average Energy for each degree of freedom is (½) kT
Assume that Each atom has 6 degrees of
Freedom: 3 translational & 3 vibrational, the
Thermal Average Energy of the vibrating solid is:
R  NA k

13. The Vibrational Molar Heat Capacity In agreement with Dulong & Petit!
Using this approximation, the Thermal Average
Energy of the vibrating solid is:
(1)
By definition, the Heat Capacity of a substance
at Constant Volume is
(2)
Using (1) in (2) gives:
In agreement with Dulong & Petit!

14. Measured Molar Heat Capacities
Measurements on many solids have shown that their
Heat Capacities are strongly temperature
dependent & that the Dulong-Petit result Cv = 3R is
only valid at high temperatures.
Cv = 3R

15. Einstein Model of a Vibrating Solid
1907: Einstein extended Planck’s quantum ideas to solids:
He proposed that energies of lattice vibrations in a solid
are quantized simple harmonic oscillators. He proposed the following model the lattice vibrations in a solid:
Each vibrational mode is an independent oscillator
Each mode vibrates in 3-dimensions
Each vibrational mode is a quantized oscillator with energy: or
En = (n + ½ )ħ

16. In effect, Einstein modeled one mole of solid
as an assembly of 3NA distinguishable
oscillators.
He used the Boltzmann distribution to calculate
the average energy of an oscillator in this model:

17. Einstein Model of a Vibrating Solid
To compute the average energy,
note that it can be written as:
where
and b = 1/kT
Z is the partition function

18. Einstein Model With b = 1/kT & En = ne, the partition function
function for the Einstein model is:
which, follows from the result

19. Einstein Model
Differentiating with respect to b:

20. Einstein Model …and multiplying by –1/Z: This is Einstein’s result for
the average energy of an
oscillator. The total energy
of the solid is just 3NA times
this result

21. Einstein Model So, the heat capacity in Einstein’s model is:
TE = e/k is called the Einstein “temperature”.
Its not a true temperature, but an energy in
temperature units.

22. Einstein Model: CV for Diamond
Einstein, Annalen der Physik 22 (4), 180 (1907)

23. The Debye Model Vibrating Solids

24. Summary
The Dulong-Petit law is expected from the equipartition theorem of classical physics. However, it fails at temperatures low compared with the Einstein temperature
If the energy in solids is assumed to be quantized, however, models (such as Debye’s) can be developed that agree with the observed behavior of the heat capacity with temperature. Einstein was the first to suggest such a model.