Classical Statistical Mechanics: 1 dimensional Simple Harmonic Oscillator

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!

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!

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

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

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

1 D Simple Harmonic Oscillator

Einstein Model for the Vibrational Heat Capacity of Solids

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”

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.

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

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

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!

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

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

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:

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

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

Einstein Model Differentiating with respect to b:

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

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.

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

The Debye Model Vibrating Solids

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.