Classical Statistical Mechanics in the Canonical Ensemble

Classical Statistical Mechanics 1. The Equipartition Theorem 2. The Classical Ideal Gas a. Kinetic Theory b. Maxwell-Boltzmann Distribution

Classical Statistical Mechanics (ONLY!) The Equipartition Theorem in Classical Statistical Mechanics (ONLY!)

The Equipartition Theorem is 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.” Note: 1. This theorem is valid only if each term in the classical energy is proportional to a momentum (p) squared or to a coordinate (q) squared. 2. The degrees of freedom are associated with translation, rotation & vibration of the system’s molecules.

Outline of a Proof Follows: In the Classical Cannonical Ensemble, it is straighforward to show that The average energy of a particle per independent degree of freedom  (½)kBT. Outline of a Proof Follows:

System Total Energy  Sum of single particle energies: Proof System Total Energy  Sum of single particle energies: System Partition Function Z  Z' , Z'' , etc. = Partition functions for each particle.

System Partition Function Z  Z = Product of partition functions Z' , Z'' , etc. of each particle Canonical Ensemble “Recipe” for the Mean (Thermal) Energy: So, the Thermal Energy per Particle is:

 = KEt + KEr + KEv + PEv + …. KEt = (½)mv2 = [(p2)/(2m)] KEr = (½)I2 Various contributions to the Classical Energy of each particle:  = KEt + KEr + KEv + PEv + …. Translational Kinetic Energy: KEt = (½)mv2 = [(p2)/(2m)] Rotational Kinetic Energy: KEr = (½)I2 Vibrational Potential Energy: PEv = (½)kx2 Assume that each degree of freedom has an energy that is proportional to either a p2 or to a q2.

Proof Continued! Plus a similar sum of terms containing the (qi)2 With this assumption, the total energy  has the form: Plus a similar sum of terms containing the (qi)2 For simplicity, focus on the p2 sum above: For each particle, change the sum into an integral over momentum, as below. It is a Gaussian & is tabulated. Ki  Kinetic Energy of particle i

Proof Continued! Finally, Z can be written: Ki  Kinetic Energy of particle i The system partition function Z is then proportional to the product of integrals like above. Or, Z is proportional to P: Finally, Z can be written:

For a Monatomic Ideal Gas: For a Diatomic Ideal Gas: Use the Canonical Ensemble “Recipe” to get the average energy per particle per independent degree of freedom: Note! u  <> For a Monatomic Ideal Gas: For a Diatomic Ideal Gas: l For a Polyatomic Ideal Gas in which the molecules vibrate with q different frequencies:

The Boltzmann Distribution Canonical Probability Function P(E): Defined so that P(E) dE  probability to find a particular molecule between E & E + dE is: Z Define: Energy Distribution Function  Number Density  nV(E): Defined  nV(E) dE  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

Simple Harmonic Oscillator

Classical Ideal Monatomic Gas For this system, it’s easy to show that The Temperature is related to the average kinetic energy. For one molecule moving with velocity v in 3 dimensions this takes the form: Also, for each degree of freedom, it can be shown that

Classical Statistical Mechanics: Canonical Ensemble Averages Probability Function: Z P(E) dE  probability to find a particular molecule between E & E + dE Normalization:

Z Average Energy: Average Velocity:

Classical Kinetic Theory Results We just saw that, from the Equipartition Theorem, the kinetic energy of each particle in an ideal gas is related to the gas temperature as: <E> = (½)mv2 = (3/2)kBT (1) v is the thermal average velocity. Canonical Ensemble Probability Function: Z In this form, P(E) is known as the Maxwell-Boltzmann Energy Distribution

Maxwell-Boltzmann EnergyDistribution n(E,T) E

P(v) = C exp[- (½)m(v)2/(kT)] Maxwell-Boltzmann Velocity Distribution Using <E> = (½)mv2 = (3/2)kBT along with P(E), the Probability Distribution of Energy E can be converted into a Probability Distribution of Velocity P(v). This has the form: P(v) = C exp[- (½)m(v)2/(kT)] In this form, P(v) is known as the Maxwell-Boltzmann Velocity Distribution

Kinetic Molecular Model for Ideal Gases Assumptions The gas consists of large number of individual point particles (zero size). Particles are in constant random motion & collisions. No forces are exerted between molecules. From the Equipartition Theorem, The Gas Kinetic Energy is Proportional to the Temperature in Kelvin.

Maxwell-Boltzmann Velocity Distribution The Canonical Ensemble gives a distribution of molecules in terms of Speed/Velocity or Energy. The 1-Dimensional Velocity Distribution in the x-direction (ux) has the form:

High T Low T

In Cartesian Coordinates: 3D Maxwell-Boltzmann Velocity Distribution a  (½)[m/(kBT)] In Cartesian Coordinates:

Maxwell-Boltzmann Speed Distribution Change to spherical coordinates in Velocity Space. Reshape the box into a sphere in velocity space of the same volume with radius u . V = (4/3) u3 with u2 = ux2 + uy2 + uz2 dV = dux duy duz = 4  u2 du

3D Maxwell-Boltzmann Speed Distribution Low T High T

Maxwell-Boltzmann Speed Distribution Convert the speed-distribution into an energy-distribution:  = (½)mu2, d = mu du

Some Velocity Values from the M-B Distribution urms = root mean square (rms) velocity uavg = average speed ump = most probable velocity

Comparison of Velocity Values Ratios in Terms of urms uavg ump 1.73 1.60 1.41

Maxwell-Boltzmann Velocity Distribution

Maxwell-Boltzmann Speed Distribution

Maxwell-Boltzmann Speed Distribution

The Probability Density Function The random motions of the molecules can be characterized by a probability distribution function. Since the velocity directions are uniformly distributed, we can reduce the problem to a speed distribution function f(v)dv which is isotropic.

The Probability Density Function Let f(v)dv  fractional number of molecules in the speed range from v to v + dv. A probability distribution function has to satisfy the condition

The Probability Density Function We can use the distribution function to compute the average behavior of the molecules: