Classical Statistical Mechanics in the Canonical Ensemble

Classical Statistical Mechanicsin the Canonical Ensemble

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

The Equipartition Theoremin 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)kBTto the thermal average energy of the system.” Note:1. This theorem is valid only ifeach 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.

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:

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:

Various contributions to the Classical Energyof 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 p2or to a q2.

Proof Continued! • 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 (½) (/bi)½

Proof Continued! Ki (½) (/bi)½ • 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:

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 FunctionP(E): • Defined so that P(E) dE probability to find a • particular molecule between E & E + dEis: 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 thatThe Temperatureis related to theaveragekinetic energy. For one molecule moving with velocity vin 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 betweenE & 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

Z • Using <E> = (½)mv2 = (3/2)kBTalong with P(E), the Probability Distribution of Energy Ecan 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. • Equipartition Theorem: • 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:

Maxwell-Boltzmann Velocity Distribution High T Low T

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) u3with u2 = ux2 + uy2 + uz2 • dV = duxduyduz = 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

Maxwell-Boltzmann Velocity Distribution

Maxwell-BoltzmannSpeedDistribution

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)dvwhich 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:

Classical Statistical Mechanics in the Canonical Ensemble