Classical Statistical Mechanics in the Canonical Ensemble

The Equipartition Theorem is Valid in Classical Stat. Mech. ONLY!!!

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

Classical Statistical Mechanics (ONLY!) It states: The Equipartition Theorem in Classical Statistical Mechanics (ONLY!) It states: “Each degree of freedom in a system of particles contributes (½)kBT to the thermal average energy Ē of the system.”

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 Ē of the system.” Note: 1. This is strictly 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.

We just finished an outline of the proof In the Classical Cannonical Ensemble, it is straighforward to show that The average energy of a particle per independent degree of freedom  (½)kBT. We just finished an outline of the proof

The Boltzmann (or Maxwell-Boltzmann) Distribution Start with the Canonical Ensemble Probability Function P(E): This is defined so that P(E) dE  probability to find a particular molecule between E & E + dE has the form: Z  Partition Function Z

The Boltzmann Distribution Define: Energy Distribution Function Canonical Ensemble Probability Function P(E): Z Define: Energy Distribution Function  Number Density  nV(E): Defined so that  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 (One dimension): Z

Equipartition Theorem Examples 1 d Harmonic Oscillator: LC Circuit: 1 d Harmonic Oscillator:

Equipartition Theorem Examples Free Particle in 3 Dimensions: Rotating Rigid Body:

1d Simple Harmonic Oscillator

Classical Ideal Monatomic Gas For this system, it’s easy to show that the Temperature T is related to the average kinetic energy. For 1 molecule moving with velocity v in 3 d, equipartition takes the form: For each degree of freedom, it’s easy to show:

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

So: 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 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

Equipartition Theorem:  Kinetic Molecular Model for Ideal Gases Due originally to Maxwell & Boltzmann Assumptions The gas consists of large number of individual point particles (zero volume). Particles are in constant random motion & collisions. No forces are exerted between molecules. Equipartition Theorem:  Gas Average 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

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 Important 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 Random motions of the molecules can be characterized by a probability distribution function. Since the velocity directions are uniformly distributed, the problem reduces to a speed distribution. The function f(v)dv is isotropic. f(v)dv  fractional number of mol ecules in the speed range from v to v + dv. Of course, 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: