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Classical Statistical Mechanics in the Canonical Ensemble

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. The Equipartition Theorem Valid in Classical Statistical Mechanics ONLY !!!.

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Classical Statistical Mechanics in the Canonical Ensemble

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  1. Classical Statistical Mechanicsin the Canonical Ensemble

  2. Classical Statistical Mechanics 1. The Equipartition Theorem 2. The Classical Ideal Gasa. Kinetic Theoryb. Maxwell-Boltzmann Distribution

  3. The Equipartition Theorem Valid in Classical Statistical Mechanics ONLY!!! “Each degree of freedom in a system of particles contributes(½)kBTto the thermal average energy of the system.” Note:1. This is valid only if each term in the classical energy is proportional either a momentum squared or a coordinate squared. 2. The possible degrees of freedom are those associated with translation, rotation &vibration of the system’s molecules.

  4. Classical Ideal Gas For this system, it’s easy to show thatThe Temperatureis related to theaveragekinetic energy. For one molecule moving with velocity v, in 3 dimensions this takes the form: Further, for each degree of freedom, it can be shown that

  5. The Boltzmann Distribution: • 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 and E + dE • . • The Canonical Probability FunctionP(E): • This is defined so that P(E) dE the probability to find a • particular molecule between E and E + dE Z

  6. Equipartition Free Particle Z Simple Harmonic Oscillator

  7. Thermal Averaged Values Average Energy: Average Velocity: Of course:

  8. Kinetic Theory of Gases & The Equipartition Theorem

  9. Classical Kinetic Theory Results • The kinetic energy of individual particles is related to the gas temperature as: (½)mv2 = (3/2) kBT Here, v is the thermal average velocity. Boltzmann Distribution of Energy • There is a wide range of energies (& speeds) that varies with temperature:

  10. The Kinetic Molecular Model for Ideal Gases The gas consists of large number of small individual particles with negligible 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.

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

  12. Low T High T

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

  14. Maxwell-Boltzmann Speed Distribution • Change to spherical coordinates: 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

  15. 3D Maxwell-Boltzmann Speed Distribution Low T High T

  16. 3D Maxwell-Boltzmann Speed Distribution Convert the velocity-distribution into an energy-distribution:  = (½)mu2, d = mu du

  17. Velocity Values from the M-B Distribution • urms = root mean square velocity • uavg = average speed • ump = most probable velocity

  18. Comparison of Velocity Values

  19. Maxwell-Boltzmann Velocity Distribution

  20. Maxwell-BoltzmannSpeedDistribution

  21. Maxwell-BoltzmannSpeedDistribution

  22. The Probability Density Function • We can then use the distribution • function to compute the average • behavior of the molecules: • 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 • which is isotropic. • Let f(v)dvbe the fractional number of molecules in the • speed range from v to v + dv. • A probability distribution function has to satisfy the condition

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

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