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Classical Statistical Mechanics in the Canonical Ensemble

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

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The Equipartition Theorem Valid in Classical Statistical Mechanics ONLY!!! “Each degree of freedom in a system of particles contributes (½)k B T to 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.

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Classical Ideal 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: Further, for each degree of freedom, it can be shown that

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The Boltzmann Distribution: Define The Energy Distribution Function (Number Density) n V (E): This is defined so that n V (E) dE the number of molecules per unit volume with energy between E and E + dE. The 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

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Equipartition Simple Harmonic Oscillator Free Particle Z

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

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Kinetic Theory of Gases & The Equipartition Theorem

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

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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.

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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 (u x ) has the form:

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Low T High T

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Maxwell-Boltzmann Distribution 3D Velocity Distribution: a (½)[m/(k B T)] In Cartesian Coordinates:

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Change to spherical coordinates: Reshape the box into a sphere in velocity space of the same volume with radius u. V = (4/3) u 3 with u 2 = u x 2 + u y 2 + u z 2 dV = du x du y du z = 4 u 2 du Maxwell-Boltzmann Speed Distribution

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3D Maxwell-Boltzmann Speed Distribution Low T High T

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3D Maxwell-Boltzmann Speed Distribution Convert the velocity-distribution into an energy-distribution: = (½)mu 2, d = mu du

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Velocity Values from the M-B Distribution u rms = root mean square velocity u avg = average speed u mp = most probable velocity

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Comparison of Velocity Values Ratio in Terms of : u rms u avg u mp 1.731.601.41

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Maxwell-Boltzmann Velocity Distribution

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Maxwell-Boltzmann Speed Distribution

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

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

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Typically, it’s easier to work with the integrals rather than the sums

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