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Chem 300 - Ch 27/#1 Today’s To Do List l Kinetic Molecular Theory Boltzmann Distribution Law Distribution of Molecular Velocities In 1, 2, 3 dimensions

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The Boltzmann Principle l The probability of a molecule having an energy is given by: p( ) = A e - /kT A = constant l Central to all statistical aspects of P-Chem. l Basis for Boltzmann Distribution Law

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Boltzmann Distribution l The System: Isolated (no matter, heat, work exchange) Total E constant Total number of molecules (N) constant Various energy states ( i ) available Many ways N molecules can be assigned energies

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Boltzmann Distribution l Goal: Look for the most probable distribution among energy states W is the probability of a particular distribution If N is very large (~ 10 23 ) W will peak sharply (W max ) at some one distribution of molecules with I N i = A e - (i)/kT

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Molec. Velocity Distribution l 1-Dimension: Just kinetic energy: = ½ mu 2 u x is ± velocity along x-axis f(u x ) = A exp(-mu x 2 /2kT) = probability distrib Fraction N(u) of total molecules (N A ) with small range of u values: dN(u x )/N A = (m/2 kT) 1/2 [exp(-mu x 2 /2kT)]du x Maxwell-Boltzmann Equation

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Summary l Symmetric Distribution l Most probable speed is zero l Spread increases with incr. T l Veloc. Distrib. For 1-dimens. Ideal Gas

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2-Dimensional Distribution l Allow velocities (u x & u y ) along x & y l p(u x,u y ) product of the 2 separate functions: dN(u x,u y )/N A =(m/2 kT) [exp(-m{u x 2 +u y 2 }/2kT)]dudu l Better in terms of a net velocity or speed: c 2 = u x 2 + u y 2 dN(c)/N A = (m/kT) [exp(-mc 2 /2kT)]cdc

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Summary l Sign of c not relevant (magnitude) l Most probable speed nonzero l ½ width increases with increasing T

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3-D Distribution l c 2 = u x 2 + u y 2 + u z 2 l dN(c)/N A = 4 (m/2 kT) 3/2 [exp(-mc 2 /2kT)]c 2 dc l F(c) = 4 (m/2 kT) 3/2 [exp(-mc 2 /2kT)]c 2 Most probable speed higher. By setting dF(c) = 0, we get c p = (2kT/m) 1/2 p(c) decreases rapidly with c But always a finite p for any c no matter how large c is.

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Averaged Speeds l Most Probable: u mp = (2kT/m) 1/2 l Average Speed: = (8kT/ m) 1/2 l Root Mean Square Speed: u rms = (3kT/m) 1/2

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Next Time l KMT (Continued) Bimolecular Collision Frequencies Mean Free Path

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