19_562ln08 - MIT OpenCourseWare http/ocw.mit.edu 5.62...

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MIT OpenCourseWare http://ocw.mit.edu 5.62 Physical Chemistry II Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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5.62 Lecture #19: Configurational Integral: Cluster Expansion Goal: For U( q ) 0, calculate Z to obtain corrections for non-ideal contributions to the equation of state. Z = dq 3N e U(q ~ )/kT ~ where U( q ) = Total Interaction Potential Energy Simplifications based on form of U( q ): 1. Assume U( q ) is sum over pairs of atoms ("pair potential") — pairwise additive interactions . 2. Assume U( q ) depends only on the distance between pairs of atoms, r i r j . U( q ) = u ij i < j r i r j ( ) r ij = distance between atoms i and j sum over pairs, i< j prevents double counting u ij pair interaction potential (Hard Sphere, Square Well, Sutherland, LJ, dipole-dipole, etc.), which is a function of r ~i r ~ j . r ~i position of i th atom Therefore — u ij kT Z = dq 3N e i < j = dq 3N e u ij kT ~ ~ i < j The next step is a USEFUL TRICK For distant particles u ij = 0 e u ij kT = 1 For convenience we define
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5.62 Spring 2008 Lecture #19, Page 2 e u ij kT ( 1 + f ij ) f ij = e u ij kT 1 giving f ij = 0 when particles have no interaction. Z = dq 3N ( 1 + f ij ) evaluate as “cluster expansion” ~ i < j Z = dq 3N ( 1 + f 12 )( 1 + f 13 )( 1 + f 14 ) ( 1 + f 1N )( 1 + f 23 )( 1 + f 24 ) ~ ( 1 + f 2N )( 1 + f 34 )( 1 + f 35 ) ( 1 + f 3N ) Z = dq 3N 1 + ( f 12 + f 13 + f 14 +… f N 1,N ) + ( f 12 f 13 + f 12 f 34 + f 12 f 56 + f 12 f 23 +… f N 2N 1 f N 1N ) + ~ ( f 12 f 34 f 56 +…+ f 12 f 23 f 34 +… ) +…
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