06_562ln08

06_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 #6: Q Corrected for Molecular Indistinguishability Transformed Q from sum over states of an entire N-molecule assembly to sum over states of an individual molecule Q = e E j kT ( −ε j kT N = q N = e ) j i sum over states sum over states of assembly of a molecule Ω n j ) e E n j ) kT = N! e −∑ i n i ε i kT ({ } ({ } n j n j n j ! { } { } j sum over all sets of occupation numbers FOR INDEPENDENT, DISTINGUISHABLE PARTICLES BIG PROBLEM: Identical molecules are INDISTINGUISHABLE!! We have implicitly been assuming that we can distinguish particle 1 from particle 2, but quantum mechanics tells us that two identical particles are not distinguishable. [Even if we could label individual atoms by their position at t 1 , then follow each of the atoms until t 2 , these positional labels will be corrupted each time there is a collision.] Exceptions? Molecules in a crystal or in spatially confined traps on the surface of a solid. Thus we will need to modify the above result for molecules, since molecules of the same type are indistinguishable. We had shown that Ω ({ }) = t N! n i n i ! i = 1 is the number of ways of putting N distinguishable particles into t states with
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5.62 Spring 2008 Lecture 6, Page 2 occupation numbers {n i }. But if all the particles are identical, the number of ways is just 1. How do we know this to be true? Do example of 3
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06_562ln08 - MIT OpenCourseWare http:/ocw.mit.edu 5.62...

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