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Unformatted text preview: CHEM 350 Lectures 16, 17, and 18 February 7, 10, and 12 2010 1 MOLECULAR PARTITION FUNCTIONS . Independent Subsystems . 2 In all cases we shall be concerned with subsystems that are statistically independent , and in the following, we shall ignore any interactions between these subsystems. • distinguishable subsystems : consider two subsystems, and consider the meaning of the above two conditions (or assumptions). – statistical independence: subsystem a with quantum numbers i ; subsystem b with quantum numbers j – weak interactions: ij = i + j . – the canonical partition function is given by Q = X ij e β ij = X i e β i X j e β j = X i e β i ! X j e β j ! and so we see that, with the above two assumptions, the canonical partition function Q for the total system is given by the product of the partition functions for the individual systems (all in thermal contact with a common bath), viz, Q = Q a Q b . – More generally: Q = Q a Q b Q c ··· . example: Air as a mixture of N 2 , ) 2 , ..., implies that Q Air = Q N 2 Q O 2 ··· 1 1 2 2 3 3 4 5 4 5 6 7 8 Distinguishable particles 1 2 3 10 4 5 6 7 8,9accidental degeneracy 11 12 • indistinguishable subsystems : – one set of quantum numbers 3 – weak interactions: e.g. ij = ai + bj for two subsystems. Q = Q a Q b ⇒ Q = X i e β i ! X i e β i ! = q 2 BUT the defining relation for Q gives Q ≡ X k e β k , with values of k given by the possible values of ai + bj ; however, as ai + bj = aj + bi . we shall have counted the energy twice in the product q a q b , rather than only once as required by the definition of Q for the entire system. For N such particles, the product Q = q N overestimates the actual value that Q must have by the number of ways of obtaining = 1 + 2 + ··· + N , i.e., by N !. Hence we have Q N = q N N ! . Evaluation of Q for N particles with translational motion only. For a single particle with translational motion (3–D box problem) the canonical partition function is given by q ( β, 1 ,V ) = Z ∞ Ω( , 1 ,V )e β d , with Ω given by Ω( , 1 ,V ) = 2 π 2 m h 2 3 2 V 1 2 . We can therefore write q ( β, 1 ,V ) explicitly as q ( β, 1 ,V ) = 2 πV 2 m h 2 3 2 Z ∞ 1 2 e β d ....
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This note was uploaded on 04/11/2010 for the course CHEM 1101 taught by Professor Leroy during the Spring '10 term at University of Toronto Toronto.
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