Lecture08A - Partition Functions and Ideal Gases (Chapter...

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Partition Functions and Ideal Gases (Chapter 18.1, 2) s we discussed last time, for gases where the number of available As we discussed last time, for gases where the number of available eigenstates is greater than N we can write , N q V β ⎡⎤ ( ) ( ) , , indistinguishable! ! q QNV N ⎣⎦ = e will explicitly determine the molecular partition function for an We will explicitly determine the molecular partition function for an ideal gas, using our available knowledge from Q.M. For an ideal monatomic gas, there can only be two contributions to the energy: . trans elec ε εε =+ These energies are independent, so: ans elec qq q =⋅ . trans () First, , i trans i qV e βε = or an atom in a cube of side 2 222 h nn = + + For an atom in a cube of side a : ( ) ,, 2 8 , , 1,2,3. .. xyz in n n x y z nnn ma == = Lecture 8 1 2 2 exp trans x y z h so q n n n a ⎛⎞ =− + + ⎜⎟ ,, 1 22 8 p exp exp y x z ma hn = ∞∞∞ ⎝⎠ =−⋅− ⋅− ∑∑ 111 exp 888 ma ma ma === ∑∑∑ ince the box is a cube each sum is identical! Since the box is a cube, each sum is identical! ( ) 3 ,e x p ans q V 2 1 8 trans n ma = ⎢⎥ As we have seen in Chem 442, the energy level spacings are very small for macroscopic values of a , the box length. Change from discrete n to continuous n: 3 2 0 x p
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This note was uploaded on 04/01/2012 for the course CHEM 444 taught by Professor Gruebele,m during the Spring '08 term at University of Illinois, Urbana Champaign.

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Lecture08A - Partition Functions and Ideal Gases (Chapter...

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