Lecture 8 - Partition Functions and Ideal Gases (Chapter...

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Lecture 8 1 Partition Functions and Ideal Gases (Chapter 18.1, 2) As we discussed last time, for gases where the number of available eigenstates is greater than N we can write () ( ) , , , indistinguishable! ! N qV QNV N β ⎡⎤ ⎣⎦ = 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: . trans elec qq q = First, , i trans i e βε = For an atom in a cube of side a : 2 222 ,, 2 8 , , 1,2,3. .. xyz in n n x y z h nnn ma == + + =
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Lecture 8 2 () 2 222 2 ,, 1 22 111 exp 8 exp exp exp 888 xyz trans x y z nnn y x z h o q n n n ma hn ma ma ma β = ∞∞∞ === ⎛⎞ =− + + ⎜⎟ ⎝⎠ ⋅− ∑∑∑ Since the box is a cube, each sum is identical! 3 2 1 ,e x p 8 trans n qV 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 8 trans d n ma ≅−
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Lecture 8 3
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Lecture 8 - Partition Functions and Ideal Gases (Chapter...

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