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Lecture07A - Going Beyond U and CV(17.6 7 8 So far we have...

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Lecture 7 1 Going Beyond U and C V (17.6, 7, 8) So far we have considered the energy and heat capacities of two systems: perfect gases (monatomic & diatomic) and perfect atomic crystal. Again, looking ahead a couple of weeks, we note ( 29 th , Pressure in the j energy state j j N E P N V V = - = By our definition of ensemble averaging: ( 29 ( 29 ( 29 ( 29 , , , , , , j E N V j j N j j j E e V P P N V p N V Q N V β β β - - = = ( 29 ( 29 , , , , Math stuff: j j E N V E N V j j j N N E Q recall Q e so e V V β β β β β - - = = - ( 29 , , , 1 j E N V j j N N E e V Q P Q V Q β β β β - - = =
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Lecture 7 2 , , 1 ln ln So, we have shown P B N N T Q Q k T V V β β = = And we equate the ensemble average <P> to the experimentally observed pressure P. ( 29 ( 29 ( 29 3/ 2 2 , 2 For our ideal monatomic gas: Q N,V, ; , ! N q V m q V V N h β π β β β = = , , ln 1 ln : B N T N Q Q so P k T V V β β = = ( 29 ( 29 , , 1 1 ln ln ! ln terms with no V N N N q N N V V V β β β β = - = + ( 29 1 1 ideal gas eqn. B Nk T nRT N V V V β = = = Note that for any ideal gas (monatomic, diatomic, and polyatomic), q(N,V) is proportional to V, so the ideal gas equation applies to any ideal gas.
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Lecture 7 3 Let’s look at things just a bit more deeply. In statistical mechanics the energies, E j (N,V), are eigenvalues of the N-particle Hamiltonian.
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