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

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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 E ( ) th , Pressure in the j energy state j j N P N V V = − = By our definition of ensemble averaging: ( ) ( ) ( ) , j E N V j j N E e V P P N V N V β β ( ) , , , , , j j j p Q N V β = = 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 = = , , ( ) , 1 j E N V j j N E e V Q β β 1 , , N P Q V Q β β = = 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. ( ) 3/2 N ( ) ( ) 2 , 2 For our ideal monatomic gas: Q N,V, ; , ! 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 = = ( ) ( ) , , 1 1 ln ln ! ln terms with no V N N N q N N V V V β β β β = = + ( ) 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 Lecture 7 2 ideal gas. 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. Remember, we couldn’t find exact eigenvalues even for a 3-body system (except H 2 + in the B.O. approximation) so what can we do for N-bodies?
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