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

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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 () th , Pressure in the j energy state j j N E P NV V ⎛⎞ =− = ⎜⎟ ⎝⎠ y our definition of ensemble averaging: By our definition of ensemble averaging: , j E NV j j E e V NV β ( ) ( ) ,, , N jj j PP N V p NV QNV == Math stuff: j j ENV j j E Q recall Q e so e VV ββ −− ∂∂ ∑∑ , , N N , j j E e V Lecture 7 1 , , 1 j N N Q P QV Q 1l n l n So, we have shown P B N T QQ kT V NN And we equate the ensemble average <P> to the experimentally observed pressure P. p ( ) 3/2 2 , 2 For our ideal monatomic gas: Q N,V, ; , ! N qV m V Nh π ⎡⎤ ⎣⎦ ln 1 ln : B NT N so P k T 11 ln ln ! ln terms with no V Nq N = + ideal gas eqn. B Nk T nRT N V = Note that for any ideal gas (monatomic, diatomic, and polyatomic), (N,V) is proportional to V, so the ideal gas equation applies to any Lecture 7 2 q( , ) p p , g q pp y ideal gas. Let’s look at things just a bit more deeply. In statistical mechanics the energies, E (N,V), are eigenvalues of the N-particle Hamiltonian. g, j (,) , g p 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? 1 st we can assume that the N-particles are independent. ( ) ( ) Then , N i V ε = 1 ji i = If the particles are distinguishable, we know which is which xamples would be our perfect crystal of N atoms at specific 2nd: (examples would be our perfect crystal of N atoms at specific coordinates).
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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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Lecture07A - Going Beyond U and CV (17.6, 7, 8) So far we...

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