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

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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 () th , Pressure in the j energy state j j N E PN V V ⎛⎞ =− = ⎜⎟ ⎝⎠ By our definition of ensemble averaging: ( ) , ,, , j E NV j j N jj j E e V PP N V p N V QNV β == , , Math stuff: j j ENV j N N E Q recall Q e so e VV ββ −− ∂∂ ∑∑ , , , 1 j j j N N E e V Q P QV Q
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Lecture 7 2 ,, 1l n l n So, we have shown P B NN T QQ kT VV β ⎛⎞ == ⎜⎟ ∂∂ ⎝⎠ And we equate the ensemble average <P> to the experimentally observed pressure P. () ( ) 3/2 2 , 2 For our ideal monatomic gas: Q N,V, ; , ! N qV m V Nh π ββ ⎡⎤ ⎣⎦ ln 1 : B NT N so P k T 11 ! terms with no V Nq N N V =− = + ideal gas eqn. B Nk T nRT N V = 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. 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. () ( ) 1 Then , N ji i EN V V ε = = If the particles are distinguishable, we know which is which (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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Lecture 7 - Going Beyond U and CV (17.6, 7, 8) So far we...

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