Lecture27A - Microscopic Connection to the Equilibrium...

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Lecture 27 1 Microscopic Connection to the Equilibrium Constant (26.8-10) Once again let’s return to our generic reaction ( 29 ( 29 ( 29 ( 29 A B Y z v A g v B g v Y g v Z g + + , We showed earlier: for a single component system T V A dA SdT PdV dn n = - - + , , A at constant T,V and n T V T V A dA dn n μ = , , ln ln Since ln B B T V T V Q Q A k T Q k T RT n N = - = - = - For a multi-component system: and A A B B Y Y Z Z i i dA dn dn dn dn dn v d ξ = + + + = ± + for products - for reactants so when 0 A = We have reached equilibrium and there is no mass flow between reactants and products. 0 Y Y Z Z A A B B v v v v = + - - =
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2 The partition function for the system, Q, is ( 29 ( 29 ( 29 ( 29 ( 29 , , , , , , , , , , , , , A B Y Z A B Y Z Q N N N N T V Q N V T Q N V T Q N V T Q N V T = ( 29 ( 29 ( 29 ( 29 , , , , for ideal gases ! ! ! ! A B Y Z N N N N A B Y Z A B Y Z q V T q V T q V T q V T N N N N = ( 29 , , ln For one component: ln ln ! j A A A A A A A V T N Q RT RT N q N N N μ = - = - - Recall Stirling's Approx: ln ! ln N N N N = - ( 29 , ln ln 1 ln ln 1 ln A A A A A A T V A A A A A A A RT N q N N N N q RT q N N RT N N = - - + = - - - + = - so 0 ln ln ln ln Y Z A B v v v v Y Z A B Y Z A B q q q q A RT N N N N ξ =
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Lecture27A - Microscopic Connection to the Equilibrium...

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