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Unformatted text preview: ( 29 C isothermal system, thermo Statistic ther LOSED Q mo N,V,T ( 29 μ Ξ isothermal system, thermo tatistic thermo OPEN V ,T, S Looking at the natural variables ( 29 ( 29 = , , ln , , A N V T kT Q N V T ( 29 μ = Ξ ln , , pV kT V T ( 29 ( 29 μ μ μ Ξ = = for a fixed value of N, we can sum over all j V,T, ( , , ) E V N kt kT N j N kT N Nj e e Q N V T e μ μ λ μ λ λ = ∆ = kT where is an absolute activity and is the difference in chemical potential for 2 states with activities and e ln ln N N kt kt 2 1 2 1 a a a a ( 29 μ λ Ξ = ( , , ) , , N N V T Q N V T N distinguishable particles, 2 possible states ( ∆ E= ε with E 1 =0) ⇒ {a j }={a 1 ,a 2 ,…a N } where a j =0, or 1 and therefore ε = j j E v a Microcanonical ensemble : degeneracy of the m th level number of ways to distribute m objects in a pool of N (i.e. distribute m quanta to obtain E total energy) ( 29 = Ω = ! ( , ) ! ! N W E N N m m ( 29 ( 29 ( 29 ε β ε β = Ω = Ω = = Ω and since , we know that when N is large enough continuous on ln , 1 1 , , ln , N N m E m E N m kT E N E E N ( 29 β ε = where we used Stirling approximation 1 ln N m m βε βε βε ε ε = = = + = + and since the energy is e 1 e 1 e 1 E m N N m N m E An example on the equivalency among ensembles ( 29 β β = we can also look at the system in the , Can ln onical , ensemble , ln E Q N V e v v ε β β ε = = = 0,1 and using ln ( , , ) ln j j j j V e E Q N j a v a a ( 29 β βε ε β β = = = + 0,1 ln ( , , ) ln ln ( , , ) ln 1 j N N j Q N V e Q N V e j a a β =  ln from here we can obtain the E Q ( 29 βε βε βε ε β + =  = + ln 1 1 N e e E N e βε ε = + 1 N E e Ensemble Constants Fundamental thermodynamics Total differentials Microcanonical N,V,E S=k b ln Ω Canonical N,V,T A=k b Tln Q(N,V,T) Grand canonical μ ,V,T pV=k b Tln Ξ ( μ ,V,T) Isothermal isobaric N,p,T G=k b Tln ∆ (N,p,T) μ = + dE pdV dN...
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This note was uploaded on 05/29/2011 for the course CHM 6461 taught by Professor Bowers during the Spring '08 term at University of Florida.
 Spring '08
 Bowers

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