Lecture 8 - ( 29 C isothermal system, thermo Statistic ther...

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Unformatted text preview: ( 29 C isothermal system, thermo Statistic ther LOSED Q mo N,V,T ( 29 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 = - ln , , pV kT V T ( 29 ( 29 - = = 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 = = 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 = ( , , ) , , N N V T Q N V T N distinguishable particles, 2 possible states ( E= with E 1 =0) {a j }={a 1 ,a 2 ,a N } where a j =0, or 1 and therefore = 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 = =- ! ( , ) ! ! N W E N N m m ( 29 ( 29 ( 29 = = = = 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 - = where we used Stirling approximation 1 ln N m m = =- = + = + 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 - = we can also look at the system in the , Can ln onical , ensemble , ln E Q N V e v v - = = = 0,1 and using ln ( , , ) ln j j j j V e E Q N j a v a a ( 29 - = = = +- 0,1 ln ( , , ) ln ln ( , , ) ln 1 j N N j Q N V e Q N V e j a a = - ln from here we can obtain the E Q ( 29 + = - = +--- ln 1 1 N e e E N e = + 1 N E e Ensemble Constants Fundamental thermodynamics Total differentials Microcanonical N,V,E S=k b ln Canonical N,V,T A=-k b Tln Q(N,V,T) Grand- canonical ,V,T pV=k b Tln ( ,V,T) Isothermal- isobaric N,p,T G=-k b Tln (N,p,T) = +- dE pdV dN...
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Lecture 8 - ( 29 C isothermal system, thermo Statistic ther...

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