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Homework Help: Chemical potential different for different systems?

  1. May 12, 2007 #1
    1. The problem statement, all variables and given/known data
    For a given phenomena of macroscopic particles.

    If we model it using the canonical ensemble then we get a certain chemical potential u.

    But if we model it using the Grand Canonical system, we get a varying chemcial potential that depends on the average number of particles in the system.

    So two different chemical potentials for the two systems. But in the thermodynamic limit of n-> infinity, both u equals each other.

    That sounds right dosen't it?
     
  2. jcsd
  3. May 13, 2007 #2

    cepheid

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    Yeah, you're basically right.

    In any of the systems in the canonical ensemble, the chemical potential is given by

    [tex] \mu(T,V,N) [/tex]

    whereas in any of the systems in the grandcanonical ensemble, it's given by

    [tex] \mu(T,V,\langle N \rangle) [/tex]

    Now, in the "thermodynamic limit" as you have called it, the number of particles in any of the grandcanonical systems is *MUCH* more likely to be equal to [itex] \langle N \rangle [/itex] than to any other number. So it's equivalent to having a closed system with a fixed number of particles [itex] \langle N \rangle [/itex].

    If I have an open system and I want it to be equivalent to a closed system with a particular number of particles [itex] N_0 [/tex], I'd have to somehow pick or set the value of [itex] \mu [/tex] such that [itex] \langle N \rangle = N_0 [/itex] for that system.
     
    Last edited: May 13, 2007
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