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Homework Help: Statistical mechanics/Thermodynamics two spin-1/2 subsystem

  1. Sep 30, 2013 #1
    1. The problem statement, all variables and given/known data
    Consider two spin-1/2 subsystems with identical magnetic moments (μ) in equal fields (B). The first subsystem has a total of NA spins with initially "a" having magnetic moments pointing against the field and (NA - a) pointing along the field, so that its initial energy is UiA = μB(a - (NA - a) = μB(2a - NA). The second subsystem has a total of NB spins with "b" having moments initially pointing against the field so that its initial energy is UiB = μB(2b - NB). Now suppose that the two subsystems are brought together so that they can exchange energy. Assume that B = constant and a, b, NA, NB >> 1. Show that in equilibrium, a0/NA = b0/NB, and this implies that the two subsystems will have the same "magnetization," i.e. total magnetic moment/spin.

    2. Relevant equations
    I'm not really sure what equations are useful in this case because I'm having trouble understanding what I need to be doing. I think I need to use multiplicity so
    Ω(N,n) = N!/(n!(N-n)!)

    3. The attempt at a solution
    I think that I have to first figure out the most probable macrostate because that is where the systems would be in equilibrium(?). So do I go along the lines of solving Ω(NA, a) and Ω(NB, b)? I don't even know if that makes sense or what a0 and b0 represent in this question.
  2. jcsd
  3. Oct 1, 2013 #2


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    Staff: Mentor

    You're going in the right direction. Have you seen entropy yet? If so, can you express the conditions for equilibrium in terms of S and U?

    My guess is that a0 and b0 are the number of spins pointing against the field at equilibrium.
  4. Oct 1, 2013 #3


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    Homework Helper

    exactly. But also, don't forget to constrain the total energy! And yeah, as DrClaude says, the a0 and b0 seem to simply be a and b once equilibrium is achieved.
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