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Microstates & Entropy of two subsystems of a 4 particle system? (ensembles?)

  1. Nov 22, 2009 #1
    1. The problem statement, all variables and given/known data

    ****************************************************************
    A four-particle system is composed of 2 two–particle subsystems. Subsystem I has par-
    ticles A and B, which can have a maximum internal energy U(AB) = 4{E}. Subsystem II
    has particles C and D, in which the internal energy U(CD) = 0.

    The subsystems are initially isolated from each other, before being brought into thermal
    contact (but still isolated from the rest of the universe). By calculating and comparing the
    number of microstates of the combined system before and after being brought into thermal
    contact, state if and how the entropy of the combined system changes.
    *****************************************************************



    2. Relevant equations

    ****************************************************************
    Let W = number of microstates
    Let kb = boltzmann constant

    W = 2^N

    S = (kb)lnW
    ****************************************************************

    3. The attempt at a solution

    ****************************************************************
    So, I've theoretically laid out:


    subsystem I, at any time t, it is likely to exist as W1(U1)
    subsystem II, at ant time t is likely to be in U2(U2) microstate


    Thus, the combined subsystem is likely to be in W(U1, U2) microstate
    where N(total) = N(I) + N(II)
    and U(total) = U(I) + U(II)


    The total number of microstates of the composite system is the product of the numbers of microstates of the 2 subsystems.

    W(U(total), N(total)) = W1(U1, N1)W2(U1,N1) = W1(U1,N1)W2(U(total) - U1, N(total) - N1)

    Thus, entropy S is
    S = S1(U1, N1) + S2(U(total) - U1, N(total) - N1)
    ****************************************************************

    However, I have no clue as to how to actually calculate W from the data and formulae given. I also do not know how to calculate S, since I do not know how to find W.

    Any help would be appreciated!
    Much love. =)

    ****************************************************************
     
  2. jcsd
  3. Nov 22, 2009 #2
    PS. N is the number of particles
     
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