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Show that the total weight w of an assembly in thermal equilibrium is

  1. Apr 8, 2014 #1
    1. The problem statement, all variables and given/known data
    Show that the total weight w of an assembly in thermal equilibrium is a maximum.


    2. Relevant equations
    S= k ln W?
    dS = dQ/ T ?
    S is maximize when thermal equilibrium is reached.


    3. The attempt at a solution
    First of all, i don't know what does "total weight w" means.
    Dose it means the number of ways W that a total of N particles can be classified into energy levels?
    I think that I have to establish a equation W = f(S) and use differentiation to prove that it is a maximum.
    However, i have no clue to start my work.
    Can someone give me some clues so that i can start my work? Thank you.
     
  2. jcsd
  3. Apr 9, 2014 #2
    I would assume that the "weight" is a statistical weight, i.e. the relative probability. Here is a related example to get you started. Suppose you have two systems A and A' with energies E and E', respectively. These systems can exchange energy, and form a total system A0 with energy E0. The probability, say Pr(E0) that the total system A0 is in a particular state (configuration of A and A') depends on how the energy is distributed between A and A'. That is Pr(E0) = Pr(E,E'). The maximum probability occurs when the derivative vanishes, i.e. dPr(E0)/dE0 = 0, or equivalently dln(Pr(E0))/dE0 = 0 because the natural logarithm is a monotonically increasing function. In general, ln(Pr(E,E')) is proportional to ln(W(E)) + ln(W'(E')) + C, where W(E) and W'(E') are the number of microstates accessible to A and A' respectively, and C is a normalization constant. We recall that dln(W(E))/dE is proportional to 1/T, and likewise for the other system. This means that maximal probability implies equality of temperatures, which is the condition for thermal equilibrium. I have glossed over some details, leaving them to you. One last point is that you must be able to generalize this to any thermodynamic system. I hope this helps.
     
    Last edited: Apr 9, 2014
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