1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted