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Statistical physiics Problem 5.5

  1. Mar 12, 2008 #1
    1. The problem statement, all variables and given/known data
    http://ocw.mit.edu/NR/rdonlyres/Physics/8-044Spring-2004/85482B93-6A5E-4E2F-ABD2-E34AC245396C/0/ps5.pdf [Broken]

    I am stuck on Problem 5 part a. They say that the relevant state variables are H,M,T, and U. Obviously the first law of thermodynamics still holds: dU = dW+dQ (does anyone know how to make inexact differentials in latex)? But does dW = -PdV here? P and V were not among the state variables they talked about so does that really make sense? How do I proceed?

    2. Relevant equations

    3. The attempt at a solution
    Last edited by a moderator: May 3, 2017
  2. jcsd
  3. Mar 13, 2008 #2
  4. Mar 13, 2008 #3


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    [itex]dW=H\,dM[/itex] and [itex]U=TS+MH[/itex]. Work terms always consist of a generalized force (an intensive quantity) and a generalized displacement (an extensive quantity). Examples: force x distance, magnetic field x magnetization, electric field x polarization, surface energy x area, stress x strain, etc.

    In this problem [itex]P\,dV[/itex] work is evidently assumed to be negligible compared to [itex]M\,dH[/itex] work. You can tell because the problem states that there are only two independent variables (recall our https://www.physicsforums.com/showthread.php?p=1645661#post1645661").
    Last edited by a moderator: Apr 23, 2017
  5. Mar 13, 2008 #4
    OK, I see why [itex]dU = \delta Q +HdM[/itex]. But why is U = TS+MH true? I am trying to express [itex]C_M \equiv \left(\frac{\delta Q}{dT} \right)_M[/tex] as a derivative of the internal energy. Can you give me a hint how to do that?
  6. Mar 14, 2008 #5


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    From what you've written, it looks like you can conclude that [itex]
    C_M \equiv \left(\frac{\partial U}{\partial T} \right)_M

    In general, [itex]
    U=TS-PV+\sum\mu_i N_i +FL+ MH+EP+\gamma A+\sigma V\epsilon\dots
    [/itex] where the terms represent the work terms I listed above. This is called the Euler form of the fundamental relation, if you want to find more information about it. Callen's Thermodynamics is a good reference.
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