Stuck on Understanding Work Terms in Statistical Physics Problem 5.5?

[ehrenfest]

Homework Statement

http://ocw.mit.edu/NR/rdonlyres/Physics/8-044Spring-2004/85482B93-6A5E-4E2F-ABD2-E34AC245396C/0/ps5.pdf

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?

Homework Equations

The Attempt at a Solution

 

[ehrenfest]

anyone?

 

[Mapes]

$dW=H\,dM$ and $U=TS+MH$. 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 $P\,dV$ work is evidently assumed to be negligible compared to $M\,dH$ 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").

 

[ehrenfest]

OK, I see why $dU = \delta Q +HdM$. 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?[/itex]

 

[Mapes]

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

In general, $U=TS-PV+\sum\mu_i N_i +FL+ MH+EP+\gamma A+\sigma V\epsilon\dots$ 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.