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 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?

 

[Mapes]

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

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