Energy equation for a magnetic system (Thermodynamics)

[BossFang]

Homework Statement

By considering the central equation of thermodynamics deduce the energy equation

$$\left(\frac {\partial{U}}{\partial{V}}\right)_T = T\left(\frac {\partial{P}}{\partial{T}}\right)_V - P$$

Write down the energy equation for a magnetic system

Homework Equations

Central equation $$\partial{U} = T\partial{S} - P\partial{V}$$

Maxwell relation $$\left(\frac{\partial{S}}{\partial{V}}\right)_T = \left(\frac{\partial{P}}{\partial{T}}\right)_V$$

The Attempt at a Solution

I am able to arrive at the energy equation above using the central equation and the maxwell relation. However my problem arises with the second part of the question.
Is writing down the energy equation for the magnetic system as simple as replacing P with -B₀(the magnetic induction in free space) and V with $$\mathfrak{M}$$(the magnetic moment) to give

$$\left(\frac {\partial{U}}{\partial{\mathfrak{M}}}\right)_T = -T\left(\frac {\partial{B_o}}{\partial{T}}\right)_V + B_o$$

 

[Mapes]

Hi BossFang, welcome to PF. As long as P-V work is assumed to be negligible, this looks good to me. You might exercise your thermo muscles a little and extend the equation to the case where P-V work is still present.