# Energy equation for a magnetic system (Thermodynamics)

1. Jul 23, 2009

### BossFang

1. The problem statement, all variables and given/known data

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

2. Relevant equations

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

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

3. 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

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

Last edited: Jul 24, 2009
2. Jul 23, 2009

### 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.

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