Transition in magnetic refrigeration

[CAF123]

Homework Statement

The graph below (see attached) shows the measured magnetization against temperature at room pressure for the material Gd and for another material for use in a magnetic refrigerator at room temperature.

1)Calculate the expected change of temperature of a thermally isolated piece of Gd when the field in which it is located slowly reduced from an induction of H1 = 1.2T to H2 = 0.8T starting at a temperature 300K. You may assume that the M versus T curve does not change appreciably with H for H in the range 0.8-1.2T. Additionally, assume that the molar specific heat capacity of Gd is C_p = 3R J/K/mol of atoms, independent of temperature and field, The formula mass of Gd is 157.25g/mol.

Homework Equations

Derived in earlier part: $$dT = - \frac{T}{C_{p,H}} \left(\frac{\partial \mathbb{M}(T,H)}{\partial T}\right)_{H,P} dH$$

The Attempt at a Solution

I think I have to use the above equation derived in an earlier 'show that'. The graph attached shows M/ρ vs T which is equivalent to $\mathbb{M} / m$ since we are also to assume the magnetization,M, is related $\mathbb{M} = MV$.
So, $$\frac{\partial \mathbb{M}/m}{\partial T} = \frac{1}{m} \frac{\partial \mathbb{M}}{\partial T} + \mathbb{M} \frac{\partial}{\partial T} \frac{1}{m},$$ where $\partial \mathbb{M}/m / \partial T$ is the slope of the given graph.

Since it says the curve does not change appreciably, I think this means the graph has a constant gradient in the particular interval. But I am not sure how to obtain a functional form of $\partial \mathbb{M}/\partial T$. I can relate this to a Maxwell relation, but I don't think this helps.


Many thanks.

 

[CAF123]

Does anyone have a hint?