Deriving Relations for Specific Heats and Susceptibilities of Magnets

[MathematicalPhysicist]

Homework Statement

This question is from Birger Bergerson's textbook on Equilibrium Statistical mechanics.

Derive the relations
$$C_H-C_M = \frac{T}{\chi_T} \bigg(\frac{\partial M}{\partial T}\bigg)^2_H$$
$$\chi_T-\chi_S= \frac{T}{C_H} \bigg( \frac{\partial M}{\partial T}\bigg)^2_H$$
$$\frac{\chi_T}{\chi_S} = \frac{C_H}{C_M}$$

Homework Equations

$$C_X = T \bigg( \frac{\partial S}{\partial T} \bigg)_X$$
$$\chi_Y = T \bigg( \frac{\partial M}{\partial H} \bigg)_Y$$

Maxwell relations:
$$\bigg(\frac{\partial S}{\partial H}\bigg)_T = \bigg(\frac{\partial M}{\partial T}\bigg)_H$$

Chain rule:
$$\bigg(\frac{\partial S}{\partial T}\bigg)_H = \bigg(\frac{\partial S}{\partial T}\bigg)_M + \bigg(\frac{\partial S}{\partial M}\bigg)_T \bigg(\frac{\partial M}{\partial T}\bigg)_H$$

The Attempt at a Solution

For the first relation I get $T^2$ instead of $T$, I'll write my solution:

$$C_H-C_M = T\bigg(\bigg(\frac{\partial S}{\partial T}\bigg)_H - \bigg(\frac{\partial S}{\partial T}\bigg)_M \bigg)=T\bigg(\frac{\partial S}{\partial M}\bigg)_T \bigg(\frac{\partial M}{\partial T}\bigg)_H$$

Notice that $\bigg(\frac{\partial S}{\partial M}\bigg)_T = \bigg(\frac{\partial S}{\partial H}\bigg)_T \bigg(\frac{\partial M}{\partial H}\bigg)^{-1}_T = \bigg(\frac{\partial M}{\partial T}\bigg)_H \frac{T}{\chi_T}$, so if I plug the last relation to the relation above for $C_H-C_M$ I get $C_H-C_M = \bigg( T^2/\chi_T \bigg) \bigg(\frac{\partial M}{\partial T}\bigg)^2_H$.

Am I right?

It seems this is a typo (if I am correct also appears in the third edition of this textbook).

Am I right?

Is this result known in the literature?

Have I missed something here?

Thanks.

 

[MathematicalPhysicist]

Never mind, I see that the mistake is in the susceptibility, the $T$ is redundant there.

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