#### MathematicalPhysicist

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

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

Derive the relations

[tex] C_H-C_M = \frac{T}{\chi_T} \bigg(\frac{\partial M}{\partial T}\bigg)^2_H [/tex]

[tex] \chi_T-\chi_S= \frac{T}{C_H} \bigg( \frac{\partial M}{\partial T}\bigg)^2_H[/tex]

[tex]\frac{\chi_T}{\chi_S} = \frac{C_H}{C_M}[/tex]

**2. Relevant equations**

[tex] C_X = T \bigg( \frac{\partial S}{\partial T} \bigg)_X[/tex]

[tex]\chi_Y = T \bigg( \frac{\partial M}{\partial H} \bigg)_Y[/tex]

Maxwell relations:

[tex] \bigg(\frac{\partial S}{\partial H}\bigg)_T = \bigg(\frac{\partial M}{\partial T}\bigg)_H[/tex]

Chain rule:

[tex]\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[/tex]

**3. The attempt at a solution**

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

[tex] 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[/tex]

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.