Question on First Law of Thermodynamics (Paramagnet)

[warhammer]

Homework Statement

For a pure Paramagnetic substance write down the differential form of first law of thermodynamics and show that C(B)-C(M)=kB^2/T^2

where C(B)=delta (Q)/delta(T) at B constant while C(M)=delta (Q)/delta (T) at M constant.

Relevant Equations

dU= delta (Q) + PdV

For the first part, I have expressed it in the following differential form- dU= delta (Q) + BdM

Now for the second part I am having major confusion. I know that B corresponds to P and M corresponds to V as generalised force and generalised displacement respectively for a Paramagnetic substance.

However I am unsure how to use the differential form as well as the possible equation M=nRT/B (from PV=nRT) in order to obtain the asked relationship. Would be indebted if someone would guide me or highlight my errors above, if any.

 

[TSny]

Relevant Equations:: dU= delta (Q) + PdV

This should be $dU = \delta Q - PdV$

For the first part, I have expressed it in the following differential form- dU= delta (Q) + BdM

OK. Note the change in sign where $+BdM$ corresponds to $-PdV$

Now for the second part I am having major confusion. I know that B corresponds to P and M corresponds to V as generalised force and generalised displacement respectively for a Paramagnetic substance.

Due to the sign change, if $B$ corresponds to $P$, shouldn't $M$ correspond to $-V$?

However I am unsure how to use the differential form as well as the possible equation M=nRT/B (from PV=nRT) in order to obtain the asked relationship. Would be indebted if someone would guide me or highlight my errors above, if any.

The correspondence $B \leftrightarrow P$ and $M \leftrightarrow -V$ in the first law does not mean that you can obtain the equation of state for the magnetic system by just changing $P$ and $V$ in the equation of state of an ideal gas(!). The equation of state of your system will be an equation that relates $M, B$ and $T$ for your particular substance. You would not expect the equation of state for a paramagnetic substance to be similar in form to the equation of state of an ideal gas. I suspect that you are supposed to assume that your paramagnetic substance obeys Curie's law.