# Question on First Law of Thermodynamics (Paramagnet)

• warhammer
In summary, the conversation discusses the use of the first law of thermodynamics for a paramagnetic substance and how it relates to generalised force and displacement. The equation of state for this substance is not the same as that for an ideal gas, and it is likely that Curie's law should be used to determine the relationship between M, B, and T.

#### 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.

warhammer said:
Relevant Equations:: dU= delta (Q) + PdV
This should be ##dU = \delta Q - PdV##

warhammer said:
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##

warhammer said:
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##?

warhammer said:
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.

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