Paramagnet: working with molar quantities

Question

I am currently working through A Modern Course in Statistical Physics by Linda E. Reichl. (2nd edition)

I am working on problem 2.13 which discusses a paramagnet.

The equation of state is given as $$m = \dfrac{DH}{T}$$ where m is the molar magnetisation, H is the magnetic field, D is a constant and T is temperature.

The molar heat capacity at constant magnetisation is given as constant: $c_{m} = c$

I am having trouble working with molar quantities.

The textbook states $C_{X, n} = \left(\dfrac{{\partial}U}{{\partial}T}\right)_{X, n}$, therefore $c_{x} = \left(\dfrac{{\partial}u}{{\partial}T}\right)_{x}$

and $\left(\dfrac{{\partial}U}{{\partial}X}\right)_{T, n}$ = $\left(\dfrac{{\partial}u}{{\partial}x}\right)_{T}$

where a lower case letter indicates we are dealing with a molar quantity ($U/n = u$).

I tried do derive the following analogously:

$C_{M, n} = T\left(\dfrac{{\partial}S}{{\partial}T}\right)_{M, n}$, therefore $c_{m} = T\left(\dfrac{{\partial}s}{{\partial}T}\right)_{m}$

And $\left(\dfrac{{\partial}S}{{\partial}M}\right)_{T, n}$ = - $\left(\dfrac{{\partial}H}{{\partial}T}\right)_{M, n}$ = $\left(\dfrac{{\partial}s}{{\partial}m}\right)_{T}$ where the first equality is a Maxwell relation

I have therefore worked out: $\left(\dfrac{{\partial}s}{{\partial}T}\right)_{m} = c/T$ and $\left(\dfrac{{\partial}s}{{\partial}m}\right)_{T} = -m/D$

I then integrated both equations to get $s(T, m) = c\ln(T) - \dfrac{m^{2}}{2D}$

This however isn't the correct answer. I am assuming I have made an error using molar quantities, however as far as I can tell I haven't done much more than repeat what the textbook says. Could someone explain why the textbook's equalities hold whereas mine don't ?

Edit : The correct answer is $$s(T, m) = (c+m^{2}/2D)\ln\left(\dfrac{T(c+m^{2}/2D)}{u_{0}}\right)$$

Answer 1

I think your answer is correct, the only thing that you need is to write the entropy in terms of its natural variables (for this case $U=U(S,M,N)$ or $u=(s,m)$). I mean, for getting the entropy that is supposed to be the correct one, the internal energy must have a 'magnetic' contribution, which is not the case, as you can easily prove by calculating $\left(\frac{dU}{dM}\right)$ at constant $T$ using that $c_m=$ constant.

On the other hand, all your molar quantities are right you haven't made any mistake. You can see Thermal physics by Morse for more details on paramagnetic substance (although they assume that $C_M=F(T)$, all the calculations are quite similar to the ones needed to solve problem 2.13). Hope this be useful, even though your question has been posted long ago.