(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Given that mechanical equation of state for a paramagnetic substance is

[tex]m=\left(\frac{DH}{T}\right)[/tex]

where D is a constant, H is the magnetic field, m is molar magnetization and the molar heat capacity

[tex]c_{m}[/tex] is constant, find entropy and enthalpy

2. Relevant equations

[tex]\left(\frac{\partial S}{\partial T}\right)_{m}=\frac{C_{m}}{T}[/tex]

[tex]\left(\frac{\partial S}{\partial m}\right)_{T}=-\left(\frac{\partial H}{\partial T}\right)_{m}[/tex]

3. The attempt at a solution

Integrating above equations I found that

[tex]S=-\frac{m^{2}}{2D}+c_{m}\ln{\frac{T}{T_{0}}}.[/tex]

Or alternatively

[tex]T=T_{0}\exp{(2DS+m^{2})/(2DC)}.[/tex]

Now to the entalphy h. By definition

[tex]\left(\frac{\partial h}{\partial S}\right)_{H}=T=T_{0}\exp{(2DS+m^{2})/(2DC)}[/tex]

and

[tex]\left(\frac{\partial h}{\partial H}\right)_{S}=-m=DH/T.[/tex]

Now, I'm a bit confused here. When integrating these kind of partial derivatives, I guess that one should express the integrand using only those variables that are used to perform the differentiation, i.e. S and H in this particular case. If this is the case, one should probably eliminate m from above equation using the equation of state. But this gives

[tex]T=T_{0}\exp{(2DS+DH^{2}/(2CT^{2}))/(2DC)}[/tex]

which makes no sense as T appears on both sides... What's going wrong?

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# Homework Help: Thermodynamics: integrating partial derivatives

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