Calc E as Fn of T for Ideal Paramagnet

[relativespeak]

Homework Statement

The entropy of an ideal paramagnet is given by S=S_{0}+CE^{2}, where E is the energy (which can be positive or negative) and C is a positive constant. Determine the equation for E as a function of T and sketch your result.

Homework Equations

[tex]
\frac{1}{T}=\frac{\delta S}{\delta U}
[\tex]

The Attempt at a Solution

I'm fairly certain I solved correctly, but the solution seems to simple. I confused about whether the E here is the same as the U in the partial derivative equation above, in which case:

[tex]
\frac{1}{T}=\frac{\delta S}{\delta U}
\frac{\delta S}{\delta U}=-2CE
\frac{1}{T}=-2CE
E=\frac{1}{-2CT}
[\tex]

In this case, the graph appears shaped like [tex] y=-\frac{1}{x} [\tex] dilated by [tex] \frac{1}{2C} [\tex].

I reasoned that in a paramagnet entropy will decrease as energy increases, so the system will more willingly give away energy, hence increasing the temperature.

 

[relativespeak]

Can someone tell me what I'm doing wrong with LaTex? I'm new to it and don't understand the problem with my code.