- #1
Telemachus
- 835
- 30
Homework Statement
Hi there. I'm having some trouble on solving this exercise, which you can find on Callen 2nd edition.
A simple fundamental equation that exhibits some of the qualitative properties of typical crystaline solids is:
[tex]u=Ae^{b(v-v_0)^2}s^{4/3}e^{s/3R}[/tex]
Where A,b, and v0 are positive constants.
a)Show that the system satisfies the Nernst theorem.
b)Show that [tex]c_v[/tex] is proportional to [tex]T^3[/tex] at low temperature.
c)Show that [tex]c_v\rightarrow 3k_b[/tex] at high temperatures.
The Attempt at a Solution
Well, I think I've solved a. And this is what I did:
[tex]\displaystyle\frac{\partial u}{\partial s}=T=Ae^{b(v-v_0)^2} \left[\displaystyle\frac{4}{3}s^{1/3}e^{s/3R}+\displaystyle\frac{1}{3R}s^{4/3}e^{s/3R}\right][/tex]
[tex]\therefore T \rightarrow 0 \Longleftrightarrow s \rightarrow 0[/tex]
I'm not sure if this is right. If there's another simple way of doing this I'd like to know.
Then I've tried with b) but I didn't get too far.
[tex]c_v=T\left(\displaystyle\frac{\partial s}{\partial T}\right)_v[/tex]
I don't know what to do from here, I've tried to get the entropic representation for the fundamental equation, but I couldn't, and I think it doesn't help. I think that I should use that for a constant volume [tex]du=Tds[/tex], but I'm not pretty much sure about this.
Help please :)
Bye there.
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