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fluidistic
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Stability, Helmholtz free energy mathematical relation "proof"
I must show that [tex]\left ( \frac{\partial ^2 F}{\partial V ^2} \right ) _T=\frac{\frac{\partial ^2 U}{\partial S^2} \frac{\partial ^2 U}{\partial V ^2} - \left ( \frac{\partial ^2 U}{\partial S \partial V} \right ) ^2}{\frac{\partial ^2 U}{\partial S^2}}[/tex]
Where F is the Helmholtz free energy defined as ##F=U-TS##.
Hint: See that ##\left ( \frac{\partial ^2 F}{\partial V ^2} \right ) _T=-\left ( \frac{\partial P}{\partial V } \right ) _T## and consider P as a function of S and V.
As you can see, it is not clear at all what variables are kept constant when he does (the book) the partial derivatives. Furthermore, I don't find any sense to considering ##-\left ( \frac{\partial P}{\partial V } \right ) _T## if P is a function of S and V rather than V and T.
I can't seem to make sense of the hint given. So I started with the definition of the Helmholtz free energy and performed twice the derivative with respect to V with T fixed to reach ##\left ( \frac{\partial ^2 F}{\partial V ^2} \right ) _T=\left ( \frac{\partial ^2 U}{\partial V ^2} \right ) _T-T\left ( \frac{\partial ^2 S}{\partial V ^2} \right ) _T##.
Now ##T=\left ( \frac{\partial U}{\partial S} \right ) _V## so that ##\left ( \frac{\partial ^2 F}{\partial V ^2} \right ) _T=\left ( \frac{\partial ^2 U}{\partial V ^2} \right ) _T-\left ( \frac{\partial U}{\partial S} \right ) _V \left ( \frac{\partial ^2 S}{\partial V ^2} \right ) _T##.
Now I multiply and divide by ##\left ( \frac{\partial ^2 U}{\partial S ^2} \right ) _T## to reach [tex]\left ( \frac{\partial ^2 F}{\partial V ^2} \right ) _T=\frac{\left ( \frac{\partial ^2 U}{\partial V ^2} \right ) _T\left ( \frac{\partial ^2 U}{\partial S ^2} \right ) _T- \left ( \frac{\partial U}{\partial S} \right ) _V \left ( \frac{\partial ^2 U}{\partial V ^2} \right ) _T}{\left ( \frac{\partial ^2 U}{\partial S ^2} \right ) _T}[/tex] so I've almost the right expression but not quite. I don't know how to proceed further. Nor do I know how to use the hint given, nor do I understand it.
Any help is appreciated. Thank you!
Homework Statement
I must show that [tex]\left ( \frac{\partial ^2 F}{\partial V ^2} \right ) _T=\frac{\frac{\partial ^2 U}{\partial S^2} \frac{\partial ^2 U}{\partial V ^2} - \left ( \frac{\partial ^2 U}{\partial S \partial V} \right ) ^2}{\frac{\partial ^2 U}{\partial S^2}}[/tex]
Where F is the Helmholtz free energy defined as ##F=U-TS##.
Hint: See that ##\left ( \frac{\partial ^2 F}{\partial V ^2} \right ) _T=-\left ( \frac{\partial P}{\partial V } \right ) _T## and consider P as a function of S and V.
Homework Equations
As you can see, it is not clear at all what variables are kept constant when he does (the book) the partial derivatives. Furthermore, I don't find any sense to considering ##-\left ( \frac{\partial P}{\partial V } \right ) _T## if P is a function of S and V rather than V and T.
The Attempt at a Solution
I can't seem to make sense of the hint given. So I started with the definition of the Helmholtz free energy and performed twice the derivative with respect to V with T fixed to reach ##\left ( \frac{\partial ^2 F}{\partial V ^2} \right ) _T=\left ( \frac{\partial ^2 U}{\partial V ^2} \right ) _T-T\left ( \frac{\partial ^2 S}{\partial V ^2} \right ) _T##.
Now ##T=\left ( \frac{\partial U}{\partial S} \right ) _V## so that ##\left ( \frac{\partial ^2 F}{\partial V ^2} \right ) _T=\left ( \frac{\partial ^2 U}{\partial V ^2} \right ) _T-\left ( \frac{\partial U}{\partial S} \right ) _V \left ( \frac{\partial ^2 S}{\partial V ^2} \right ) _T##.
Now I multiply and divide by ##\left ( \frac{\partial ^2 U}{\partial S ^2} \right ) _T## to reach [tex]\left ( \frac{\partial ^2 F}{\partial V ^2} \right ) _T=\frac{\left ( \frac{\partial ^2 U}{\partial V ^2} \right ) _T\left ( \frac{\partial ^2 U}{\partial S ^2} \right ) _T- \left ( \frac{\partial U}{\partial S} \right ) _V \left ( \frac{\partial ^2 U}{\partial V ^2} \right ) _T}{\left ( \frac{\partial ^2 U}{\partial S ^2} \right ) _T}[/tex] so I've almost the right expression but not quite. I don't know how to proceed further. Nor do I know how to use the hint given, nor do I understand it.
Any help is appreciated. Thank you!