[TA1LGUNN3R]

Okay, I've worked through a handful of proofs, and have a few problems with a couple of 'em.  I'll ask about one at a time but keep them in this thread.

For my first question, I have to prove:

[tex](\frac {\partial S}{\partial T})_V= \frac {C_V}{T}[/tex]

I started by rearranging via cyclic rule:

[tex](\frac {\partial S}{\partial T})_V=-(\frac {\partial S}{\partial V})_T(\frac {\partial V}{\partial T})_S[/tex]

Then, I subbed in for for dS=(dU+PdV)/T, which gives:

[tex](-(\frac {\partial U}{T\partial V})_T+\frac {P}{T})(\frac {\partial V}{\partial T})_S[/tex]

Using cyclic rule again for (dU/dV)_T gives:

[tex](\frac {1}{T}(\frac {\partial U}{\partial T})_V(\frac {\partial T}{\partial V})_U+\frac {P}{T})(\frac {\partial V}{\partial T})_S= \frac {C_V}{T}+\frac {P}{T}(\frac {\partial V}{\partial T})_S[/tex]

Here's where I'm stuck.  Somehow the (P/T)(dV/dT)_S should go to zero.  I've tried subbing in R/V for P/T and cyclic rule rearrangements and Maxwell relations.  Not sure where to go.

Thanks for any assistance,
TG

[mjc123]

Why not omit the first step and substitute straight away?
(dS/dt)_V = 1/T{(dU/dT)_V + P(dV/dT)_V} = C_V/T as the second term is self-evidently zero.
You're tying yourself in knots in your last equation. IINM (dT/dV)_U = 0 (U defines T), while (dV/dT)_S = C_V/P

[TA1LGUNN3R]

Why not omit the first step and substitute straight away?
(dS/dt)_V = 1/T{(dU/dT)_V + P(dV/dT)_V} = C_V/T as the second term is self-evidently zero.
You're tying yourself in knots in your last equation. IINM (dT/dV)_U = 0 (U defines T), while (dV/dT)_S = C_V/P

I don't know why I put in the first step.  Probably because the previous few started out with cyclic rule.

Okay.  I thought (dV/dT)_S might be zero, but I didn't see it in the book anywhere.  Thank you.

-TG

[TA1LGUNN3R]

Alright, with this one, I'm very close, but I'm not sure a term can just cancel out (it should go to zero, but I'm getting an undefined term).

I mus prove [itex]C_P(\kappa_T-\kappa_S)=TV\alpha_P^2[/itex]

Here's how I worked it out:

[tex]C_P(\kappa_T-\kappa_S)=T(\frac {\partial S}{\partial T}_P)(-\frac {1}{V})[(\frac {\partial V}{\partial P})_T-(\frac {\partial V}{\partial P})_S)][/tex]

Doing some cyclic rule, I get:

[tex]T(\frac {\partial S}{\partial T}_P)(-\frac {1}{V})[(\frac {\partial V}{\partial P})_T-(\frac {\partial V}{\partial P})_S)]=\frac {T}{V}(\frac {\partial S}{\partial T})_P(\frac {\partial V}{\partial T})_P[(\frac {\partial T}{\partial P})_V+(\frac {\partial V}{\partial S})_P][/tex]

Now, if I multiply the (dS/dT)_P term through the part in brackets, I get a nice (dV/dT)_P and the confusing (dT/dP)_V(dS/dT)_P.  Does this simplify to (dS/dP)_P,T, and if it does, wouldn't this be undefined (divided by zero)?  Does this mess up the equation?  Because otherwise I just have to multiply by V/V to get my final answer.

Thanks,
TG

[msk034]

In fact (dV/dT)s can not be zero, it is temperature change at adiabatic volume contraction.