Thermodynamics - Maxwell Reactions

[Philistine]

We have been asked the following: "For fixed compositions, show that T = (partial U/partial S)_V = (partial H/partial S)_P". There are 7 other equations after this one, but I think once I know how to solve the first one I will be ok with the rest.

We have not been provided with any relevant equations. My knowledge of partial differential equations is a bit rusty, but I have been doing some reading today to brush up. I initially tried to differentiate H = U + PV and substitute that in, but I ended up with an extra Pdv/S, which with constant P I am not sure how to cancel out. I suspect it has something to do with C_v = T(ds/dt)_v and C_p = T(dt/ds)_p but am not sure how to apply them or how to convert from equations at constant volume to equations at constant pressure.

I have ordered Classical Thermodynamics by Van Ness and Abbott at the recommendation of our professor, but it won't arrive until next week. The homework isn't due in for another 2 weeks, but if anyone could help me get started, I would really appreciate it. Maybe just some basic pointers so that I had an idea of what path to follow.

Thanks in advance.

 

[stewartcs]

We have been asked the following: "For fixed compositions, show that T = (partial U/partial S)_V = (partial H/partial S)_P". There are 7 other equations after this one, but I think once I know how to solve the first one I will be ok with the rest.

We have not been provided with any relevant equations. My knowledge of partial differential equations is a bit rusty, but I have been doing some reading today to brush up. I initially tried to differentiate H = U + PV and substitute that in, but I ended up with an extra Pdv/S, which with constant P I am not sure how to cancel out. I suspect it has something to do with C_v = T(ds/dt)_v and C_p = T(dt/ds)_p but am not sure how to apply them or how to convert from equations at constant volume to equations at constant pressure.

I have ordered Classical Thermodynamics by Van Ness and Abbott at the recommendation of our professor, but it won't arrive until next week. The homework isn't due in for another 2 weeks, but if anyone could help me get started, I would really appreciate it. Maybe just some basic pointers so that I had an idea of what path to follow.

Thanks in advance.

http://en.wikipedia.org/wiki/Maxwell_relations

CS

 

[Ygggdrasil]

For most thermodynamics problems of these sorts, you can start with one of these basic equations and go from there:

dU = T dS - P dV
dH = T dS + V dP
dA = -S dT - P dV
dG = -S dT + V dP

For example, with the first, you can start with the differential for U. Since you want the partial derivative where V is constant, dV = 0. You should be able to take it from there.

 

[Philistine]

Thanks!

So:

dU = T dS - PdV
As V is constant, dV = 0. Hence, - PdV = 0
dU = T dS
dU/dS = T

dH = T dS - V dP
As P is constant, dP = 0. Hence, - V dP = 0
dH = T dS
dH/dS = T

Hence
T = dU/dS @ constant V = dH/dS @ constant P

?

Also, when you use "d" are you referring to partial differential or ordinary differential? If the former, does it matter that I need to find the partial differential?

 

[Ygggdrasil]

The four equations have the ordinary differential. However, by fixing P or V, you convert the ordinary differential into a partial differential (since a partial derivative is an ordinary derivative with one or more of the other variables treated as constant).