Understanding Thermal Radiation: Entropy, Gibbs Function, and Heat Capacity

[bon]

Homework Statement

Thermal radiation can be treated thermodynamically as a gas of photons with internal energy U = u(T) V and pressure p = u(T)/3 where u(T) is the energy density. Show that

(a) entropy density s is given by s = 4p/T
(b) Gibbs function = 0
(c) heat capacity at constant volume C_v = 3s per unit volume
(d) heat capacity at constant pressure C_p is infinite.

Homework Equations

The Attempt at a Solution

I've done (a), (b), but am having trouble on (c). I guess I am trying to find T(Ds/DT) where capital D is partial.. but I am having touble arriving at 3s..

any hints? Thanks! :)

 

[G01]

Can I see your work up to this point for c)? I can't figure out where your going wrong unless I can see your calculation.

 

[bon]

Can I see your work up to this point for c)? I can't figure out where your going wrong unless I can see your calculation.

Ok sure so for (c) I need Cv

Cv = T (DS/DT)v (i.e. at constant volume)

I guess I am trying to work out Cv/V i.e. heat capacity per unit volume

Now since V is constant and S = sV

I can write Cv = T(Ds/DT)v

I just can't seem to see where to go from here..whatever i try i can't get 3s out... :S

 

[bon]

any ideas?

 

[G01]

OK. Here's something to get you started:

I don't know the constants off the top of my head, but let's just call them all A:

$$u(T)=AT^4$$

Using this, what is p in terms of only T? Then, what is s in terms of only T?

Take the derivative of this result and plug it into the equation for Cv. You should then be able to do some algebra to show it is equal to 3 times the result you got for s in terms of T.

 

[bon]

Thanks but how can i get to this using only the results I've been given?

 

[bon]

a-n-y-o-n-e?