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

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Homework Help Overview

The discussion revolves around the thermodynamic treatment of thermal radiation as a gas of photons. The original poster presents a series of statements regarding entropy density, Gibbs function, and heat capacities, seeking to demonstrate specific relationships and values based on given equations.

Discussion Character

  • Exploratory, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • The original poster attempts to derive the heat capacity at constant volume (Cv) but encounters difficulties in reaching the expected result of 3s. They express a need for hints rather than complete solutions.
  • Some participants request to see the original poster's calculations to identify potential errors in their reasoning.
  • One participant suggests using a specific form for the internal energy to derive pressure and entropy in terms of temperature, indicating a possible path forward.
  • Another participant questions the feasibility of using only the provided results to reach the desired conclusions.

Discussion Status

The discussion is ongoing, with participants actively engaging in exploring the problem. Some guidance has been offered regarding potential approaches, but no consensus or resolution has been reached. The original poster continues to seek clarification and assistance.

Contextual Notes

The original poster is constrained by the need to work within the parameters of the problem statement and is looking for hints rather than complete solutions. There is an emphasis on deriving results based on provided equations and relationships.

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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 Cv = 3s per unit volume
(d) heat capacity at constant pressure Cp 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! :)
 
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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.
 
G01 said:
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
 
any ideas?
 
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:

[tex]u(T)=AT^4[/tex]

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.
 
Thanks but how can i get to this using only the results I've been given?
 
a-n-y-o-n-e?
 

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