What is the Explanation for d=1/3 in the Thermodynamic Relationship?

[mrjohns]

Homework Statement

Consider relationship for a thermodynamic system:

S=A[UVN]^d , where A is a constant and d a real number.

I need to explain why d=1/3 is the only allowed value consistent with the postulates of thermodynamics.

The Attempt at a Solution

I'm having a hard time determining why this is the case from the postulates.

 

[clamtrox]

You should start by writing down the postulates you think could be relevant.

 

[berkeman]

Homework Statement

Consider relationship for a thermodynamic system:

S=A[UVN]^d , where A is a constant and d a real number.

I need to explain why d=1/3 is the only allowed value consistent with the postulates of thermodynamics.

The Attempt at a Solution

I'm having a hard time determining why this is the case from the postulates.

Please check your PMs. You must write out the relevant equations and show your attempt at solving this problem.

 

[Chestermiller]

This is a problem involving units. The units on both sides of the equation must match.

 

[mrjohns]

I don't have an attempt because I'm completely stumped.

The units on the left hand side are J/K, and on the right they are J^(1/3) m - which don't match.

I can't see anything in the postulates that helps either:

P1 - There exist equilibrium states characterised completely by U, V, N.
P2 - There exists a function of the macroscopic variables, the entropy, which is maximised when a constraint is removed
P3 - Entropy is additive over subsystems, and is a continuous and differentiable and increasing function of the total internal energy U

 

[vela]

Considering you don't know the units of A, it's pointless to match units between sides.

Can you give an example of what the third postulate means?

 

[Jorriss]

While you can't match units directly because of the A term you can still make progress by requiring that the entropy be extensive.

 

[clamtrox]

P3 - Entropy is additive over subsystems

This is important.

 

[Chestermiller]

U, V, N, and S are all extensive properties. So, if you double U, V, and N, what has to happen to S?

 

[mrjohns]

I know that for a constant:

S(kN,KV,KU)=kS(N,V,U)

But I'm not sure how that restricts the power to a third.

 

[Chestermiller]

Substitute kU, kV, and kN, and kS into your thermodynamic relationship, and see what you get.