# Thermodynamic Postulates

## 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.

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

berkeman
Mentor

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

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
Staff Emeritus
Homework Helper
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?

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

P3 - Entropy is additive over subsystems

This is important.

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

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