Derivation from the 0th Law of Thermodynamics

[PhiJ]

Homework Statement

I really don't know if I'm in the right subforum...
I've started reading this text on statistical mechanics from MIT, but I'm stuck on page 2. Here's the statement:

Let the equilibrium state of systems A, B, and C be described by the coordinates
{A₁, A₂,· · ·}, {B₁, B₂,· · ·}, and {C₁, C₂,· · ·} respectively. The assumption that A and C are in equilibrium implies a constraint between the coordinates of A and C, i.e. a change in A₁ must be accompanied by some changes in {A₂,···;C₁, C₂,···} to maintain equilibrium of A and C. Denote this constraint by

f_AC(A₁, A₂,···;C₁, C₂,···) = 0. (I.1)

The equilibrium of B and C implies a similar constraint
f_BC(B₁, B₂,···;C₁, C₂,···) = 0. (I.2)

Each of the above equations can be solved for
C₁ to yield
C₁ = F_AC(A₁, A₂,···; C₂,···) (I.3a)
C₁ = F_BC(B₁, B₂,···; C₂,···) (I.3b)​

Thus if C is separately in equilibrium with A and B we must have
F_AC(A₁, A₂,···; C₂,···) = F_BC(B₁, B₂,···; C₂,···) (I.4)

However, according to the zeroth law there is also equilibrium between A and B, implying
the constraint
f_AB(A₁, A₂,···;B₁, B₂,···) = 0. (I.5)
Therefore it must be possible to simplify eq.(I.4) by cancelling the coordinates of C.​

Homework Equations

(I.1) through (I.5) above.

The Attempt at a Solution

I understand up to I.2. For I.3 I'm going to assume there is a unique solution - I'm sure the solution isn't always unique, but it means their argument is correct and I don't have to play mind twister to agree. So given that assumption I'm happy with I.3. I.4 is then obviously true, and I.5 is obviously true by the zeroth law.

I suppose 'cancel the Cs' means substitute some constants into the values for C_X, so define:

F_A(A₁, A₂,···) = F_AC(A₁, A₂,···; c₂,···)
F_B(B₁, B₂,···) = F_BC(B₁, B₂,···; c₂,···)
where c_x is a constant.

But then:

F_A(A₁, A₂,···) - F_B(B₁, B₂,···) = 0

And we can define f_AB = F_A - F_B

i.e. we already have a constraint in terms of A and B. Therefore the zeroth law is not a foundational assumption, it's something you can prove. I'm sure I'm wrong on that count, but I can't see why. I realized I've made an assumption to accept I.3, but is I.3 wrong? It's stated in the text.

 

[DrClaude]

F_A(A₁, A₂,···) - F_B(B₁, B₂,···) = 0

And we can define f_AB = F_A - F_B

i.e. we already have a constraint in terms of A and B. Therefore the zeroth law is not a foundational assumption, it's something you can prove.

You arrived at that equation because you could cancel out the coordinates of C, which was possible by invoking the 0th law. Proving the 0th law using this would be circular reasoning.

 

[PhiJ]

I think I 'cancelled out' the coordinates of C without using the zeroth law.

Substituting constants into the equation is standard mathematical practise. That gives me:

F_AC(A₁, A₂,···; c₂,···) = F_BC(B₁, B₂,···; c₂,···)

Then I defined two functions:

F_A(A₁, A₂,···) = F_AC(A₁, A₂,···; c₂,···)
F_B(B₁, B₂,···) = F_BC(B₁, B₂,···; c₂,···)
where c_x is a constant.

then substituted those defined functions in:

F_A(A₁, A₂,···) = F_B(B₁, B₂,···)

Then subtracted:

F_A(A₁, A₂,···) - F_B(B₁, B₂,···) = 0

Then defined again f_AB = F_A - F_B

And I'm done. Those steps require the normal rules for functions and arithmetic, and that there is a solution to f_AC and f_BC. The existence of a solution is implied by the question: if A and C are in equilibrium, then there is a constraint between A and C, but the implication is that this is a constraint with a solution. If A and C are in equilibrium, and B and C are also in equilibrium, then there are two constraints, and the implication is that there is a solution that fits both constraints.

I suppose without the zeroth law, we'd still have the constraint, but it wouldn't be a constraint that defines a thermal equilibrium between A and B?

 

[DrClaude]

Then I defined two functions:

F_A(A₁, A₂,···) = F_AC(A₁, A₂,···; c₂,···)
F_B(B₁, B₂,···) = F_BC(B₁, B₂,···; c₂,···)
where c_x is a constant.

I think that the crux is that this should work for any c_x, not a particular one.

 

[PhiJ]

I think that the crux is that this should work for any c_x, not a particular one.

Of course! F_A - F_B has to have the same trace, whichever c you pick, or your constraint is only true under certain conditions. I think I was getting confused by the concept of cancelling variables within an arbitrary function, because it's obvious in hindsight.

Thanks for the help.