The zeroth law of thermodynamics

[QuasarBoy543298]

let's assume I have 2 systems A and B. the surface that describes when the 2 systems are in equilibrium is given by F(a1,a1,...,b1,b2,...) = 0.
assuming we can write this surface as A(a1,a2,..)=B(b1,b2,...) why do A and B describes the temperature function of the systems?

in class, we defined the temperature of a system by the value of some coordinate c1 of some system c, when the other coordinates c2,... were determined, when the system is in equilibrium with c.
so for some system A ,I would get some function T1 = c1(a1,a2,...,c2,...) = c1(a1,a2,...)
and from the equilibrium with system B, I would get T2 = A(a1,a2,...)
how do I know T1 is the same as T2? it doesn't even seem to relate

the motive for asking this question was an exercise where some surface f=0 that describes equilibrium between A and B was given, and the goal was
to find the temperature functions of A and B only from the connection f=0.

 

[Christopher Grayce]

What you do know, from general thermodynamic principles, is that A and B must be intensive parameters (e.g. T or P for the simplest of systems). Which they are depends on what kind of equilibrium you have established. If you only allow A and B to exchange energy, then, yes, at equilibrium the temperatures must be equal and therefore the functions you list must be the equations of state of the temperature in A and B.

The proof that the temperature must be equal is not hard. Imagine an infinitismal transfer of energy dU from A to B. Write the differential change in the entropy dS as a function of dU, making use of the fact that dS/dU = 1/T and the First Law. Set dS = 0, as required by the system being at equilibrium, and you will find T_A = T_B is required.