What would this be?
I believe that the law states that if a thermodynamic system A is in thermal equilibrium with another system C, and a thermodynamic system B is also in thermal equilibrium with C, then A and B are in thermal equilibrium. This applies even if A and B are separated by an adiabatic barrier (one through which NO heat can flow)
It essentially says that there is a useful measure we call "temperature". IOW, the numbers make sense; If A is warmer than B and B is warmer than C, then A is warmer than C.
It sounds like basic math, if A is greater than B and B is greater than C than A is greater than C.
From a University Physics Book (Young and Freedman):
Again, if C=A=B, A=B
True, but remember that "temperature" is some real property.
This means, that the physicist must explicitly assume that it is meaningful to measure as a number this real property.
There exist other properties in the world which it is doubtful can ever be quantified, for example, "kindness".
A physicist who wish to develop a mathematical model of the world, is limited in talking about structures of the world which is "mathematizable".
If we have two thermodynamic systems which are in thermal equilibrium, there is a function of their state variables which has the same value in both systems. That function is called temperature.
So the Zeroth principle says: Let have three thermodynamic systems, we say that if A is in thermal equilibrium with C, and also with B, C and B are at thermal equilibrium.
There are several things that make this actually non-trivial:
1. Remember that you are equating the temperature parameter. There's nothing that says that the heat content of these A,B, and C are the same. They each could have a different heat capacity, mass, etc., but yet, all at the same temperatures. So establishing what we mean by "thermal equilibrium" is crucial. It means that only temperature is the valid parameter that is relevant, not heat content, etc, etc.
2. That equality will not be true if the 2nd Law is violated. So this is indirectly an entropy manifestation. I have seen many quacks who implicitly make use of the thermal equilibrium principle and yet, attack the validity of the 2nd law.
3. The 0th law is the very reason why, when metals and semiconductors of various kinds come in contact with each other, everything is scalled according to the location of the Fermi energy (or the chemical potential for semiconductors to be exact).
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