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My textbook states that

The Zeroth law clearly suggests that when two systems A and B are in thermal equilibrium there must be physical quantity {later stated to be the temperature} that has the same value for both.

My question is why?

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  • $\begingroup$ Check out the following link.hyperphysics.phy-astr.gsu.edu/hbase/thermo/thereq.html $\endgroup$
    – Bob D
    Mar 5, 2019 at 14:16
  • $\begingroup$ According to my textbook the state of a system is an equilibrium state if the macroscopic variables that characterise the system do not change in time. $\endgroup$
    – Harsh Sahu
    Mar 5, 2019 at 14:27
  • $\begingroup$ Yes of course. But equilibrium can refer to various macroscopic properties (volume, pressure, temperature, mass, etc.). The Zeroth law pertains to thermal equilibrium. $\endgroup$
    – Bob D
    Mar 5, 2019 at 14:44
  • $\begingroup$ Here it is stated that the variables do not change with time not that they are the same then why temperatures are said to be the same? $\endgroup$
    – Harsh Sahu
    Mar 5, 2019 at 15:04
  • $\begingroup$ Did you look at the link? It explains that. $\endgroup$
    – Bob D
    Mar 5, 2019 at 15:08

1 Answer 1

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Mathematically, we would say that the zeroth law means that thermal equilibrium is a transitive relation. We already know that thermal equilibrium is reflexive (a system is in thermal equilibrium with itself) and symmetric (if A is in thermal equilibrium with B then B is in thermal equilibrium with A). These three properties tell us that thermal equilibrium is an "equivalence relation". This in turn means that it defines a "partition" of systems - we can divide the universe of systems into non-overlapping subsets called "equivalence classes" such that any pair of systems are in thermal equilibrium if and only if they are in the same equivalence class.

We then say that all the systems in a given equivalence class have a physical attribute in common and this physical attribute is what we call temperature.

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  • $\begingroup$ Sorry but I am not familiar with these mathematical terms. $\endgroup$
    – Harsh Sahu
    Mar 5, 2019 at 14:43
  • $\begingroup$ en.wikipedia.org/wiki/… $\endgroup$
    – gandalf61
    Mar 5, 2019 at 14:44
  • $\begingroup$ How does the zeroth law suggest this? $\endgroup$
    – Harsh Sahu
    Mar 5, 2019 at 14:59
  • $\begingroup$ @HarshSahu Did you read the Wikipedia article about equivalence relations ? Which part is not clear ? $\endgroup$
    – gandalf61
    Mar 5, 2019 at 15:11

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