In my thermodynamics course (and in other places on the internet) it is asserted that the Zeroth Law of Thermodynamics can be used to define the concept of temperature. One statement of the Zeroth Law I have seen states that the relation thermal equilibrium on two closed systems brought into diathermal contact defined is in fact an equivalence relation.
The argument continues by saying that if we call the equivalence classes so defined isotherms, then we can assign arbitrary numbers to these isotherms and these numbers are what we call temperature.
We now have a set of numbers assigned to a set of equivalence relations. But what I don't see is how this numerical assignment bears any relation to physical temperature. Where is order defined? For example, if one of the classes gets the number "1", another gets the number "2", and a third "3", what in the above derivation shows that the isotherm "2" comes between the isotherm "1" and the isotherm "3"?
Maybe more is needed than just the Zeroth Law. If so, what is necessary in order to complete the argument?
Best Answer
The zeroth law posits the existence of temperature by stating that if A is in equilibrium with B and A is in equilibrium with C, then B is in equilibrium with C. We can then assign an intensive property to A, B and C that we call "temperature". They are in equilibrium == they have the same temperature.
As soon as they are NOT in equilibrium, the zeroth law is silent. Thus, as you observed in your question, we cannot derive an ordering of temperatures based on the zeroth law alone.
Here, the second law comes to the rescue. The formulation I am familiar with states
If we have two objects that are not in thermal equilibrium, then when we bring them into contact we expect heat to flow between them. Now according to the second law, if we move heat $\delta q$ from $A$ to $B$ (at temperatures $T_B$ and $T_B$ respectively), the change in entropy is
$$\delta S = T_A \delta q - T_B \delta q\\ = \delta q (T_A - T_B)$$
Now if the entropy of the system cannot decrease, then if $\delta q$ is positive we know that $T_A - T_B$ must be positive.
This is where we find the ordering of temperature: heat travels from hotter to cooler until thermal equilibrium is reached. Thus when we have two objects in unequal states we can tell which is hotter by looking at the direction in which heat flows between them. That direction is always from hotter to colder - and to prove this you need the second law.
There is an amusing (although somewhat dated - 50 years old this year) song by the duo of Flanders and Swann that touches on this topic. See http://youtu.be/VnbiVw_1FNs