# Zeroth Law of Thermodynamics, temperature, and ordering

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?

• If you want to derive the physical temperature scale, you also need the second and third law. Aug 21, 2014 at 1:27
• "Heat cannot of itself pass from one body to a hotter body". Flanders and Swann, 1964 Aug 21, 2014 at 2:40

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

the entropy of a closed system never decreases

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

• This appears a bit circular without independently defining what is "gaining" or "losing" heat, which involves the concept of "internal energy" and some form of Carnot engines (to establish heat/work conversion). Jan 27, 2020 at 22:14

The rational behind the definition of temperature is exactly what is stated in the 0th law of thermodynamics.

As such:

Since objects into contact will reach a thermal equilibrium (and heat transfers from hotter to colder), this means one can use a device that can measure this difference of this quantity that changed as follows:

We have a device (called a thermometer) which has an indication. When the "thermometer device" comes into contact with another object, 0th law states that they wil reach thermal equilibrioum (which is an equivalence relation as you stated). Then the indication of the device will change to reflect the difference of this quantity that changed or transfered (refered to as temperature).

One such device is the thermometer based on mercury (which is used as the indicator due to its special heat expansion factor).

As such, the measurements of temperature are relative, not absolute. So this is the (relative) order the question is about. One can order objects by temperature using the difference measured with the same thermometer under same conditions. And this becomes an ordering relation. This is possible of course because thermal equilibrium is possible and thus a thermometer is possible, which provides the scale, on which finaly this relative ordering is possible.

• Heat flows from hotter to colder - that provides the ordering. Aug 21, 2014 at 2:41
• @Floris, yes, but heat is not temperature, which is the content of the 0th law Aug 21, 2014 at 17:36
• Right - you need more than the 0th law to get the ordering. The zeroth law provides the existence of temperature; the second law tells you how to order them. Aug 21, 2014 at 18:10
• @Floris, no i dont think, and according to this answer, no. The possibility of thermal equilibrium makes a thermometer possible , that's all there is to it. Aug 21, 2014 at 18:16
• I disagree. Zeroth law: if A is in equilibrium with B and A is in equilibrium with C, then B is in equilibrium with C. This does NOT tell me what happens when they are NOT in equilibrium. For that you need the second law - which you slip in when you say "and heat transfers from hotter to colder". The zeroth law alone says NOTHING about systems that are not in equilibrium and thus does NOT provide the ordering. Aug 21, 2014 at 18:20

To have an equivalence relation ~ on a set $X$ you need three things: reflexivity, symmetry, and transitivity:

Reflexivity - for any $x \in X$, $x$~$x$

Symmetry - for any $x, y \in X$, if $x$~$y$, then $y$~$x$

Transitivity - for any $x, y, z \in X$, if $x$~$y$, and $y$~$z$, then $x$~$z$

Do do this right, we need to first understand that thermal equilibrium is said to be established when two bodies in contact do not change over time. This is the order relation you were wondering about in disguise, but more on that in a little bit

Now, the Zeroeth Law can be stated as follows (taken from the Wikipedia page on the topic):

If a body A, be in thermal equilibrium with two other bodies, B and C, then B and C are in thermal equilibrium with one another.

This is transitivity. The other two requirements for a equivalence relation are taken care of pretty much a priori. A body is always going to be in thermal equilibrium with itself (reflexivity) - it cannot just spontaneously change over time for no reason that would violate energy conservation - and if a body is in thermal equilibrium with another body, then obviously that other body is in thermal equilibrium with the first (symmetry) - thermal equilibrium only makes sense in the case of two bodies in contact both unchanged over time.

Now, that we have the equivalence relation for temperature, order is fairly easy. If two bodies come in to contact, and they do change over time: (1) They are not in thermal equilibrium but you could define each their temperatures separately based on what other bodies they are in thermal equilibrium with, and (2) when the two are brought together, the body which gains energy is said to be at a lower temperature and the body which loses energy is said to be at a higher temperature. "Heat" is just a name the exchange of energy when two bodies at different temperatures are in contact.

This is the order you were wondering about