## Defining temperature from thermal equilibrium

The zeroth law of thermodynamics is actually a definition of temperature. We see here how thermal equilibrium can be used to define temperature.

Work, pressure, internal energy, volume, temperature, heat, etc are some words we’ll repeatedly use throughout our understanding of thermodynamics. And we won’t remind you of their meaning every time they’re used in the description of the theory. Moreover, you’ll gain but a partial understanding of the concepts, until you’ve understood their meanings. So, it is very essential for you to first understand what each of these jargons mean precisely, before delving deeper into the game. In this article, we begin with defining **temperature** from the concept of thermal equilibrium.

# The zeroth law revisited

Earlier, we’d called temperature, pressure and volume; the state variables. And we’d seen towards the end, the remarkable yet commonsensical law – the Zeroth law of thermodynamics. This law simply says that *if you have two systems that are in thermal equilibrium with a third system, separately, then all the three systems are in thermal equilibrium with one another*. Now, assuming you have an intuitive grasp of what is meant by volume and pressure (if you don’t, we’ll define them soon), consider two systems A and B, having pressures and volumes, , and , respectively. If you bring the systems A and B in contact so that they are separated by a metal sheet (so the contents of A and B can’t pass through and exchange, but “heat” can), you’ll observe (in a general situation) that the values of , , , and will start changing.

To make things simpler, let us assume the two systems have initially fixed volumes and . Therefore, on contact between A and B, only the pressures and start changing. Now, if you let the two systems stay in contact for a while and *wait*, you’ll observe that there will come a time when and will stop changing – they will assume constant values. When this happens, we say that the systems A and B have attained** thermal equilibrium**.

Next, we take the systems apart (they’re no longer in contact). Leaving system A as it is, we change the pressure and volume of system B to new values , . What would you expect if we again brought the systems A and B in contact with one another (with the metal sheet separation)? The values of , , , will change until…thermal equilibrium is achieved. But we claim that if we had initially set the values of and to a special pair of values, nothing would happen when the systems are brought into contact. I.e., the systems A and B would already be in thermal equilibrium (there won’t be a need to wait for the systems to attain thermal equilibrium). Therefore, we say that the states and are in thermal equilibrium.

Now, repeating the above procedure, we don’t disturb the system A at all, but change the volume and pressure of system B to , . And we again claim to have made this change to a special set of values, so that and are directly in thermal equilibrium. We could go on repeating this procedure, and we obtain a series of values for the states of system B: , , ,… which are all in thermal equilibrium with the state of system A.

# What thermal equilibrium is

Therefore, from the statement of the zeroth law, we may conclude that the states , , , … are themselves in thermal equilibrium with one another. So, we may speak in terms of there existing some property that all these states have in common, which is leading to them being in mutual thermal equilibrium. We call this common property, the **temperature**.

Hence, we may conclude simply that two systems are said to be in thermal equilibrium with one another, if their temperatures are the same.

# Degree of hotness

Now, all of the above procedures may seem a little elaborate, just for defining the temperature. But that is the thermodynamical definition of temperature. For a more intuitive understanding, you could call temperature the *degree of hotness* of an object. The hotter an object is, larger is its temperature. So, what really is happening when you bring two systems with different general states and in contact with one another is, the two systems are initially at different temperatures and . If is greater than , during the attainment of thermal equilibrium, drops and rises. That is, the hotter body loses “heat” and becomes cooler, while the colder body receives the “heat” and becomes hotter. And this happens until both the bodies become of the same temperature.

But that wasn’t the case when you chose the state for system B. There was no change of pressure on contact with system A, because the values of and were chosen *wisely*, so that the corresponding temperature equaled the temperature of system A, . Similarly, all the following states were chosen so that the temperature of system B was always equal to that of system A and hence, the two systems were already in thermal equilibrium (even before contact).

So, the next time you’re holding a pair of ice cubes in your palms, remember that as you hold, the ice is gaining the warmth of your palms and so is melting, while your hands are losing heat to the ice cubes, and making your hands feel colder. And the ice will keep melting until it has attained a thermal equilibrium with your palms. (And by then it’d all have turned water.)