What is Temperature? The Zeroth Law of Thermodynamics

Physical chemistry can be complicated and hard to understand. Fully grasping canonical partition functions, Gibbs samplers, or quantum mechanical anharmonic oscillators, can take years. These are very detailed, interesting subjects, but their direct application to the understanding of all, my ultimate goal for Free Range Science, can unfortunately be extremely difficult.

However, reaching back into the 19th century, chemists and physicists developed a set of theories for understanding some aspects of the everyday: thermodynamics. From a 21st-century perspective, the early thermodynamicists seemed obsessed with steam engines--understandably, because most of scientific progress reflects the desire to improve the technological innovations of the day. But what they leared about engines, which are devices that turn heat flow into work, has consequences for a myriad of seemingly different topics. Today, we use thermodynamics less to understand engines, and more to understand modern issues such as how tightly a drug molecule binds to a protein molecule, how much light energy is required to drive a photocatalyst, and a myriad of other issues. (Although these days the discussion focuses more upon the microscopic manifestation of thermodynamics, called statistical mechanics.)

Thermodynamics deals with everyday objects as well as extraordinary ones, and it can answer a question that has to do with almost everything there is: what is temperature?

Everyone is familiar with the use of a thermometer to test the temperature of an object. Simply, the thermometer is placed in suitable contact with the object, then we wait a while, and after some time the thermometer registers what we call the "temperature" of the object.

In order to understand what measuring a temperature really means, we will make an analogy. At a very deep level, the analogy is exact, since the way we understand, mathematically, what happens when we try to measure temperature with a thermometer exactly the same way we understand the process I'm about to introduce: the stretching of rubber bands.

Imagine two rubber bands hooked, each hooked over its own nail. Both nails are pounded partway into a board. Then we stretch the rubber bands until they meet, and we tie them together or to some connector.

We should be careful to state some conditions that we think might matter in the situation at hand, since if they don't hold the analogy fails. Let's assume that the rubber bands are strong enough so they won't break, we've attached the ends together so that they won't slip apart, and that the nails are firmly implanted in the board. If there is breakage or slippage then the way we understand this situation will be much less helpful to our ultimate goal, to make an analogy to temperature and heat flow.

The next imporant concept is of equilibrium, the state in which nothing changes. In the rubber band situation, this is called mechanical equilibrium, where all of the forces balance out. Because they're stretched out and tied to one another, the rubber bands exert forces on one another. In the situation where the knot is not moving, the system is at mechanical equilibrium; each rubber band pulls on the other one, but the force each rubber band exerts on the other is the same.

If the forces didn't balance out, such as the situation where we pull the knot towards one nail or the other and let it go, then there would be motion. If we let the motion of the system evolve, then eventually it would find the state of mechanical equilibrium. While not in equilibrium, one rubber band is pulling harder than the other one. This leads to motion in the form of the overstretched band shrinking some, whereas the understretched rubber band extends a little. Since the distance between the nails is fixed, this can be described as the rubber bands exchanging lengths with one another: if the length of one rubber band decreases by 1 cm, then the length of the other increases by 1 cm.

It should be obvious that the exchange of length doesn't necessarily occur until both rubber bands are the same length; what if one were just shorter than the other? On the contrary, the exchange of length occurs until the forces balance out--until mechanical equilibrium is reached.

A similar process occurs when we bring a thermometer into contact with an object--a little kid, the outside air, or a pot of boiling sugar--in order to say something about its temperature.

Temperature is like the forces in the rubber band situation: something will be exchanged between two objects of different temperatures just as length will be exchanged between two rubber bands. This "something" will be exchanged until the temperatures of the two objects are the same, just as in the rubber band case length is exchanged until the forces are equal. Once the exchange is complete, for temperatures, the objects will have reached thermal equilibrium. The thing exchanged is often called "entropy," but it is useful to think of probability being exchanged instead. Probability and entropy are deeply related, but probability is much simpler to understand, physical scientist or not.

The probabilities in question are those for the different possible arrangements of particles and energy in the two objects. There are huge numbers of states for particles to arrange themselves in. A good analogy is a bag of rice; there are thousands of rice grains, and each one can be oriented in any direction. The rice grains also "interact" with each other; since they're solid, they can't overlap, which limits the number of orientations of any rice grain, much like microscopic particles in a very dense system. Opening a bag of rice reveals one of the possible states that the system can be in. Not all arrangements of rice grains are equally probable: for instance, a jumbled-up arrangement is much more likely than an arrangement where all of the rice grains are pointing in the same direction.

The analogy of probability flowing to the exchange of length in the rubber band situation is even deeper. Like the total length of the rubber bands, which is a constant, the probability has a fixed total value for two objects brought into contact--it has to add up to 1, or 100 %. Each of the possible arrangements of particles and energy in each system has some probability; the probability that the particles and energy are arranged in some way is 1. How the probability is partitioned among all of the immense number of arrangements--how some states are more probable than others--depends upon the details of how the particles interact, which can get very complicated, and upon the temperature.

When two objects at two different temperatures are not in contact, they tend to spontaneously arrange the matter and energy that constitute them into the most probable ways. This spontaneous arrangement is summarized by the Second Law of Thermodynamics, which says that entropy cannot decrease in an isolated object, but it really just means, quite trivially, that one is more likely to find matter and energy arranged in probable ways than in improbable ways.

However, when two objects of different temperatures, themselves having attained their most probable arrangements, are brought into contact, the two objects considered together are no longer in a probable arrangement, even though they were in their most probable states before being brought into contact. There is more energy per particle in the hotter object than in the colder object, so the particles of the hotter object will tend to transfer their energy to the particles of the colder object. (This happens because the objects are in contact, so it happens where the objects are in contact. Surface area matters--the energy transfer will be faster if the contact surface is larger.) As the two objects exchange energy, the arrangement spontaneously evolves until it's in a more probable state. In consequence, the originally hotter object assumes a less probable state, from its point of view, in order that the originally cooler object can assume, from its point of view, a more probable state. This is all done in such a way that, considering the probabilities of different arrangements of both objects together, the ultimate arrangement--both objects the same temperature--is more probable than the initial state with different temperatures.

So temperature is the force that causes probability to flow, until thermal equilibrium is reached, when the temperature-forces balance. But this really only explains thermal equilibrium, rather than temperature. To finally understand temperature, however, we consider what is called the Zeroth Law of thermodynamics, which states that if object A is in thermal equilibrium with object B, and that C is in thermal equilibrium with B, then A and C are also in thermal equilibrium with each other. This might sound trivially silly, almost as silly as calling something "zeroth," but the reason for the law was one of logical consistency; thermodynamicists realized that the mathematical structure of thermodynamic theory, already well-developed, required an assumption that is logically prior to the other three laws in order to make the theory more consistent. This is also why it's called the "zeroth" law: the first, second, and third laws had already been invented.

(We have now seen the zeroth, first, and second laws of thermodynamics. The third law is not very interesting.)

We call an object like object B, above, a thermometer. We can look at the behavior of the thermometer as it interacts with objects of different temperature, and use our results to define a numerical scale. Put into contact with A, until thermal equilibrium is reached, we obtain a numerical value of the temperature of A. If we repeat with object C and we find that the numerical value of the temperature is the same, then we say A and C are the same temperature (obvious, right?), without actually having to put them into contact to see if any heat flows. This is effectively a prediction that if we put A and C into contact, that no heat would be exchanged--they would already be in their most probable states, without having to exchange any probability.