11.1 Temperature

This book frequently uses the word temperature, but what does that really mean? It is often said that temperature is some measure of the kinetic energy of the molecules, but that is a dubious statement. It is OK for a thin noble gas, where the kinetic energy per atom is $\frac32{k_{\rm B}}T$ with $k_{\rm B}$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1.380,65 10$\POW9,{-23}$ J/K the Boltzmann constant and $T$ the (absolute) temperature in degrees Kelvin. But the valence electrons in an metal typically have kinetic energies many times greater than $\frac32{k_{\rm B}}T$. And when the absolute temperature becomes zero, the kinetic energy of a system of particles does not normally become zero, since the uncertainty principle does not allow that.

In reality, the temperature of a system is not a measure of its thermal kinetic energy, but of its hotness. So, to understand temperature, you first have to understand hotness. A system A is hotter than a system B, (and B is colder than A,) if heat energy flows from A to B if they are brought into thermal contact. If no heat flows, A and B are equally hot. Temperature is a numerical value defined so that, if two systems A and B are equally hot, they have the same value for the temperature.

The so-called “zeroth law of thermodynamics” ensures that this definition makes sense. It says that if systems A and B have the same temperature, and systems B and C have the same temperature, then systems A and C have the same temperature. Otherwise system B would have two temperatures: A and C would have different temperatures, and B would have the same temperature as each of them.

The systems are supposed to be in thermal equilibrium. For example, a solid chunk of matter that is hotter on its inside than its outside simply does not have a (single) temperature, so there is no point in talking about it.

The requirement that systems that are equally hot must have the same value of the temperature does not say anything about what that value must be. Definitions of the actual values have historically varied. A good one is to compute the temperature of a system A using an ideal gas B at equal temperature as system A. Then $\frac32{k_{\rm B}}T$ can simply be defined to be the mean translational kinetic energy of the molecules of ideal gas B. That kinetic energy, in turn, can be computed from the pressure and density of the gas. With this definition of the temperature scale, the temperature is zero in the ground state of ideal gas B. The reason is that a highly accurate ideal gas means very few atoms or molecules in a very roomy box. With the vast uncertainty in position that the roomy box provides to the ground-state, the uncertainty-demanded kinetic energy is vanishingly small. So ${k_{\rm B}}T$ will be zero.

It then follows that all ground states are at absolute zero temperature, regardless how large their kinetic energy. The reason is that all ground states must have the same temperature: if two systems in their ground states are brought in thermal contact, no heat can flow: neither ground state can sacrifice any more energy, the ground state energy cannot be reduced.

However, the ideal gas thermometer is limited by the fact that the temperatures it can describe must be positive. There are some unstable systems that in a technical and approximate, but meaningful, sense have negative absolute temperatures [4]. Unlike what you might expect, (aren’t negative numbers less than positive ones?) such systems are hotter than any normal system. Systems of negative temperature will give off heat regardless of how searingly hot the normal system that they are in contact with is.

In this chapter a definition of temperature scale will be given based on the quantum treatment. Various equivalent definitions will pop up. Eventually, section 11.14.4 will establish it is the same as the ideal gas temperature scale.

You might wonder why the laws of thermodynamics are numbered from zero. The reason is historical; the first, second, and third laws were already firmly established before in the early twentieth century it was belatedly recognized that an explicit statement of the zeroth law was really needed. If you are already familiar with the second law, you might think it implies the zeroth, but things are not quite that simple.

What about these other laws? The “first law of thermodynamics” is simply stolen from general physics; it states that energy is conserved. The second and third laws will be described in sections 11.8 through 11.10.