6.5 About Temperature

The previous section discussed the wave function for a macroscopic system of bosons in its ground state. However, that is really a very theoretical exercise.

A macroscopic system of particles is only in its ground state at what is called absolute zero temperature. Absolute zero temperature is $\vphantom0\raisebox{1.5pt}{$-$}$273.15 $\POW9,{\circ}$C in degrees Celsius (Centigrade) or $\vphantom0\raisebox{1.5pt}{$-$}$459.67 $\POW9,{\circ}$F in degrees Fahrenheit. It is the coldest that a stable system could ever be.

Of course, you would hardly think something special was going on from the fact that it is $\vphantom0\raisebox{1.5pt}{$-$}$273.15 $\POW9,{\circ}$C or $\vphantom0\raisebox{1.5pt}{$-$}$459.67 $\POW9,{\circ}$F. That is why physicists have defined a more meaningful temperature scale than Centigrade or Fahrenheit; the Kelvin scale. The Kelvin scale takes absolute zero temperature to be 0 K, zero degrees Kelvin. A one degree temperature difference in Kelvin is still the same as in Centigrade. So 1 K is the same as $\vphantom0\raisebox{1.5pt}{$-$}$272.15 $\POW9,{\circ}$C; both are one degree above absolute zero. Normal ambient temperatures are near 300 K. More precisely, 300 K is equal to 27.15 $\POW9,{\circ}$C or 80.6 $\POW9,{\circ}$F.

A temperature measured from absolute zero, like a temperature expressed in Kelvin, is called an “absolute temperature.” Any theoretical computation that you do requires the use of absolute temperatures. (However, there are some empirical relations and tables that are mistakenly phrased in terms of Celsius or Fahrenheit instead of in Kelvin.)

Absolute zero temperature is impossible to achieve experimentally. Even getting close to it is very difficult. Therefore, real macroscopic systems, even very cold ones, have an energy noticeably higher than their ground state. So they have a temperature above absolute zero.

But what exactly is that temperature? Consider the classical picture of a substance, in which the molecules that it consists of are in constant chaotic thermal motion. Temperature is often described as a measure of the translational kinetic energy of this chaotic motion. The higher the temperature, the larger the thermal motion. In particular, classical statistical physics would say that the average thermal kinetic energy per particle is equal to $\frac32{k_{\rm B}}T$, with $k_{\rm B}$ $\vphantom0\raisebox{1.5pt}{$=$}$ 1.38 10$\POW9,{-23}$ J/K the Boltzmann constant and $T$ the absolute temperature in degrees Kelvin.

Unfortunately, this story is only true for the translational kinetic energy of the molecules in an ideal gas. For any other kind of substance, or any other kind of kinetic energy, the quantum effects are much too large to be ignored. Consider, for example, that the electron in a hydrogen atom has 13.6 eV worth of kinetic energy even at absolute zero temperature. (The binding energy also happens to be 13.6 eV, {A.17}, even though physically it is not the same thing.) Classically that kinetic energy would correspond to a gigantic temperature of about 100,000 K. Not to 0 K. More generally, the Heisenberg uncertainty principle says that particles that are in any way confined must have kinetic energy even in the ground state. Only for an ideal gas is the containing box big enough that it does not make a difference. Even then that is only true for the translational degrees of freedom of the ideal gas molecules. Don’t look at their electrons or rotational or vibrational motion.

The truth is that temperature is not a measure of kinetic energy. Instead the temperature of a system is a measure of its capability to transfer thermal energy to other systems. By definition, if two systems have the same temperature, neither is able to transfer net thermal energy to the other. It is said that the two systems are in thermal equilibrium with each other. If however one system is hotter than the other, then if they are put in thermal contact, energy will flow from the hotter system to the colder one. That will continue until the temperatures become equal. Transferred thermal energy is referred to as “heat,” so it is said that heat flows from the hotter system to the colder.

The simplest example is for systems in their ground state. If two systems in their ground state are brought together, no heat will transfer between them. By definition the ground state is the state of lowest possible energy. Therefore neither system has any spare energy available to transfer to the other system. It follows that all systems in their ground state have the same temperature. This temperature is simply defined to be absolute zero temperature, 0 K. Systems at absolute zero have zero capability of transferring heat to other systems.

Systems not in their ground state are not at zero temperature. Besides that, basically all that can be said is that they still have the same temperature as any other system that they are in thermal equilibrium with. But of course, this only defines equality of temperatures. It does not say what the value of that temperature is.

For identification and computational purposes, you would like to have a specific numerical value for the temperature of a given system. To get it, look at an ideal gas that the system is in thermal equilibrium with. A numerical value of the temperature can simply be defined by demanding that the average translational kinetic energy of the ideal gas molecules is equal to $\frac32{k_{\rm B}}T$, where $k_{\rm B}$ is the Boltzmann constant, 1.380,65 10$\POW9,{-23}$ J/K. That kinetic energy can be deduced from such easily measurable quantities as the pressure, volume, and mass of the ideal gas.


Key Points
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A macroscopic system is in its ground state if the absolute temperature is zero.

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Absolute zero temperature means 0 K (Kelvin), which is equal to $\vphantom0\raisebox{1.5pt}{$-$}$273.15 $\POW9,{\circ}$C (Centigrade) or $\vphantom0\raisebox{1.5pt}{$-$}$459.67 $\POW9,{\circ}$F (Fahrenheit).

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Absolute zero temperature can never be fully achieved.

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If the temperature is greater than absolute zero, the system will have an energy greater than that of the ground state.

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Temperature is not a measure of the thermal kinetic energy of a system, except under very limited conditions in which there are no quantum effects.

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Instead the defining property of temperature is that it is the same for systems that are in thermal equilibrium with each other.

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For systems that are not in their ground state, a numerical value for their temperature can be defined using an ideal gas at the same temperature.