# Why aren't the units for temperature related to kinetic energy, if temperature is defined as average kinetic energy?

Temperature, as far as I know, is a measure of the average kinetic energy of the particles in a substance. For instance, if a bar of pure gold is sitting at room temperature ($$273.15 \text{ K}$$), then the gold molecules within the lattice will be vibrating to some extent at some average kinetic energy in J. If the temperature of that bar of gold were somehow reduced to $$0 \text{ K}$$, then its particles would not be under any motion at all, and the average kinetic energy would be 0 J.

Because of this, I would imagine that temperature could be measured in units of energy, like Joules. But as far as I know, no such unit has been assigned to temperature. Why not?

• Your title premise is wrong. Feb 15, 2023 at 21:43
• chemistry.stackexchange.com/questions/129290/… Feb 15, 2023 at 22:19
• – Karsten
Feb 16, 2023 at 9:15
• Imagine you would not measure water temperature since today, but the mean energy per a degree of freedom of water molecules instead. Good luck! Feb 16, 2023 at 14:10
• if temperature is defined as average kinetic energy is wrong. Temperature is not defined as average kinetic energy. E.g. at the same T, the mean kinetic energy of atom of Ar, molecules of N2, H2O and CO2 are different for each substance. Feb 16, 2023 at 14:40

In thermodynamics temperature describes how much the entropy of a system changes with a change in its internal energy, at constant volume:

$$T^{-1} = \left(\frac{\partial S}{\partial U}\right)_V$$

(well, as the equation shows the inverse of T describes that).

A change in entropy can be related to a reversible transfer of heat between system and surroundings. Temperature describes the extent to which the entropy of a system changes when non-mechanical energy (heat, energy associated with random atomic motions) is transferred to the surroundings.

Therefore temperature is a rate measured in units of energy per unit of entropy, and, as explained by other answers, is an intensive property, meaning it does not change with the size of the system if other intensive properties (such as pressure) remain the same. This is different from energy which is an extensive property (if you double the size of the system, you double the amount of energy in it).

• And I thought temperature was a fundamental. That entropy was defined in terms of energy and temperature Feb 1 at 20:24
• As thermodynamic concepts both entropy and temperature are fundamental and linked. However (I think) entropy is more fundamental. Boltzmann's entropy formula $S=k_B \log \Omega$ does not include T, and energy can be defined without reference to T. Feb 2 at 11:21
• @Aurelius The second law relates a change in S to a change in energy for a given T: $dS=dq_{rev}/T = dU_V/T$. So they change hand in hand at constant V. That is the origin of the equation in my answer. Feb 2 at 11:27

Kelvin is the SI unit for temperature. In the modern definition, it is defined as 1 Joule divided by the Boltzmann constant, or more exactly, such that the Boltzmann constant takes the value $$\pu{1.3806505E−23 J K-1}$$. So kinetic energy and temperature have different dimensions, but they are closely related (as are their SI units, via the set value of the Boltzmann constant).

For instance, if a bar of pure gold is sitting at room temperature (273.15 K), then the gold molecules within the lattice will be vibrating to some extent at some average kinetic energy in J.

Yes, the energy is $$k_B T$$, and the temperature is $$T$$. Related, but different quantities and different units.

To give another example where two quantities are related but have different dimensions and units, think of electromagnetic waves in a vacuum. You can characterize them by specifying either their wavelength or their frequency. That does not mean wavelength and frequency are the same, or have the same dimension. They are related, however, in this case via the speed of light.

Temperature is not energy and energy is not temperature. This problem of understanding heat, temperature and energy plagued the 19th century physicists for a long time. Indeed it is/was a tough problem with philosophical connotations. For instance can we ask what is the unit of blue color because blue color corresponds to a certain wavelength and that corresponds to certain energy in the visible spectrum. One should be able to say, how many Joule of blue color are there in a blue T-shirt? This is how temperature and energy are related. Temperature is a measurable property of matter but it is not energy. It is a measure of average kinetic energy.

The "temperature" of a particle can be expressed in eV or K. There are convenient references and calculators for that. Indeed, for plasma physics, the two units may be used interchangeably.

However, the temperature of a bulk substance is not expressed in eV, because one must take into account the energy of all particles. Is there more energy available in 1 µg of hydrogen 1,000 K hotter than the surroundings, or in 1 kg of hydrogen 1 K hotter than the surroundings?

Some properties of a material system, like temperature, are independent of the amount of matter under investigation. These properties are called intensive. Other intensive properties are color, density and flavor. However much material you have makes no difference.

In contrast, properties such as total mass, volume and energy (relative to some standard), will vary depending on how large the system is. The bigger the system, the greater will be the measure of the property. These properties are called extensive.

Energy and temperature are not the same kind of property. They are related like roses and the fragrance of roses. More roses, whether by mass, weight, or number, is a measure of an extensive property. The fragrance of roses is intensive, because it is not quantified, but identified.

There is a possible confusion if you claim that more roses equals more fragrance of roses. But, no. The fragrance is identified as the same. What you do have more of is the rose oil which is the source of the fragrance.

If you have a higher temperature, you do not have more temperature. You still have just a temperature; but it's a higher temperature - in a different system.

A good discussion can be found in many textbooks or Wikipedia (https://en.wikipedia.org/wiki/Intensive_and_extensive_properties).