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Kelvin: Thermodynamic Temperature

In everyday life, we usually think of temperature in terms of comparisons: How hot or cold something is relative to some physical property or reference, as in “Bundle up, it’s below freezing out there!” or “My daughter is running a 103-degree fever.”

translational motion
Translational motion
Credit: Sean Kelley/NIST
Thermodynamic temperature, by contrast, is an absolute measure of the average total internal energy of an object or objects—namely its kinetic energy (energy of motion) plus contributions from other factors.

It applies to the average energy of a collection of atoms or subatomic particles—for example, the atoms in a block of iron, or the air molecules in a room. It is expressed in number of kelvins above absolute zero, the theoretical point at which nothing can get colder.

Usually, most of an object’s kinetic energy is embodied in what is called “translational motion,” which occurs when it moves around in space. In gases, atoms or molecules fly around in all directions, colliding with barriers and with each other. (Their individual speeds are different, but for any given set of objects at a given temperature, one range of speeds is more probable than others.) The average speed of the atoms and molecules depends on their energy content.

Collective motions called phonons.
Credit: Sean Kelley/NIST

In solids, where atoms are constrained by bonds and cannot move independently, kinetic energy takes the form of collective motions called phonons. Thermal energy can also flow through solids in the motion of unbound, mobile electrons. In addition, objects or parts of complex objects can have kinetic energy in the form of vibration and rotation, and these, too, add to the total energy content.

Thermodynamic temperature is proportional to the average of all the energies in all the ways in which it is possible for an object to move. These are known as “degrees of freedom.” So, for a single helium atom, there are only three degrees of freedom: motion in the up-down, left-right, back-forth directions. A two-atom molecule of nitrogen, however, there are two additional degrees of freedom: one of rotation and one of vibration. In general, the more components there are in a complex object such as a large molecule, the greater the number of possible motions and degrees of freedom.

It is extremely difficult to measure this internal energy directly. Instead, scientists measure its effect when it moves as heat (thermal energy in transit) to or from a system of objects. When heat no longer flows between these objects—in other words, when they are in thermal equilibrium—that is their thermodynamic temperature.

Thus, internal energy and temperature are different, though directly related. The SI unit of energy is the joule. A “derived” SI unit, the joule itself is defined in terms of three SI base units — the kilogram, the meter and the second. But thermodynamic temperature is expressed in kelvins. There needs to be a way to connect the two.

The bridge between those two realms is the Boltzmann constant (kB, or often just k), which relates the kinetic energy content (E) of matter to its temperature (T): E = kBT. For the simplest collection of particles such as atoms, the average kinetic energy is ½mv2 distributed over the three degrees of freedom, where m is the mass and v is the velocity, so the total translational energy is 3/2 kBT.

Created May 15, 2018, Updated June 5, 2019