By Andy May
In my last post, “Defining Temperature,” we saw that temperature is a simple measurement that has little meaning without context. If one is referring to the equilibrium system state variable or “thermodynamic equilibrium temperature,” they should state that to provide clarity because there are many types of temperature measurements and they mean different things. Thermodynamic equilibrium temperature is defined in the previous post. There are many definitions of temperature, I will discuss the most common ones used in physics and everyday life in this post and perhaps in subsequent posts. The fundamental meaning of thermodynamic equilibrium temperature and the temperature measured with a thermometer are very different and they should not be confused.
Measuring Kinetic Temperature
Whereas the thermodynamic temperature state variable of a system in equilibrium is proportional to the total internal energy of the system, kinetic temperature is the temperature proportional to the average translational kinetic energy in the system. It can be measured using thermal expansion (liquid in glass thermometers), electrical resistance (thermistors, etc.), or infrared emissions (pyrometers). Normal thermometers respond to total internal energy, not just translational kinetic energy. They correlate with kinetic temperature in gases, but in solids and liquids they respond to vibrational energy, which is not purely kinetic. Common thermometers do not measure translational kinetic energy directly (except in gases); they measure a bulk property that correlates with it under certain conditions (Reif, 2009).
Kinetic temperature in gases
Kinetic temperature can be measured in gases, liquids, and solids. However, in liquids and solids, the relationship between kinetic energy and temperature becomes more complicated because molecular motion is constrained by bonds and intermolecular forces. In gases the direct connection between kinetic energy and temperature is cleaner, so we will start there.
The ideal gas law: [Eq. 1] PV = N·kB·T
In the ideal gas law, P is pressure, V is volume, N is the number of molecules in the volume considered, kB is Boltzmann’s constant, and T is temperature. Now, divide both sides by volume (V) and we get:
The kinetic theory version of the ideal gas law: [Eq. 2] P = n·kB·T
Now “n” is the number density, or the number of molecules per unit volume. Writing the ideal gas law as in Eq. 2 shows that pressure depends only on number density (n) and kinetic temperature. This relation holds for gases in local thermodynamic equilibrium. In non-equilibrium gases, temperature must instead be inferred from the particle velocity distribution, not from pressure.
In solids, vibrational modes dominate. In most solids, atoms sit in a repeating lattice and behave like they are connected by tiny springs. When they vibrate collectively, they create waves that travel through the material. The movement increases when the solid warms. In quantum mechanics, these vibrational waves come in discrete packets of energy called a phonon. A phonon is a quantized unit of sound-like mechanical energy in a solid.
In both solids and liquids, the kinetic part of thermal energy is often less than half the total thermal energy. Most of the energy is stored in intermolecular potential wells. Intermolecular wells are an energy valley created by attractive forces between molecules. The valley gets deeper or shallower depending upon the energy added to the system. The molecules of liquids and solids are trapped in these wells. Most thermal energy is stored as vibrational potential energy, not just kinetic energy. The temperature of the solid or liquid reflects how vigorously molecules oscillate inside the well. This is why the simple “temperature – average kinetic energy” idea works well for gases but fails for condensed phases.
The depth of the intermolecular potential well determines the boiling point and the melting point of the substance. It also defines the surface tension and viscosity of the material. This concept is not theoretical; it explains everyday physical properties.
So, if you tried to define temperature purely from kinetic energy in condensed phases, sometimes you’d get the wrong value. Temperature is proportional to average kinetic energy only for ideal gases. In even more complicated situations, like plasmas or in astrophysics, kinetic temperature is inferred from Doppler broadening of spectral lines and time-of-flight measurements.
Doppler Broadening
Atoms moving toward you emit light that is Doppler‑shifted slightly higher in frequency. Atoms moving away emit light slightly lower in frequency.
Here we need to define the Maxwell–Boltzmann velocity distribution (“Maxwellian distribution”). It gives the probability that a gas molecule has a particular speed. At any temperature, some molecules move slowly, some extremely fast, and most cluster around a characteristic speed. The distribution’s shape depends only on temperature and molecular mass. Thus, if the mass is known, the temperature can be determined from the distribution but only if the atoms or molecules have this characteristic Maxwellian distribution. Maxwellian distributions are shown in figure 1 for -100°C, 20°C and 600°C.
If the atoms have a Maxwellian distribution, they are likely in local thermodynamic equilibrium or “LTE”, but this is not guaranteed. If a gas is in LTE, collisions enforce a Maxwellian distribution. The line‑of‑sight component of that distribution is Gaussian, even though the full 3‑D distribution is not. Because Doppler shifts depend linearly on velocity, the resulting spectral line profile is also Gaussian. This is the origin of thermal Doppler broadening. The width of the Gaussian is proportional to kinetic temperature, and in anisotropic systems the width depends on viewing angle, allowing direction‑dependent temperatures to be measured.
Time-of-flight
Time-of-flight is kinetic temperature in its purest form. Let the particles fly freely and see how long they take to get to the detector. Faster ones arrive first. The spread in arrival times tells the spread in velocities. The spread in velocities tells you the temperature.
Time-of-flight measurements measure the time it takes for molecules escaping from a plasma, a small aperture in a gas container, or desorbed from a surface, to reach a detector. Molecules or particles arrive at different times and form a distribution which can be converted into a time-of-flight velocity spectrum. Fit this velocity distribution to a Maxwellian distribution and the kinetic temperature can be computed. Time-of-flight gives the full velocity distribution, not just an average. In anisotropic systems, it can measure the various temperatures in all directions. It is also effective at determining species specific temperatures, for example electron, ion, and neutral temperatures in plasmas and gases.
Discussion
Almost daily we discuss the outdoor and indoor temperature, and our body temperature. All are intended to be kinetic temperatures as discussed in this post and all can have problems, especially if they assume LTE. The indoor temperature is not in LTE most of the time since most houses are either heated or air conditioned. The outdoor temperature certainly isn’t, since it varies locally depending upon the location and which side of the Stevenson screen the Sun is on and which way the wind is blowing. Our body temperature is usually taken by placing a thermometer in our mouth, where the saliva and the flesh have different temperatures. The examples are endless.
The kinetic temperature of a system is convenient since there are many, relatively easy, ways of determining it and full system equilibrium is not required. Unfortunately, as a measure of temperature, it does not always work (Bormashenko, 2020). The thermodynamic equilibrium temperature as defined in the previous post, is not related to the average kinetic motion of the particles in the system. Thus, the very common kinetic temperature is unrelated in concept and meaning to the thermodynamic definition (Bormashenko, 2020).
What is temperature?
As Bormashenko admits in his essay on “What is Temperature?” in Entropy:
“The operational definition of temperature is shaped as follows: Temperature is what we measure with a thermometer.” (Schroeder, 2000)
This is an acceptable and quite logical everyday definition; it certainly defines how the word “temperature” is normally used. Common thermometers respond to bulk properties (expansion, resistance, radiation) that correlate with internal energy. In gases this correlates directly with translational kinetic energy, but in solids and liquids the signal comes primarily from vibrational energy. I recommend that in technical writing the author should be more specific and say “kinetic temperature” if that is what he or she means.
Bormashenko goes on to show that the kinetic temperature is a very narrow definition of temperature, and it does not always work. This is true and I list some problems with it in this post. However, this does not mean that the “thermodynamic equilibrium temperature” definition discussed in my previous post is an appropriate definition of “temperature” or that the two are somehow synonymous. They are not, and to do so is disingenuous, since the reader will assume the kinetic temperature, aka the thermometer temperature.
In the twitter (or X) discussion referred to in my previous post (see the link at the bottom of this post) many commentors wanted to change the common definition of “temperature” to mean the very uncommon thermodynamic equilibrium temperature, but that is nonsense. That temperature can only be used in a laboratory where it is possible to accurately determine the internal energy and entropy of a system that must be in equilibrium. This is clearly not the thermometer temperature, and it is an awkward definition to use. No one would know what you are talking about, the thermodynamic temperature means something completely different to the everyday definition.
Future
The requirement for thermodynamic equilibrium temperature discussed in my previous post is lessened for the kinetic temperature which does not require equilibrium, although we often assume LTE when using thermometers. To eliminate the equilibrium requirement completely we need to recognize that the particle velocity distribution may not be Maxwellian and the system may not be isotropic. It is also possible for different system components to have different temperatures, for example a porous rock can have a different temperature than the fluid flowing through it. In some systems, the pressure may not be isotropic. In these circumstances you often cannot use the kinetic equation described above and you certainly cannot use the thermodynamic equilibrium temperature. Non-equilibrium temperature will be discussed in the next post.
Some additional non-equilibrium temperature references are cited and discussed here.
The original X thread that started all this can be seen here.
Works Cited
Bormashenko, E. (2020). What Is Temperature? Modern Outlook on the Concept of Temperature. Entropy, 22(12). https://doi.org/10.3390/e22121366
Reif, F. (2009). Fundamentals of Statistical and Thermal Physics. Waveland Press, Inc.
Schroeder, D. V. (2000). Thermal Physics. San Francisco: Addison Wesley Longman.
