By Andy May
This is part 3 of a series on defining temperature. In part one I covered various temperature definitions used and pointed out that “temperature,” unmodified is only a measurement of an emergent statistical property with little meaning beyond that. It is important in physics and daily life, but not a primitive well defined property like mass or energy. In part two I covered kinetic temperature, which is “what we measure with a thermometer” (Schroeder, 2000). In the original X discussion that spurred me to write these posts, some argued that the “real” definition of “temperature” was the thermodynamic equilibrium temperature. Of course this is not true. For nearly everyone on Earth, thermometers measure temperature.
There are a number of unique conditions required to measure a thermodynamic equilibrium temperature and a separate set of conditions to measure a kinetic (or thermometer) temperature, and they are not the same. While the kinetic temperature may be the same as the thermodynamic equilibrium temperature in a system at equilibrium, they are not the same thing. There is also another category of temperatures that are very different from kinetic or thermodynamic temperatures, called non-equilibrium temperatures. Non-equilibrium temperatures do not meet all the conditions required for kinetic and thermodynamic temperatures. Examples:
- The particle velocity distribution may not be Maxwellian
- Different directions in the system may have different temperatures (anisotropic)
- Different system components may have different temperatures (electron vs ion temperature in plasmas)
- System pressure may not be isotropic
Thermodynamic equilibrium temperature
Let’s briefly review what thermodynamic equilibrium temperature is, some illustrative plots are presented in figure 1.
At the top of figure 1 I show the formal mathematical definition of thermodynamic equilibrium temperature; it applies to the middle and right-hand plots. It is described in detail in post 1. In the equation, T is temperature, U is the system internal energy, S is system entropy, ∂U/∂S is the partial derivative of internal energy with respect to entropy, N is the number of particles, and V is the volume.
Figure 1 illustrates how thermodynamic temperature arises from the geometry of the equilibrium state space. The left panel shows the equilibrium manifold in the thermodynamic state space. The horizontal axis is entropy S, the vertical axis is volume V, and the shading and contour lines represent internal energy U(S,V). For this plot N (the number of particles) is held fixed. Entropy may be interpreted as the number of microscopic configurations compatible with the macroscopic constraints, although in equilibrium thermodynamics it is treated as a state variable. For a system with fixed internal energy, volume, and particle number (a “microcanonical” ensemble), the equilibrium state is the point on this manifold where entropy is maximized. This point is the red dot. The blue arrow projects this equilibrium point down to the -axis, indicating how the full two‑dimensional manifold reduces to a one‑dimensional slice when volume is held fixed.
This projection shows how the manifold reduces to the slice shown in the middle and right-hand illustrations. The middle panel shows the curve U(S) at fixed N and V. The equilibrium point lies on this curve. The right panel zooms in on the equilibrium point and shows the tangent line to the curve U(S) at equilibrium. The slope of this tangent, (∂U/∂S)N,V, is the thermodynamic equilibrium temperature. Thus, temperature is not a coordinate in the state space but a geometric property.
Kinetic Temperature
Kinetic temperature was discussed in the last post, it is the conventional thermometer measured temperature. Whereas the thermodynamic temperature comes from the slope of the equilibrium surface U(S,V,N), Kinetic temperature comes from the average translational kinetic energy of the particles in the system being measured. It is defined through the velocity distribution of the particles, not through an equilibrium manifold. Figure 2 illustrates the development of kinetic temperature.
Comparing figure 1 to figure 2 shows that, while thermodynamic equilibrium temperature and kinetic temperature of a system at equilibrium might be equal, it is a superficial equality, the two temperatures are defined differently.
Thermodynamic temperature
- Defined on the macroscopic equilibrium manifold .
- Temperature is the slope .
- Requires equilibrium and a well‑defined entropy.
Kinetic temperature
- Defined from microscopic particle motion.
- Temperature is proportional to the average translational kinetic energy.
- Does not require entropy or equilibrium surfaces.
- Emerges from the velocity distribution.
Kinetic temperature arises from statistical mechanics rather than equilibrium thermodynamics.
Non-equilibrium temperature
There are number of techniques for measuring temperatures in non-equilibrium systems, but the commonly used translational temperature in a shock front is a very good example. It illustrates how important it can be to measure non-equilibrium temperatures. It is defined from the non‑Maxwellian velocity distribution immediately behind a shock front. The shock front temperature is widely used in aerospace, combustion, and atmospheric entry and it differs from thermodynamic and kinetic temperature in almost every way.
In a shock front, the particle velocity distribution is distorted. It often has a high‑energy tail, is anisotropic, and contains mode‑dependent temperatures, like translational, rotational, vibrational, and electron. The development of a shock front temperature is illustrated in figure 3. A shock front is the advancing edge of a shock wave, a propagating disturbance that moves faster than the speed of sound in a fluid and causes an abrupt, nearly discontinuous change in pressure, temperature, density, and other flow properties (Wikipedia).
The left illustration in figure 3 shows an example shock wave velocity space, it is clearly not at equilibrium. The middle illustration compares this distribution to a Maxwellian distribution in blue. The right illustration computes the shock front temperature using only the core region of the distribution.
The shock-front non-equilibrium temperature is only one of many. There are others, for example the brightness temperature used by Spencer and Christy (Spencer & Christy, 1990) at the UAH to determine atmospheric temperature for several intervals.
Radiation (brightness) temperature
This temperature measurement is ubiquitous in the atmospheric sciences, astrophysics, and remote sensing. Whereas the previous examples of temperature measurements used either entropy or particle velocity to determine temperature, this measure uses brightness or radiation intensity (I) to determine temperature. In essence, brightness temperature is the temperature you would infer if you assumed the radiation intensity was coming from a blackbody. You simply take the measured intensity at a given frequency (Iv), plug it into Planck’s law and solve for temperature. Crucially, it does not require the radiation field to be Planckian.
Brightness temperature is not necessarily the real temperature of the emitting body. A real radiation field may be a mixture of temperatures, and it may be far from equilibrium and if so, the spectrum is not a Planck curve. But, at any given frequency, radiation intensity always increases with temperature (see the right-hand plot in figure 4) and you can always find a unique temperature for any brightness. So, brightness temperature is valuable because it expresses radiance on a temperature scale, making radiative transfer relationships intuitive even when the radiation field is far from equilibrium. This is convenient in microwave remote sensing, such as that done at UAH.
However, that said, brightness temperature can equal physical temperature, or be very close to it, when the emitting medium is optically thick and in local thermodynamic equilibrium (LTE) at that frequency. The O₂ microwave bands from the atmosphere are close to this ideal, which is why UAH can treat brightness temperature as physical temperature after instrument corrections and calibration. UAH made a very good choice when they picked the oxygen microwave bands (~50-60 GHz) to use in their work. Oxygen molecules (O2) in the atmosphere are well mixed and behave like a blackbody in local thermodynamic equilibrium. Thus, the brightness temperatures in the O2 frequencies are very close to the real atmospheric temperature.
The satellite radiation measurements must be corrected for various instrument effects and orbital drift, and the satellites contain a two-point internal calibration to make their calculation of brightness temperature more accurate. Periodically the instrument is pointed to space (about 2.7K) and then to a warm target inside the satellite with a known temperature and these readings are used as part of the process of calibrating the brightness temperature to real temperatures (Spencer & Christy, 1990). These calibration points and the instrument corrections are good enough that both UAH and RSS use the final brightness temperature as is, they don’t try and calibrate the brightness temperatures to any ground-based data, but they do use the ground-based data to estimate the accuracy at chosen points, mainly weather balloon launch sites. The accuracy is quite good (Christy et al., 2018). The development of brightness temperature is illustrated in figure 4.
The left panel in figure 4 shows an equilibrium Planck curve versus frequency in black and a distorted brightness (I) versus frequency (v) curve in red. The distortion could be due to the lack of equilibrium or other complicating factors, like optical thinness, mixtures of temperatures, or anisotropic radiation fields. Real spectrums are more like the red distorted curve than the black Planckian curve. The middle panel zooms into a portion of the left plot and computes two brightness temperatures at two different frequencies. The two estimated temperatures are far from the Planckian curve. Finally, the right-hand plot inverts the Planck curve and shows a new plot of brightness versus temperature for one frequency. The curve is monotonic, with temperature increasing as brightness increases.
While the brightness temperature is not always equal to actual temperature, it can be very close, as it is in the UAH oxygen brightness measurements discussed above.
Besides brightness temperature, there are other non-equilibrium temperature measurements, these include:
- Vibrational temperature: Used for combustion and plasmas. It depends upon internal state populations and is defined with Boltzmann plots of excited populations of particles.
- Electron temperature: Used in plasmas, determined from the slope of a distorted, often non-Maxwellian energy distribution. It is defined by the high energy tail of the electron energy distribution.
Discussion
The key point is that the term “temperature” is just a measurement arbitrarily scaled with increasing energy. It has little meaning, until the type of temperature is specified with a modifier, like “thermodynamic,” “kinetic,” “shock-front,” “brightness,” etc. In everyday usage, “temperature” is assumed to be a temperature measured with a thermometer, which is a kinetic temperature, not a thermodynamic equilibrium temperature as some argued in the subject X thread.
Temperature does not have one definition or one meaning, either in physics or everyday use of the word. The examples discussed and illustrated above all have very different meanings and definitions. All are useful temperature measurements, but they do not all fit into one definition.
The illustrations were all made with R, the programs can be downloaded here.
H/T to Mike Chillit who suggested I make the illustrations to make the definitions clearer.
Works Cited
Christy, J. R., Herman, B., Sr., R. P., Klotzbach, P., McNider, R. T., Hnilo, J. J., . . . Douglass, D. (2010). What Do Observational Datasets Say about Modeled Tropospheric Temperature Trends since 1979? Remote Sensing, 9, 2148-2169. https://doi.org/10.3390/rs2092148
Christy, J. R., Spencer, R. W., Braswell, W. D., & Junod, R. (2018). Examination of space-based bulk atmospheric temperatures used in climate research. International Journal of Remote Sensing, 39(11), 3580–3607. https://doi.org/10.1080/01431161.2018.1444293
Schroeder, D. V. (2000). Thermal Physics. San Francisco: Addison Wesley Longman.
Spencer, R., & Christy, J. (1990). Precise Monitoring of Global Temperature Trends from Satellites. Science, 247. Retrieved from https://science.sciencemag.org/content/247/4950/1558.abstract
nice details. Whilst it is very useful to estimate/calculate temperature from radiation brightness, it confirms my position of being far more sceptical of doing the reverse, like on the pathway to creating energy budgets.. for the reasons posted. The multiple frequencies aren’t additive like colours of a rainbow.
I am at a loss for words.
Andy, your definition makes it seem that kinetic temperature comes from statistics, rather than from physics.
A clearer way to express it, revising your sentence, is kinetic temperature arises from the connection between statistical thermodynamics and kinetic molecular theory rather than equilibrium thermodynamics. Discussion here.
LiG thermometers register the KE of a surrounding medium. Temperature is determined when the KE of the particles composing medium in the thermometer are in local equilibrium with the the KE of the particles in the surrounding medium, where ‘particles’ is atoms or molecules.
That is, temperature is established when a local equilibrium prevails between the KE of the internal and external mediums. In a meteorological field station, environmental variables may cause an equilibrium condition to never be attained.
The temperature scale is determined by the almost arbitrary scoring of the glass column of the LiG thermometer. Researchers eventually settled on the LiG Centigrade scale, and that now generally pertains to PRTs and other solid-state thermometers.
Wrong place initially. See below in reply to Andy
Pat, I have no problem with what you wrote, it is all true and in agreement with what I’ve written and think, but this sentence is a little confusing:
It seems you object to my emphasis on statistical mechanics, or maybe the tone of my article. I was simply emphasizing the differences between how a thermometer measures temperature in the real world and the thermodynamic equilibrium temperature. They are not the same and never will be.
I do realize that kinetic temperature is simultaneously statistical and physical. It is defined as ensemble averages, but the averages directly correspond to physical processes (kinetic theory). I hope I interpreted your comment correctly.
AM, thanks for this multipart series. I admit I had not thought about any of this before. Fascinating. Shock front temperature for space vehicle reentry is obviously very important, and different than ‘thermometer’ temperature—even tho NASA press releases do not make the distinction.
Reminded me of a legal joke about ‘originalist’ constitutional interpretation at SCOTUS:
’Originalism is asking the question, “What would the Founders have thought about something about which they did not think (because it did not yet exist)?” ‘ In law school I was taught a different way of reasoning, by distinction and extension (as the world evolves). Had that ‘older’ reasoning style been used, I think the recent SCOTUS birthright citizenship decision would have come out differently concerning things like birthright tourism.
“Temperature does not have one definition or one meaning, either in physics or everyday use of the word. The examples discussed and illustrated above all have very different meanings and definitions. All are useful temperature measurements, but they do not all fit into one definition.”
Maybe the problem, then, is careless expropriation of “temperature” in discussing different sorts of physical states.
Science aims to be monosemous. Of Singular meaning. Using temperature to mean many things is to violate the rule of explicit meaning that governs scientific theory.
Let temperature be T = (𝞭𝐔/𝞭𝐒)ɴ,ⅴ, and then coin new terms to mean what’s now called brightness temperature or kinetic temperature. Doing so would go a long way to clearing up the confusion.
Even a standard subscript, e.g., Tᵦ for brightness temperature or T𝛋 for kinetic temperature to distinguish them from T itself would help solve the confusion.
I get a bit waxed at the equally careless use of “entropy” in the Shannon entropy, which allows people to conflate corruption of transmitted information with a Thermodynamic state function.
“Let temperature be T = (𝞭𝐔/𝞭𝐒)ɴ,ⅴ, and then coin new terms to mean what’s now called brightness temperature or kinetic temperature. Doing so would go a long way to clearing up the confusion.”
If you were the king of words you could do that, but you are not. Words are defined by how they are used, not by how we think they should be used.
I aspire to no kingship. Scientific terms are not drift-subject dictionary usages.
Scientific societies regularly use physical meaning to set terms. E.g. Celsius.
Various constants are subscripted for specificity, such as the Boltzmann constant kᵦ. Why not temperature T, so as to specify meaning? Doing so would halt confusion.
Subscripting T would help tremendously. I recommend it in all academic writing, but when T is used alone, especially thermometer T, I and most of the world will assume kinetic T, not thermodynamic T.
Entropy is defined as: “Entropy may be interpreted as the number of microscopic configurations compatible with the macroscopic constraints” what is wrong with that definition? Be specific.
I have no immediate problem with your definition of entropy. I merely object to the appropriation of entropy to mean Shannon’s information loss.
Shannon entropy isn’t an appropriation. Thermodynamic entropy is the Shannon entropy of the equilibrium distribution. The two quantities differ in domain, not in mathematical structure. Thermodynamic entropy, the way I use the term, is a state function for systems in equilibrium. Shannon entropy is a measure of uncertainty in a probability distribution, not what I meant.
Shannon entropy is the expected value of the log multiplicity when the microstate probabilities are not uniform. Thermodynamic entropy is a special case where all microstates are equally probable.
This is why Shannon entropy as the general form and thermodynamic entropy is the uniform‑distribution limit. They are related, but not the same.
The real difference in what we are saying is not in the physical details, it is in the linguistics. You are saying what you want temperature to mean, I am saying what it actually means.
My point is that “temperature” is not a primitive well defined physic property, it needs a modifier.
T = (𝞭𝐔/𝞭𝐒)ɴ,ⅴ seems pretty well-defined to me.
Temperature would need no modifier if distinct terms were assigned to each of the present several meanings.
It might be well defined but it is impossible to measure. There is no device that measures entropy nor can you take the derivative of internal energy (again something that can’t be measured) with respect to Entropy to give a meaningful answer.
T = (𝞭𝐔/𝞭𝐒)ɴ,ⅴ is well defined, but it is not “temperature,” it is “thermodynamic temperature at equilibrium.” Temperature is what is measured with a thermometer.
For Celsius/Centigrade there is nothing arbitrary about the scale and it is intimately connected to molecules in a fluid. The low point is where ice and liquid water coexist in mixture and the upper point is where steam and water coexist while at both ends dynamic equilibrium is physically established. Omitted are other variables, the main one being atmospheric pressure at sea-level.
What I do not know is where the scale was first established, was it Upsalla and under what conditions, high pressure system or low pressure system or dead calm balmy weather around the laboratory. Probably fixed by some authoritative official body of boffins somewhere else, maybe Paris or Greenwich.
The point being, it is part of the legacy classical physics of yesteryear and retains a cultural and historic meaning very likely unalterable.
Pat,
We are talking past each other, I think. You are arguing for terminological monosemy; I am arguing for descriptive accuracy about how scientists actually use the word temperature. Those are not the same project, and that’s why the exchange keeps circling.
I am saying: Temperature is not a primitive physical property; it needs a modifier.
You are saying: Temperature should be a primitive physical property, and everything else should be renamed.
Good luck with that.
Robert Mulliken coined orbital to describe electronic states around a nucleus. Had he kept orbit, confusion among and with mesoscopic or planetary motions is invited.
We’re not talking past each other, Andy. I’m just noting a dissatisfaction that a very important concept in physics is confused by application of the one term to multiple and distinguishable physical variables.
Doing so has produced exactly the need to clarify among them, as you’ve done here. The confusion among the sorts of temperature is unnecessary, self-generated, and self-imposed.
Not really. The definition of temperature has always been the same. This strange idea that thermodynamic principles should overrule language is unnecessary confusion that is self-imposed and self-generated.
Of course Thermodynamics should overrule the careless use of language in science. How strange that you should think not.
Well, I think we agree that careless use of language is a bad thing and we should all be careful how we use vague terms like “temperature.” In many ways, I think the misuse of “temperature” in climate science is at the root of many problems in both meteorology and climate science. One must remember that the laws of thermodynamics and the equation of state only apply to equilibrium conditions, a rare situation.
Pat, It is fine for academic types to think that the “real” temperature is the thermodynamic definition. But, in the real world, it isn’t. The real definition is the quantity measured by a thermometer and always will be. Temperature is not a real thing, like mass, it is an abstraction.
Actually academic types now define temperature in terms of Boltzmann’s constant which now has the exact value (in SI units) of 1.380649×10^23 J/K. So experiments that previously was used to measure Boltzmann’s constant are now used to measure temperature. Optical Raman spectroscopy for example can be used to measure the absolute temperature by looking at the ratio of the stokes and anti-stokes signals.
But doesn’t the SB equation rely on a constant temperature, in terms of the surface (blackbody) and in the relation of radiation which is used in atmospheric physics?
This is a complicated subject and not very pretty. Historically Kelvin was defined by the triple point of water, now it is artificially defined in meteorology using an assumed and fixed KB (Boltzmann’s constant). Historically acoustic, noise, raman, and doppler, measurements were used to determine KB, but now that KB is defined, they become primary thermometers. The methodology has flipped with this definition.
However, this does not change anything, even if we assume a KB, there are still multiple definitions of temperature, and they do not all mean the same thing.
Andy,
This has been a good series of articles and associated comments. I must have been comatose during my ChE Thermo courses, as I’ve always labored under the impression that ‘T’ related to a kinetic temperature that one would could actually measure using off the shelf instruments. Subsequently, the results of that, and other measurements could then be used to ascertain all kinds of useful information about a process from various charts and tables.
Anyways, as my horizons have since been expanded, I thought I’d drop off this (very short) item that seems pertinent to both this discussion on temperature, as well as the assumption of LTE for purposes of climate modeling.
https://scienceworld.wolfram.com/physics/LocalThermodynamicEquilibrium.html
Hi Frank, That quote is mostly correct and illuminating. Certainly, in the upper atmosphere, where density is low and radiative processes dominate the infrared, visible, and ultraviolet brightness temperatures differ from the kinetic temperature and the radiation field becomes non-Planckian -> thus not in LTE. In parts of the lower atmosphere, the collision rates are high, molecular activity equilibrates rapidly and LTE is a decent assumption. Not sure how common that is though, the lower atmosphere is complex.
CO2 in the lower atmosphere still radiates so the page Frank from NoVA shows is still correct. If the object is subject to conduction flux or evaporative flux as well as radiative flux then it can’t be considered to be in local thermal equilibrium.
I am enjoying this little to and fro..🙂
Energy state is a real thing. Temperature is no more an abstraction than velocity.
Energy state is certainly real, and equilibrium temperature is indeed a thermodynamic state variable. The key point is that not every measured temperature is a state variable.
A thermometer reading only corresponds to a true thermodynamic temperature when the thermometer and the surrounding medium achieve local thermal equilibrium.
In everyday environmental measurements—especially meteorological ones—that equilibrium often does not exist. The reading is still a useful physical measurement, but it is not a thermodynamic state variable unless one makes strong assumptions about equilibrium and homogeneity.
This distinction is frequently glossed over in climate science discussions, where “temperature” is treated as if it were always a well‑defined state variable rather than a context‑dependent measurement.
“This distinction is frequently glossed over in climate science discussions, where “temperature” is treated as if it were always a well‑defined state variable rather than a context‑dependent measurement.”
Dead-on right, Andy. The cli-sci temperature measurement model assumes extrema precision, high accuracy, random error, and perfect rounding to the correct integer.
Which assumptions inspired the LiG Metrology paper.
Good paper! I liked these comments in the conclusions:
Well said!
Thank-you, Andy. I’m very appreciative.
but, but but, its increasing CO2, not all this clever physics stuff. The Guardian said so…..
🙂
Thanks for finally ditching the twitter reference.
So what exactly are the guys on the other side referring to when they say CO2 will cause an increase in temperature. Second are the words and scales Celsius, Fahrenheit and Kelvin only used for kinetic temperature or are they also used for the other variations?
My point is that thermometers measure temperature. When people say “the temperature is” they mean that is what they measured with a thermometer. All thermometers measure kinetic temperature, they do not measure thermodynamic temperature, except in a system at thermodynamic equilibrium, which only exists in laboratories.
It is not that simple. I can buy a cheap infra-red temperature sensor that is calibrated to what you call radiation temperature. It does not measure kinetic temperature for all the reasons you stated above. Such thermometers work well for cooking and are especially useful for things like pizza ovens.
True, I always forget about IR thermometers, and they are in common use today. They are not kinetic.
IR thermometers are not measuring temperature, not directly.
They measure emitted IR field strength and using a model calculate an equivalent temperature.
So we should be able use an IR camera and take pictures of CO2?
It would have to be a very long exposure to get anything at all.
And just to be pedantic which seems to be the whole point of these articles, nowhere in the universe is in thermodynamic equilibrium nor will anywhere be so until the heat death of the universe in the far distant future. However it is easy enough to create a good approximation of thermal equilibrium — just put some water in a thermos flask for instance.
Temperature is the relative kinetic energy of stuff per LoT 0.
There are Celsius units on the Celsius scale.
There are Celsius units on the Kelvin scale.
Ther is no such thang as “Kelvins” of Kelvin units.
The satellite brightness measurements incur an uncertainty due to path loss through the atmosphere. As far as I can find, path loss is *not* measured per observation but is estimated using an absorption coefficient that is a model that is dependent on temperature and humidity of the atmosphere.
I simply cannot find any measurement uncertainty factor for this anywhere. No observation versus model output validation – probably for a primary reason that no observation network exists. It would require placement of a network of transmitters in the 50-60Ghz range that the satellites could observe on a per measurement basis to calculate a more exact path loss factor.
Since temperature and humidity at any measurement point can have significant variance temporally, use of a long term “average” factor value
in a model introduces the same measurement uncertainty as the variance of the factors. When temperature differences in the hundredths digit are being calculated it’s not obvious how the satellite data uncertainty, due just to humidity variance alone, can be prevented from overwhelming the supposed “differences” due to a modeled path loss factor instead of one that is actually measured.
I get the feeling that this is just one more of the instances in climate science where it is assumed that the measurement uncertainty cancels by assuming it is Gaussian and random thus allowing the use of a long-term “average” with no degradation in accuracy.
Spot on.
Oxygen is 500 (20%)times more abundant than CO2 (.04%). A CO2 emission is only 330 times stronger than an Oxygen. So is O2 a GHG?
😀
A GHG? Not by definition. But it is involved in heat transport. Is kinetic energy transfer from other molecules considered to be conduction? Does this “collision” energy then get transported by O2 through convection and advection?
O2 also absorbs and emits micro waves.
Really critically astute observation, Tim.
Bond albedo is brightness viewed at a distance.
It includes emitted EM and reflected EM.
To use it to accurately calculate the surface temperature, one needs to have a precise measurement of the reflection of the incoming EM and the emitted EM needs to account for thermal energy transfers both into and out of the surface.
The article address those in different terms, but it is critical that local thermodynamic equilibrium constantly changes. The earth rotates.