This claim is made by people trying to “disprove” anthropogenic global warming. It is based on misleading ideas about the significance of comparing the 15-micron primary emission wavelength of CO₂ to the peak emission wavelength of an ideal black-body[1] radiator at a temperature of -80ºC.

There is mistaken idea that the frequency or wavelength of a photon corresponds to a unique temperature. However, look at the plot below showing the spectrum of radiation associated with black-body objects at temperatures ranging from -50℃ to 15℃.

The chart reveals if we encounter photons with wavelengths of 8, 15, or 50 microns, any of those photons could have originated with a source with a temperature anywhere in the range from -50℃ to 15℃. There is no way of knowing the temperature of the source that originated a photon of thermal radiation, simply based on its wavelength.

Yet, some people have the misconception that wavelength and temperature have a simple, rigid relationship. Let’s look at all this more closely.

The above chart shows how spectral radiance (energy per time per solid angle per unit wavelength)[2] of infrared radiation varies with wavelength.

The upper (blue) line shows the emission spectrum of an ideal “black-body” radiator at -80ºC emitting thermal radiation according to Planck’s Law.[3]

The lower (red) line shows the spectrum of radiation spontaneously emitted by CO₂ gas at -80ºC. (This assumes low pressure, because at sea-level atmospheric pressure, CO₂ at -80ºC would be a solid, i.e., dry ice.[4])

CO₂ emits and absorbs radiation in a number of distinct wavelength bands, including bands around 4.3 μm (microns) and 15 μm.[5] Only the latter band is shown, since it is the band most important with regard to its atmospheric impact on the temperature of the Earth.

(Technical note: I am not attempting to show the actual shape or width of the CO₂ 15-μm spectral band, which actually consists of many closely-spaced spectral lines. However, the chart does accurately reflect that, in isolation, the peak spectral energy density of a gas’s emission line is expected to match the spectral energy density of a blackbody radiator at the same temperature, per equation 49 in [6])

Notice that the center wavelength of the CO₂ 15-μm emission line matches the peak of the thermal radiation spectrum for at black-body at a temperature of -80ºC. Based on this similarity of wavelengths, some people argue that thermodynamically, CO₂ “must” have an effect similar to that of an object at a temperature of -80ºC.

This argument might at first seem superficially plausible. But, let’s draw a few more charts.

This is the same as the preceding chart, except that two more curves have been included, to show the spectra for a blackbody at 0ºC and for CO₂ at 0ºC.

Comparing the two black-body radiation curves, we see that the peak wavelength has shifted somewhat. But, the largest difference is that the 0ºC black-body emits 4 times as much total radiance (energy per time per solid angle) as does the -80ºC black-body. Temperature changes are associated with changes to both wavelength and radiance.

Notice that the radiation emitted by CO₂ is still centered around a wavelength of 15 μm. However, for CO₂ at 0ºC the total radiance emitted is 4.4 times as large as it is for CO₂ at -80ºC. Surely, radiation from CO₂ at 0ºC will have more warming effect than radiation from CO₂ at -80ºC given this more than four-fold increase in power!

People who deny that CO₂ can warm things seem to argue that both the (green) radiation curve for CO₂ at 0ºC and the (yellow) radiation curve for CO₂ at -80ºC will have an effect similar to the (blue) radiation curve for a black-body at -80ºC.

The (green) radiation curve for CO₂ at 0ºC matches the peak wavelength of the (blue) radiation curve for a black-body at -80ºC, but matches the spectral radiance of the (red) radiation curve for a black-body at 0ºC.

How is it rational to argue that the radiation from CO₂ at 0ºC will have an effect equivalent to one black-body curve but not the other?

The dubious nature of the argument becomes even more clear if one plots the spectra differently. One can plot a spectrum of electromagnetic radiation in a variety of different ways. How you choose to plot the spectrum affects where the “peak” turns out to be.

Let’s consider a chart just like the last one, except that we plot spectral radiance with regard to frequency instead of wavelength. (Planck’s Law can be formulated with regard to wavelength or with regard to frequency.[7])

Note that in this chart, the spectral radiance is energy per time per solid angle per unit frequency. The difference between “per unit wavelength” and “per unit frequency” means that what is being plotted is a subtly different quantity than what was being plotted in the prior charts. This difference allows the peak value to be in a different place.

When we plot things in terms of frequency, the CO₂ emission band at 20 THz (which corresponds to a wavelength of 15 μm) is nowhere near the peak of the emission spectrum for a black-body at -80ºC. Plotted this way, it seems totally implausible that the CO₂ emission would have an effect equivalent to that of the -80ºC black-body.

The familiar form of Wien’s Displacement Law allows one to calculate the wavelength where the Planck radiation spectrum peaks with respect to wavelength. However, there are also variants of the law for calculating the peak of the spectrum with respect to frequency.[8] Alternatively, there are on-line tools that allow us to figure this out. Using such a tool, one finds that a black-body at a temperature of 67ºC has a spectral radiance with respect to frequency that peaks at a frequency corresponding to a wavelength of 15 μm.[9]

As the above chart illustrates, the CO₂ 20 THz (i.e., 15 μm wavelength) band coincides with the peak of the Planck emission curve (with respect to frequency) for a 67ºC black-body.

So, if we were to follow the logic of those who claim CO₂ can’t warm things beyond a certain temperature, but plot our spectra with respect to frequency instead of wavelength, we would come to the conclusion that **CO₂ “must” have the same thermodynamic effect as a black-body radiator at 67ºC.**

So, which is it? Does the electromagnetic radiation from CO₂ always act like it’s from an object at -80ºC or does it always act like it’s from an object at 67ºC?

The answer is: neither is correct. It is a fundamentally flawed way of thinking, to think that you can predict the thermal effect of a gas just by looking at its emission wavelength, and determining the temperature of an “equivalent” Planck-radiator (or black-body).

It’s not just wavelength that matters. Intensity also matters.

How “hot” something is depends on how much energy it contains. Temperature depends on quantity of energy, not just kind of energy (or wavelength).

* * *

Part of the story that goes along with the claim that CO₂ can’t warm things is the assertion that 15-μm wavelength microns are “too weak” to warm things beyond -80ºC.

This claim seems to involve a misunderstanding of the significance of photon energies.

The wavelength (or frequency) of electromagnetic radiation determines how much energy each photon carries. This photon energy determines what quantum transitions in a molecule the photon can interact with. This in turn determines how likely it is that the photon will be absorbed. It also determines how the energy of the photon initially affects the material that absorbs it. Energy may enter the material as an excitation of electrons, or as a vibration or rotation.

However, after the photon is absorbed, in most cases the energy is quickly redistributed to other “modes” (distinct places energy can reside) within the material. This energy redistribution is what makes the notion of temperature meaningful. Being at a particular temperature means that energy is distributed among the various modes in a way that is characteristic of a “thermal” energy distribution at that temperature. (In particular, energy is distributed according to the Boltzmann distribution.[10])

When a photon is absorbed, under most circumstances and for most forms of matter, this redistribution of energy into a “thermal” (Boltzmann) distribution happens rapidly. This happens for solids, liquids, and gases (unless the gas pressure is so extremely low that collisions are infrequent).

If this didn’t happen, the idea of “temperature” wouldn’t be nearly as useful as it is, and materials would not radiate according to Planck’s Law.

(I’ve seen people both claim that energy redistribution doesn’t happen and claim that materials are Planck radiators; but it’s not logically consistent to claim both things. Energy redistribution into a Boltzmann distribution is the mechanism that makes things behave like Planck radiators.)

Given that energy redistribution happens, **after a set of photons is absorbed, it no longer matters how much energy the individual photons carried.** All that matters is the total amount of energy carried by the set of photons. That is not dependent on the wavelength (or frequency) of the radiation. Any wavelength can be associated with any total amount of energy.

**If a material is capable of absorbing electromagnetic radiation of a particular wavelength, then that wavelength is capable of heating the material, without any temperature limit.**

* * *

Here is where those who think the radiation emitted by CO₂ can’t warm things are almost certain to jump in with another bit of muddled logic.

They’ll say, “That can’t be true, because then the radiation from an object at one temperature can’t raise the temperature of a warmer object.”

The flaw in the logic of this has to do with an implicit false belief that the only thing that characterizes the Planck radiation for a given temperature is the wavelength of the radiation. No. Planck’s Law says that temperature impacts the intensity (radiance) as well as the wavelength distribution of the thermal radiation that is emitted.

It is **the ***combination*** of intensity (or more properly radiance) and wavelength distribution that leads to things like heat flowing from hot objects to cold ones**, rather than the other way around. It’s not just about wavelength.

* * *

Someone really committed to the idea that the radiation emitted by CO₂ can’t warm things might now say, “Temperature must only depend on wavelength and not intensity, because otherwise you could concentrate the light of the Sun to heat an object to a temperature higher than that of the Sun’s surface, and we know that’s thermodynamically impossible.”

That’s a valiant attempt, but still gets things wrong.

It turns out that any optical system (of the sort one might use to try to concentrate sunlight) conserves something called “étendue.” A corollary of the conservation of étendue is that “basic radiance” never increases.[11] This guarantees that optically concentrating sunlight can never result in more radiance than what was present at the source. *This is what ensures that concentrated sunlight can never produce a temperature hotter than that of the surface of the Sun.* It’s not about wavelengths at all.

* * *

I think the idea that radiation emitted by CO₂ cannot warm things to warmer than -80ºC is, to some extent, the result of honest confusion about how nature enforces the principle that “heat flows from hot to cold.”

The belief seems to be that the wavelength at the peak of the spectrum of radiation somehow sets an upper limit on the temperature that the radiation can produce.

Ultimately, the main problem with the hypothesis is that **it’s not consistent with the existing theory of how temperature works, which was well established and well tested long before climate change became an issue.**

Aside from that, this hypothesis has a number of problems:

- As I’ve mentioned, where the “peak” of the spectrum is depends enormously on how the spectrum is plotted. (I pointed out two distinct ways of defining a “peak”, but there are even more possibilities.[12])
- For this to be a real theory of how the direction of heat flow gets honored, there would need to be an underlying mathematical model that is consistent with other laws of physics and which correctly predicts the thermal behavior of complex systems. There does not seem to be such an underlying mathematical model.
- The predictions being offered don’t properly account for conservation of energy. If after some point, absorbed photons no longer increase temperature, one needs to account for what happens to the energy.
- When translated to assertions about what particularly types of photons can or cannot do, the hypothesis mistakenly assumes the idea of temperature to be meaningful at the level of individual particles. Temperature is a parameter linked to the statistical properties of a system, and is meaningless at the level of individual particles. There is a reason that thermodynamics is rooted in
*statistical mechanics*.[13]

The idea that “it’s all about wavelength” is not consistent with the Second Law of Thermodynamics. The core of thermal physics is the principle that entropy tends to be maximized. This happens when energy is distributed according to the Boltzmann distribution.[14] Every means of heat transfer, whether it involves conduction, convection, or radiation, can be understood as involving the Boltzmann distribution in one place trying to equalize with the Boltzmann distribution in another place, thereby maximizing entropy.

The exchange of radiation between a solid and a gas is no exception to this pattern. It doesn’t matter that the solid is likely to act as a Planck radiator, while the gas emits and absorbs on distinct spectral lines. *The Boltzmann distribution in the solid and the Boltzmann distribution in the gas will still tend to equalize (i.e., adjust to have similar characteristic temperatures).*

CO₂ gas can be at temperatures greater than -80ºC because energy within the modes of the gas molecule can be distributed according to a Boltzmann distribution associated with a higher temperature.

Once the CO₂ is associated with some higher temperature, then **the basic laws of thermodynamics (relating to the tendency to maximize entropy) guarantee that CO₂ exchanging radiation with another object will tend to warm that other object to its own temperature** (in the context of an isolated system).

* * *

After I had published the answer above, a lead proponent of the idea that radiation from CO₂ can’t significantly warm things (“can’t even melt an ice cube”), started working on a new justification for his claim.

He shared in a comment (in relation to another essay of mine) that the Boltzmann constant, k,[15] “literally equates the temperature imparted to a molecule by a photon to its kinetic energy divided by the constant. Once it has been absorbed, the photon ceases to exist, and the temperature imparted is all the molecule has to work with, meaning more photons can’t raise its temperature but only keep imparting the same temperature.”

(To be clear about where this narrative goes wrong, it assumes “temperature” is a concept that is meaningful at the level of the energy imparted by an individual photon. It’s not. At that level, only the total energy imparted matters.)

So, he was now apparently claiming that a photon can’t raise temperatures higher than T=E/k where E is the photon energy and k is Boltzmann’s constant. (Photon energy is given by E=h𝞶 where h is Planck’s constant and 𝞶 is the photon frequency.)

I pointed out that this formula predicted that a 15-μm wavelength photon from CO₂, can’t raise temperatures higher than 959 Kelvin = 686ºC.

This was clearly not the desired result. So, the response came back that “the energy distributes among 5 degrees of freedom,” so you need to “divide by 5 and you’ll get -80ºC.” He added this claim to a Quora answer that he routinely updates.

I asked where this “5 degrees of freedom” figure came from, and he offered an “explanation” that wasn’t remotely plausible. (He talked about a hypothetical type of gas that would have 5 degrees of freedom per molecule, but which couldn’t physically exist. He didn’t explain how this was supposed to relate to any real material.) It seemed clear that 5 was simply the number that was needed to generate the answer that was wanted.

Of course, regardless of what divisor one chooses, the idea of computing a “maximum temperature” for a given wavelength of photons which “can’t be exceeded no matter how many such photons arrive,” is *not justified by, and is completely inconsistent with, the established and tested physical explanations of how temperature works.*

Since I published the text above, the person in question has moved on to new, entirely fabricated, easily disproven, justifications for claiming that CO₂ can’t warm things.

What this episode demonstrates to me is that these claims about 15-μm radiation from CO₂ are strongly driven by “motivated reasoning.”[16] There seems to be a** committed focus on finding justification for a predetermined answer, rather than a balanced effort to understand scientific reality.**

The goal seems to be to find as many justifications as possible for asserting that radiation from CO₂ can’t warm things—not to determine the truth about whether radiation from CO₂ can warm things.

**That’s not the way legitimate scientific analysis works.**

* * *

Bottom line: **There is no basis for the claim that CO₂ can’t warm things to temperatures higher than -80ºC.**

## Addendum

Those who misinterpret Wien’s Displacement Law seem to imagine that there is a unique correspondence between wavelength (or frequency) and temperature. Yet, that’s simply not the case.

There are many *different* ways of associating a wavelength (or frequency) with temperature, as depicted in the chart below.

The temperature of a perfect emitter, i.e., “black body”, is associated with a *distribution*of wavelengths/frequencies. That distribution can be characterized in a number of different ways.

Looking at where the distribution has its peak value, as is done in Wien’s Displacement Law, is a rather poor way of characterizing the distribution, since the location of that peak value varies, depending on how the distribution is plotted. As shown in the figure, there is a different value depending on whether one looks at the peak value relative to wavelength (orange curve at bottom), relative to frequency (green curve towards top), or relative to the logarithm or wavelength or frequency (violet curve).

It’s perhaps more meaningful to consider percentiles, since those are independent of how the distribution is plotted. See the quartile and median curves in the chart.

The *peak by wavelength* is actually a highly questionable way of characterizing the distribution, since the peak wavelength curve is essentially the same as the 25% quartile curve. Only 25% of the emitted radiation has wavelengths shorter than the peak wavelength, while 75% of the emitted wavelength has wavelengths longer than the peak.

Some have argued that it would be most sensible to focus on the average photon energy (light red dashed curve). The energy of a photon with a 15-micron wavelength (1.3×10^{−20} Joules) corresponds to the average energy of photons emitted by a black-body with a temperature of 355K / +82℃. That’s 162℃ warmer than the -80℃ temperature that climate skeptics try to associate with this radiation!

There is no rational reason to consider the peak wavelength curve to be any more important than the other curves. Climate skeptics do this only because (a) they don’t understand the actual physics, and (b) they can convince themselves that it justifies their false belief that 15-micron radiation corresponds to a low (-80℃) temperature—though it doesn’t.

## Footnotes

[5] Absorption coefficient of carbon dioxide across atmospheric troposphere layer

[6] Radiation and Heat Transfer in the Atmosphere: A Comprehensive Approach on a Molecular Basis

[8] Wien’s displacement law – Wikipedia

[9] Peaks of Blackbody Radiation Intensity