Can 400 ppm CO2 Provide the Heating Required by the AGW Hypothesis?

This is a thought experiment.  The idea is to make a simple test of the plausibility or even possibility of the claim of the catastrophic man-made or anthropogenic global warming (AGW) hypothesis that the increase from about 300 ppm (parts per million) of carbon dioxide in the atmosphere over the last 120 years or so has caused a surface temperature increase of 1°C or 1K.

Now I know that this is a concept very much ignored by the advocates of catastrophic man-made global warming, but I am going to assume that the Law of Conservation of Energy, which they have not explicitly refuted, still holds.  I have a high level of confidence in this law of physics.

Let us start out with the simplifying assumption that every air molecule has the same heat capacity at constant pressure.  This means that the same amount of heat is required to raise the temperature of each kind of molecule by 1K.  We will discuss the correction for this below.  I will also make the simplifying assumption that the carbon dioxide molecules are heated fully to the necessary temperature and then dumped into the air whose temperature is raised by 1 K upon reaching equilibrium.  The limitations on this assumption will also be examined later.

If the addition of 100 molecules of CO₂ to the prior 300 molecules/ 1 million air molecules has caused all air molecules near the surface of the Earth to have a 1K temperature increase, then given Conservation of Energy, each of the CO₂ molecules has to gain an additional temperature of T_a which is transferred by collisions to all the other gas molecules until they are all at an equilibrium temperature 1K higher than the air had previously been.  T_a is given by:

1,000,000 (1K) = 400 T_a

T_a = 2500 K,

because of the proportional relationship of heat, a form of energy, to the temperature of the gas molecules.

Now let us allow for the fact that the heat capacities per mole at constant pressure of the molecules in air are not equal.  We want the values per mole because that gives them for equal numbers of molecules, while the per gram values of heat capacity do not.  We have for each gas heat capacity values of:

N₂, 29.12 J/K mol.

O₂, 29.38 J/K mol.

Ar, 20.79 J/K mol.

H₂O, 28.03 J/K mol.

CO₂, 36.94 J/K mol.

Air, 29.07 J/K mol.

We discover now that a 1 K increase in the temperature of a molecule of CO₂ gives it more additional energy than the average air molecule needs to raise its temperature by 1K.  So, the temperature per CO₂ molecule that we calculated above will not need to be quite so large.  The required temperature T_a will only be:

T_a = (2500 K) / (36.94 / 29.07)

T_a = 1967 K

Now, recall that this is only the additional temperature of each CO₂ molecule required to cause a 1K temperature increase spread evenly over every air molecule in equilibrium at the Earth’s surface.

The same infra-red absorbing properties claimed to give CO₂ molecules this amazing power, would also be acting in an atmosphere with a mere 100 ppm of CO₂.  In fact, each additional 100 ppm of CO2 has a diminishing effect on heating other gases according to the proposed mechanism for such heating.  The mean free path length for each infra-red absorption event by a molecule of CO₂ becomes exponentially shorter and shorter as more CO₂ molecules are added.  The size of the effect for additional molecules becomes less and less.  Consequently, the first 100 ppm of CO₂ molecules would heat the air by more than 1K given that 400 ppm of CO₂ molecules heat it by 1K upon the increase in concentration from 300 ppm to 400 ppm.  Note that this means that the first 100 ppm of molecules of CO₂ would have to achieve a temperature more than four times 1976 K or 7868 K.

The further increase in the temperature upon going from 100 ppm to 200 ppm would also be greater than that for going from 300 ppm to 400 ppm, as would be the temperature increase on going from 200 ppm to 300 ppm.  Each concentration increase of 100 ppm would cause a smaller temperature increase than the previous one did.  But upon arriving at 200 ppm concentration of carbon dioxide, there are half as many molecules of it as one has upon arriving at 400 ppm of CO₂, so each molecule has to be twice as hot.  Upon arriving at 300 ppm from 200 ppm, each molecule has to be 4/3rds as hot as it did at 400 ppm.

So let us put together the effect that the wonder molecule of CO₂ has had on heating the Earth's surface temperature to date, we calculate the additional minimal temperature each CO₂ molecule must have to be to be consistent with the catastrophic AGW claim:

0 to 100 ppm:   T_a > 4(1976 K) = 7868 K

100 to 200 ppm:  T_a > 2(1976 K) = 3952 K

200 to 300 ppm:  T_a > (4/3)(1976 K) = 2635 K

300 to 400 ppm:  T_a = 1976 K

We can now calculate the cumulative effect of  CO₂ molecules given that we have very conservative lower limit values for the heating of the atmosphere provided by CO₂ molecules above.  The heating due to the miracle molecule at lower concentrations has not gone away.  The energy being dumped into the atmosphere is still there, though it is now shared by a larger number of CO₂ molecules.  Let T_t be the cumulative necessary temperature of our present 400 ppm of carbon dioxide to provide this total base of energy.  Invoking Energy Conservation once again, we must conservatively have:

400 T_t > 100 (7868 K) + 200 (3952 K) + 300 (2635 K) + 400 (1976 K)

T_t > 7895 K

Consequently, the temperature required per CO₂ molecule to be transferred to all air molecules by collisions to establish an equilibrium temperature based on the claims of the advocates of the catastrophic AGW hypothesis is substantially greater than 7895 K.

This required temperature per CO₂ molecule is then substantially greater than 7895 K or 7622°C.  The sun is the source of this energy and its surface temperature is only 5778K or 5505°C.  A cooler body cannot heat another body to a temperature greater than its own temperature.  The highest temperature anything could be raised to on Earth by solar radiation is further diminished by the distance of the Earth from the sun and the fact that radiation emitted by the Earth is mostly directed at temperatures in space of about 4K.

There is, of course, another problem.  The carbon dioxide molecules would cease to be molecules long before they could reach these hypothetical temperatures.  The very collisions that spread the heat from the infra-red active molecules to nitrogen and oxygen molecules would be so energetic that carbon dioxide molecules upon collision would be ionized and destroyed.  The unfortunate molecules struck so violently would be ionized.  You cannot imagine how bad the ozone problem would be as a result of the ionization of the oxygen molecules in the air!

If the claimed CO₂ heating effect were even a small fraction of what is being claimed, then there might be a large fraction of carbon dioxide, water, and methane molecules flying about in the air at velocities much greater than those of the Maxwell-Boltzmann velocity distribution.  However, that distribution is a fairly accurate description of actual air molecule velocity distributions and it has been tested for many decades. So why do we see no substantial such violation of the Maxwell-Boltzmann equation for the velocity of gas molecules?

It might be due to a combination of a minimal temperature increase in the molecule for each absorption event and a rapid collision rate preventing the accumulation of energy from more than one such absorption event in the molecule.  Almost every time a carbon dioxide molecule absorbs an infra-red photon emitted by the Earth's surface, that molecule might commonly have several collisions with other air molecules, losing that absorbed energy.  There is very good reason to believe this is the case that the energy gained by one absorption event is quickly lost in large part by collisions with other molecules.  Only occasionally does the absorbed infra-red energy get re-emitted as an infra-red photon before it has a collision.  So most infra-red active gas molecules are in equilibrium or nearly so with the other gases in the air surrounding them.  In other words, the number of absorption events is actually rather small for a given molecule and it does not build up very great amounts of energy to transfer to the non-infra-red active gas molecules of the air.

The primary infra-red wavelength emitted by the surface of the Earth which is absorbed by carbon dioxide molecules is the 15 micron wavelength.  This is the equivalent of a photon with an energy of 0.0827 eV or 1.325 x 10^-20 J.   Recall that the heat capacity of carbon dioxide at constant pressure is 36.94 J/K mol.  There are 6.023 x 10²³ (Avogadro's Number) carbon dioxide molecules in a mole, so we can divide the heat capacity per mole by Avogadro's number to find the heat capacity of a single molecule of carbon dioxide.  The heat capacity of a single molecule is then 6.135 x 10^-23 J/K.  The absorption of one 15 micron infra-red photon by one carbon dioxide atom would then raise its temperature by

(1.325 x 10^-20 J) / (6.135 x 10^-23 J/K) = 216.0 K

So while the frequency of collisions with other molecules may prevent the accumulation of energy in the carbon dioxide molecule at multiples of the single absorption event energy increase, even a single energy absorption event does raise the velocity of a carbon dioxide molecule a great deal.  The scale of the claimed catastrophic AGW effect requires each carbon dioxide molecule to transfer the heat equivalent of at least a 7895 K molecule to the air around it.  This is 36.55 absorptions of a 15 micron photon per molecule to be transferred by collisions.  Somewhat more absorption events are required to accommodate the lower probability that the energy will be given up in the emission of another photon.

The Maxwell-Boltzmann velocity distribution curve is well-known for an ideal gas and it has been well confirmed by experiment.  The number of higher velocity carbon dioxide molecules due to the absorption of infra-red radiation emitted from the Earth's surface and their time duration in the excited state should be a measurable parameter that can be used to test the claims of the catastrophic AGW hypothesis.  If the advocates of that hypothesis, foremost being the U.S. government, were serious about determining its validity, these measurements would have been performed.  What is more, those measurements would be known and discussed as a function of altitude.  The fact that little attention has been devoted to such issues is a warning flag that the U.S. government and other advocates of this hypothesis are not serious about wanting it to be tested.

So, we have performed a simple thought experiment based on what we are told is the central fact of the catastrophic man-made global warming hypothesis and found that the hypothesis does not require an impossible effect from this point of view, however a hoot it at first appeared by the first part of the argument above.  However, the basic parameters that would define the effect should have long since been measured to define the basic physics.  Discussions of the mean free path of infra-red radiation in the atmosphere at various altitudes should be common.  It should be common knowledge what fraction of the carbon dioxide molecules are in an excited state at any given moment.

I have made several other arguments that do explain why the physics behind catastrophic AGW is wrong.  It is in fact very wrong, but one has to develop other arguments than the one above to show that to be the case using readily available data.  It is a shame that the data discussed above is not readily available.

Substantial re-writes were made on 18 and 19 October 2015 of this post.  A serious error due to an unreliable energy converter on-line existed in the 18 October version and is corrected in the 19 October version.