The Maxwell Boltzmann Distribution Effect

How carbon dioxide cools the atmosphere

Vincent J. Curtis

16 Oct 23

Maxwell-Boltzmann statistics are used in Statistical Thermodynamics.  The Maxwell-Boltzmann partition equation that we shall use is as follows:

N_i = N[exp – (ΔE/kT)] = N[exp – (hυ/kT)];

where N is the total number of molecules; Ni is the number of molecules in state i; ΔE is the difference in energy between the ground state and excited state i; h is Planck’s Constant, υ is the frequency, k is the Boltzmann constant, and T is absolute temperature in degrees Kelvin.  N_i/N is the ratio of the number of molecules in excited state i over the total number of molecules..

This equation shows the partition into ground and excited states for the temperatures relevant to the earth’s atmosphere.  Excitation arises from thermal collisions, and the one excited state of CO₂ is all that’s relevant to the problem at hand.

We obtain υ for the transition of a CO₂ molecule from the ground state to the vibrationally excited state represented by the absorption at 667 cm^-1 from the universal wave equation:

c = υƛ; 

where c is the velocity of light, υ the frequency, and ƛ the wavelength

Rearranging and substituting:

υ= c/ƛ = (3 x 10⁸ ms^-1/15 x 10^-6 m) = 2 x 10¹³ s^-1

Substituting into the Maxwell-Boltzmann equation:

N_i = N{exp – [(6.626 x 10^-34 x 2 x 10¹³)/(1.38 x 10^-23 x T)]}

N_i = N[exp – (960/T)]

For properly normalized results, we have to sum over the two states: the ground and excited state.  Since the ground state is deemed to have ΔE = 0, the exponential of that value gives 1.

Stated precisely, the normalized results for our two state system is given by:

            N_i/N = [exp –(960/T)]/[1 + exp –(960/T)]

Hence, the proportion of the excited state (N_i/N) at various temperatures are as follows:

T = 220K: 1.25 %

T = 288K: 3.45 %

T = 303K: 4.04 %

The values hold good at any pressure; at atmospheric pressure, this state is thermally quenched before emission.[1]  At low pressures, say, at 0.02 Torr, which obtains at 250,000’ altitude, radiation becomes the dominant[2] means by which an excited CO₂ molecule returns to the ground state; and in the upper atmosphere, this radiation can be emitted to space.

The effect of the Maxwell-Boltzmann distribution is the basis for the 220K curve in the blackbody radiation of the earth, as seen from outer space.  The CO₂ of the lower atmosphere completely absorbs all the radiation at 15 microns (667 cm^-1); but that in the upper atmosphere emits radiation at that frequency to space.  One can observe a spike in the middle of the big absorption peak at 667 cm^-1 that arises from the emission at high altitudes.  Emission of radiation to space is ultimately how the earth cools itself from solar radiation received during the day.

Because the proportion of CO₂ in the upper atmosphere is nearly the same as in the lower atmosphere, an increase in atmospheric CO₂ would eventually increase the amount of emission by CO₂ in the upper atmosphere to space, cooling the upper atmosphere more efficiently.  A cooler upper atmosphere would speed up the cooling of the lower atmosphere; hence, any warming from the increase of CO₂ (which I deny as a violation of the law of conservation of energy[3]) would be off-set by a faster rate of heat transport from the lower to the upper atmosphere.  And if the cooling effect in the upper atmosphere occasioned by a higher concentration of CO₂ was all there was, then increasing CO₂ would cool the atmosphere faster, and this does not violate the law of conservation of energy.

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[1] See 17 September 2023 entry “Carbon Dioxide Saturation and Global Warming.”

[2] See 28 September 2023 entry “There is no greenhouse effect, Part 1”

[3] See “There is no greenhouse effect, Part 3” to be published.