A Hypothetical Earth Atmosphere of Carbon Dioxide and Comments on Vertical Mixing

There is a claim by some doubters of the catastrophic man-made global warming hypothesis that because CO₂ is a heavier molecule at 44.01 atomic mass units (amu) than the mean mass of air at about 28.96 amu, carbon dioxide will settle into higher concentrations at low altitude and be much rarer at higher altitudes than the lighter molecules making up nearly 100% of our atmosphere.  In other words, it will not maintain the same ratio of number density to other air molecules as the altitude increases.  I will show why this is a reasonable claim, but I will also caution that there are powerful forces stirring the atmosphere and we should not be surprised that the case for a molecule at less than 400 ppm concentration is very different than when it is 100% of the atmosphere.

In fact, the vertical distribution of carbon dioxide is a very reasonable measurement to make experimentally.  The claim is that the actual measurements show it to be well mixed vertically through the troposphere.  There is apparently a slight decrease in relative density in the stratosphere, but that is thought to be due to a slow diffusion rate through the tropopause in response to the increase in carbon dioxide in the troposphere.  These are measurements I would expect atmospheric scientists to get right, so I am inclined to accept that the forces mixing the heavy atmospheric gases with the lighter atmospheric gases are highly effective.  Note that water vapor is not so well-mixed, but this is due to its high transition temperature from the liquid to the gas phase.

Still, it is interesting to look at some properties of a carbon dioxide atmosphere with a few boundary conditions applied from our present atmosphere.  Let us assume an average surface temperature of 14.5° = 287.65K, even though the solar insolation incident upon the Earth’s surface would undoubtedly be different with a 100% carbon dioxide atmosphere.  The cooling mechanisms would also be affected.  Let us also assume that the atmosphere has the same total number of molecules in it.  Then we are going to ask how the sea level pressure, the sea level number density of atmospheric molecules, and the temperature gradient with increasing altitude change.  We will ignore any effects due to absorption of solar insolation in the carbon dioxide atmosphere as well.  By making these assumptions, I am not implying that these are unimportant effects.  I am not implying that my assumptions are realistic.  Nonetheless, we will see some interesting properties of the hypothetical atmosphere.

At sea level, the hypothetical carbon dioxide atmosphere will exert a pressure (44.01)/(28.96) = 1.520 times that of the air atmosphere we now have, given the same number of gas molecules.  The U.S. Standard Atmosphere says that sea level pressure is 1.01325 bar.  In comparison, the carbon dioxide atmosphere will have a sea level pressure of 1.540 bar or 1.54 x 10⁵ N/m².

Let us find out what the number density of carbon dioxide molecules is at sea level.  For a perfect or ideal gas, P = (2/3) (N/V) , where P is the pressure, N is the number of molecules, and V is the volume.  N/V is the number density of the molecules, which is what we want to calculate.

To do that, we need to calculate the mean kinetic energy of translation.  Because the temperature is proportional to the mean kinetic energy for a perfect gas, we can calculate the mean kinetic energy knowing the gas temperature.  So, at sea level, the carbon dioxide gas was posited to have the present Earth surface temperature of 287.65K.

Now, a monatomic gas, such as the argon that makes up about 0.93% of our atmosphere, has a temperature (T) to total kinetic energy (E_t) relationship of E_t = (3/2) kT, where k is the Boltzmann Constant of 1.381 x 10^-23 J/K.  Nitrogen and oxygen molecules have two rotational degrees of freedom in addition to the three translational degrees of freedom, so E_t = (5/2) kT.  Molecules such as CO₂ with 3 or more atoms have 3 degrees of rotational freedom and a total kinetic energy to temperature relationship given by E_t = (6/2) kT = 3 kT.  Consequently, carbon dioxide at sea level with T = 287.65K has a total mean kinetic energy of 1.192 x 10^-20 J.  Half of this energy is in the rotational modes and half is translational kinetic energy.  Therefore, the mean translational kinetic energy at sea level is 5.96 x 10^-21 J.

Now, using the pressure to mean translational kinetic energy relationship three paragraphs up:

P = (2/3) (N/V)

1.54 x 10⁵ N/m² = (2/3) (N/V) (5.96 x10^-21 J)

N/V = 3.876 x 10²⁵/m³

The U.S. Standard Atmosphere sea level molecular number density is only 2.547 x 10²⁵, so the carbon dioxide atmosphere has a number density 1.522 times greater according to this calculation.  The number density increase factor is the same as the pressure increase factor.

Now, if this perfect gas atmosphere of carbon dioxide has an increased number density of molecules at sea level and the same total number of molecules in the atmosphere, then the number density of such an atmosphere must decrease with increasing altitude more rapidly than does the number density in our air atmosphere.  It is this fact that has excited some who doubt the catastrophic man-made global warming hypothesis based on man’s emissions of carbon dioxide.  They believe there ought to be a much lower ratio of carbon dioxide to the other gases in the atmosphere at higher altitudes in the troposphere.

However, in our present atmosphere, the frequency of molecular collisions at sea level is 6.92 x 10⁹/s and at 10 km altitude it is still 2.06 x 10⁹/s.  At sea level, the mean free path length between collisions is only 66 nm.  At 10 km altitude, the mean free path is 197 nm.  In other words, the almost 100% of other molecules are slamming into the carbon dioxide molecules constantly and apparently do a very good job of stirring them into the mix of the dominant gases.  Carbon dioxide molecules apparently do not have the opportunity to settle out to lower altitudes as a result.

Before we leave our hypothetical carbon dioxide atmosphere, it is interesting to calculate the static temperature gradient in the atmosphere with increasing altitude.  We have for altitude h:

E_{t, h=0} = E_{t , h=10000m} + mgh

3 kT_h=0 = 3 kT _h=10000m + mgh

1.192 x 10^-20 = 3 kT _h=10000m + (44.01 amu)(1.660 x 10^-27 kg/amu)(9.776 m/s²)(10000 m)

T _h=10000m = 115.33K at 10,000m altitude

This is a much lower temperature than we have in the case of the perfect gas and static and dry U.S. Standard Atmosphere of 223.25K.  This hypothetical carbon dioxide atmosphere would have a cooling temperature gradient of -17.23K/thousand meters, instead of the U.S. Standard Atmosphere temperature gradient of -6.49K/thousand meters.  Of course this much greater temperature gradient ignores solar radiation absorbed by the carbon dioxide directly and was based on the unlikely condition that the Earth surface temperature would be unchanged.  Still, this is an interesting property of an atmosphere with a very heavy carbon dioxide concentration.  It is a consequence of its greater mass, its added rotational degree of kinetic energy freedom, and Conservation of Energy in the Earth's gravitational field.