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Carbon Dioxide and Henry’s Law

by James R. Barrante, Ph.D.

 

There have been a number of unsuccessful attempts to relate the level of carbon dioxide in the atmosphere to the concentration of dissolved CO2 in the oceans using Henry’s Law.  The Henry’s Law equation is a simple linear relationship between the partial pressure of CO2 in the atmosphere to the activity of dissolved CO2 gas

CodeCogsEqn-2

Dissolved carbon dioxide reacts with water to form a weak acid, carbonic acid.  Consequently, the Henry’s Law constant k is not affected significantly by the high ionic strength of seawater.  The activity coefficient of undissociated carbonic acid is close to unity, so the activity of dissolved CO2 can be replaced by its concentration in moles per liter.  This, of course, is not true for the dissociation constants of carbonic acid.  (See Ocean Acidification I and II).   Like all equilibrium constants, the Henry’s Law constant is temperature dependent.  Therefore, an average partial pressure of atmospheric CO2 must be carefully calculated.  That is, it is not acceptable to determine the partial pressure of atmospheric CO2 at various points around the globe and average them together.

One must first recognize that the temperatures of the oceans are not randomly distributed around the globe, but are banded in zones following closely to the latitude lines.  Moreover, the latitude zonal surface seawater areas differ at least two ways.  First, from the equator to each pole the total surface area around the globe is different, and second, each area contains a different amount of land mass.  Consequently, before any calculations can be done, the total relative surface area of each zone must be determined.  This author found that a zonal width of 5˚ was adequate.  Since we are concerned with relative surface area, it is not necessary to use actual values.  The surface area of a zone is easily calculated using the equation

S  =  2 π r h

where is the radius of the globe and h is the height between successive latitude lines.  Table 1. shows the surface area of each zone on a globe of radius 50.0 units.  Notice that the difference in surface area between the first three zones is for all practical purposed zero.  We find that is it not necessary to go much beyond 80˚ north and south.

Table 1.

Latitude Range      Difference in Latitude          Surface Area
        (degrees)                                       Lines                          (square units)                                                    
0˚ to 5˚                                           4.2                                               1319
5˚ to 10˚                                          4.2                                               1319
10˚ to 15˚                                         4.2                                                1319
15˚ to 20˚                                         4.1                                                1288
20˚ to 25˚                                         4.1                                                1288
25˚ to 30˚                                         3.8                                                1194
30˚ to 35˚                                         3.8                                                1194
  35˚ to 40˚                                         3.5                                                1100
  40˚ to 45˚                                         3.4                                               1068 
  45˚ to 50˚                                         3.0                                                 942  
 50˚ to 55˚                                          2.5                                                  785
 55˚ to 60˚                                         2.4                                                  754
  60˚ to 65˚                                         2.1                                                  660
  65˚ to 70˚                                          1.7                                                  534
   70˚ to 75˚                                          1.4                                                  440
     75˚ to 80˚                                          0.9                                                  283   
 

Once the surface area of each zone is found, it is necessary to determine the fraction of that area that is water and the average temperature of that water.  Figure 1. shows a grid of the globe designed to do this.   The average temperature of

SS Area(2)

Figure 1.  Planar grid of globe.

each zone was found by layering the grid shown in Figure 1 on top of a SST map such as those produced by the National Weather Service Environmental Modeling Center for 2014.  Table 2. shows the fraction of each zone that is water and the average temperature of each zone.

Table 2.
Table2
 

Knowing the average sea surface temperature of each zone and using the equation describing the variation of the Henry’s Law constant with respect to temperature, we can determine the partial pressure of CO2 associated with each zone.  To do this, we must assume a level of dissolved CO2.  From the post Ocean Acidification II, a reasonable average value is 1.36 x 10-5M. By multiplying the fraction of ocean by the surface area of a particular zone, we can determine the zone’s contribution of partial pressure CO2 to the average total pressure of CO2.    The results of these calculations are shown in Table 3.

Table 3.

Table 3

We see that the total weighted average pressure of CO2 is 347.2 ppm, not too bad, considering the uncertainty in the concentration of dissolved CO2.  Keep in mind that this is an equilibrium value that may take a couple hundred years to be established. If the instantaneous level of CO2 is measured to be around 400 ppm, this would indicate that the contribution of CO2 from all other sources, including burning fossil fuels, is about 50 ppm.

It is important to note the silliness in measuring CO2 levels around the globe and assuming that the average has some statistical meaning.  Look at the data in Table 3.  Simply based on ocean temperature around the equator, CO2 levels are well over 400 ppm, while levels in Antarctica are similar to those found during ice ages.  This, obviously, has to be the case to get an average somewhere between these two extremes.  This really puts into question the validity of ice-core data as a proxy for “global” CO2 levels.  It is difficult to understand how samples of ice taken from Antarctica could represent global CO2 levels in violation of Henry’s Law.

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