Appendix II The
Effect of Carbon Dioxide on pH
Carbon dioxide (CO_{2}) is an acid anhydride, which means that
when it is dissolved in water it lowers the pH of the solution. In fact
pure rainwater has a pH of 5.6 because of the 360 ppm atmospheric
concentration of CO_{2}. During World War II, atmospheric
concentrations of CO_{2} were closer to 330 ppm, but that is not
an important difference for the purposes addressed here. One
can think of the process as having three steps all of which involve a
chemical equilibrium. The first step is the dissolution of
CO_{2} in water; its equilibrium constant is the Henry's
Law constant of CO_{2}, K_{hc}:
CO_{2} + H_{2}O <=> H_{2}CO_{3}
The second step is the acid base reaction between carbonic acid and
water; its equilibrium constant is denoted here as K_{c1}:
H_{2}CO_{3} + H_{2}O <=>
H_{3}O^{+} + HCO_{3}^{}
The third step is reaction between the bicarbonate ion
(HCO_{3}^{}) and water whose equilibrium constant is here
denoted as K_{c2}:
HCO_{3}^{} + H_{2}O <=>
H_{3}O^{+} + CO_{3}^{2}
Also, one needs to consider the autoprotolysis of water:
2H_{2}O <=> H_{3}O^{+} + OH^{}
In aqueous solution, one can treat the concentration of water as a
constant and derive a constant for the equilibrium, K_{w}.
These constants and their associated enthalpies are reported in
Seinfeld. ^{1}
Here is a table of the relevant quantities:
Table II.1: Equilibrium Constants and Associated
Enthalpies for the Absorption of Carbon Dioxide by Water.

Constant 

Value 

Enthalpy at 298 K (kcal/mole) 
K_{hc}   3.4 x 10^{2}   4.846 
K_{c1}   4.283 x 10^{7}   1.825 
K_{c2}   4.687 x 10^{11}   3.55 
K_{w}   1.008 x 10^{14}   13.345 
The slight temperature dependence of these constants can be accounted
for by means of the enthalpies given. The pH of a solution is defined
as log[H^{+}] where [H^{+}] is the concentration of
hyronium ions in moles/liter. From the information given it is possible
to derive a cubic equation expressing the relationship between
[H^{+}] and the partial pressure of CO_{2},
P_{CO2}. This equation is also presented in
Seinfeld: ^{1}
[H^{+}]^{3}  (K_{w}+K_{hc}K_{c1}
P_{CO2})[H^{+}]2+K_{hc}K_{c1}
K_{c2}P_{CO2}=0
I solved this equation by means of a simple computer program employing a
rootfinding method similar to Newton's method.
^{2}
The results are shown in figure II.1.
Figure II.1
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