Alkalinity

This is the second of who knows when it will end series of post whose goal is to provide some of the background needed to understand the mechanism by which increasing the concentration of CO2 in the atmosphere acidifies the oceans, leading to a decrease in dissolved inorganic carbon which is needed by sea creatures such as corals and oysters to build shells. 

Continuing what Eli started about a week ago, a Revelletion as it were, discussing how the carbonate concentration of the oceans buffers changes in acidity (expressed as pH = - log10[H]⁺(aq)], because much of the excess hydrogen ions produced in the forward reaction (2)

(R2)  CO2(aq)+ H2O = HCO3^-(aq) + H⁺(aq)

 when carbon dioxide dissolved in the ocean are consumed in the reverse reaction (-3).  

(R-3) H⁺(aq) + CO3^2-(aq) =  HCO3^-(aq)

The bunnies know that the pH as  - log10([H⁺](aq)) where the square brackets are the concentration of whatever is in between them.  The concentration of dissolved inorganic carbon (DIC) in the ocean is

 (1) [DIC] = [CO2(aq)] + [HCO3^-(aq)] + [CO3^2-(aq)]

where the carbonate ions, CO3^2-(aq), are the stuff from which the shells are made.  Decrease the carbonate ion concentration even at constant or increased DIC and the little critters shells never form or dissolve.  Since some of the little critters are beautiful (corals) or tasty (lobsters), this is not good. 

Alkalinity is defined as the total concentration of species which can neutralize acids, that is combine with H⁺(aq)

(2) alk = [HCO3^-(aq)] + [CO3^2-(aq)] + [OH^-(aq)] - [H⁺(aq)] + [B(OH)₄^-(aq)]

This pretty much follows Revelle Revisited by Egleston, Sabine and Morel, who, in turn are following, somewhat more elegantly, a lot of folk.  The definition of alkalinity offers a hint of why decreases of pH, even above pH 7 are referred to as acidification not dealkalification, because alkalinity includes a lot more than hydroxide, OH^- concentration.

At this point, some chemistry is needed. A weak acid, call it HA,  is an acid that does not completely dissociate into  H⁺(aq) + A^- (aq).  The degree of ionization is  captured by the acid ionization constant Ka

(3)  Ka = ([H⁺(aq)] [A^-(aq)]) / [HA(aq)]

looking at Reaction (2) we can write

(4)  Ka = ([H⁺(aq)] [HCO3^-(aq)]) /( [CO2(aq)][H2O])

since the concentration of water does not change during the reaction it remains pure water this can be written as

(5) Ka = ([H⁺(aq)] [HCO3^-(aq)]) /[CO2(aq)]

and rearranged to

 (6) [HCO3^-(aq)]= Ka [CO2(aq)] / [H⁺(aq)]

The same idea goes for Reaction 3 from the earlier post

(R3) HCO3^-(aq) = H⁺(aq) + CO3^2-(aq)

where

 (7) Ka2 = ([H⁺(aq)] [CO3^2-(aq)]) /[HCO3^-(aq)]

and again rearranging in the same way as in (6)

(8) [CO3^2-(aq)] = Ka2 [HCO3^-(aq)] / [H⁺(aq)]

 Subtituting (6) into this we get

(9) [CO3^2-(aq)] = Ka Ka2 [CO2(aq)]/ ([H⁺(aq)]²

The reason for all this algebra is to write the DIC, the carbon in the ocean accessible for forming shells as a function of  [CO2(aq)] and [H⁺(aq)] (or pH if someone prefers)

(10) [DIC] = [CO2(aq)] + Ka [CO2(aq)]/[H⁺(aq)] + Ka Ka2 [CO2(aq)]/[H⁺(aq)]²

or

 (11) [DIC] = [CO2(aq)](1 + Ka /[H⁺(aq)] + Ka Ka2 /[H⁺(aq)]²)

 The concentration of hydrogen ions, [H⁺(aq)], can be precisely measured by measuring the pH.  Measuring [DIC] or [CO2(aq)] is a story for later.