Percentage of each carbonic acid component in DIC in seawater

　Write down the equation for each carbonic acid component concentration in terms of 【DIC】 to determine its coefficients a, b, and c.

【DIC】=a【H₂CO₃】、【DIC】=b【HCO₃^-】、【DIC】=c【CO₃^2-】

　Find the ratio (1/a, 1/b, 1/c ) of each component to the 【DIC】 of seawater (pH 8).

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Transforming the previous equations (1), (3), and (4).

　【DIC】=a【H₂CO₃】

　【DIC】=b【HCO₃^-】

   【DIC】=c【CO₃^2-】

The reciprocal of each value of a, b, and c is the ratio of each component to 【DIC】.

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For now, let's lay out all the equations so far.

Equation (1)　【DIC】 = 【HCO₃^-】 + 【CO₃^2-】 + 【H₂CO₃】 

Equation (2)　　　K0 = 【H₂CO₃】／pCO₂

Equation (3)　　　K1 = 【HCO₃^-】・【H⁺】／【H₂CO₃】 

Equation (4)　　　K2 = 【CO₃^2-】・【H⁺】／【HCO₃^-】 

Equation (5)　　　Kw = 【H⁺】・【OH^-】 　　

　K0 = 10^-1.489、K1 = 10^-5.882、K2 = 10^-9.035、Kw = 10^-13.409

There are equations (1) through (5), but let's strategize which one to use. Since pCO₂ is not marked in the solution clue, it is likely that Equation (2) will not have to be used.

Since both Equations (3) and (4) have 【HCO3-】, substituting equations (3) and (4) into Equation (1), we obtain 【HCO3-】 = b[DIC]. First, transform equations (3) and (4).

Equation (3)'　　【H₂CO₃】= 【HCO₃^-】・【H⁺】／K1

Equation (4)'　　【CO₃^2-】= 【HCO₃^-】・K2／【H⁺】

Substitute this into Equation (1).

【DIC】=   【HCO₃^-】・【H⁺】／K1+【HCO₃^-】+【HCO₃^-】・K2／【H⁺】

             = (【H⁺】／K1 +1 + K2／【H⁺】 )【HCO₃^-】　　　Equation (7)

b = ( 【H⁺】／K1 +1 + K2／【H⁺】 )　was obtained.

 【DIC】=b【HCO₃^-】　→　【HCO₃^-】=【DIC】／b    Equation (H)   Substituting this into Equations (3) and (4), we also obtain a and c.

Substituting Equation (H) into Equation (3)　K1 = (【DIC】／b )・【H⁺】／【H₂CO₃】 

then, 【H₂CO₃】 = (【DIC】／b )・【H⁺】／K1  =  (【H⁺】/( b・K1 ) )・【DIC】　　Equation (6)

1/a =   (【H⁺】/( b・K1 ) )　　was obtained.

Substituting Equation (H) into Equation (4)　K2 = 【CO₃^2-】・【H⁺】／（【DIC】／b）

then,【CO₃^2-】= ( K2／(b・【H⁺】) ) ・【DIC】　　　Equation (8)

1/c = ( K2／(b・【H⁺】) ) 　　was obtained.

　Find the values of a, b, and c, assuming K1 = 10^-6, K2 = 10^-9, pH = 8, and 【H⁺】 = 10^-8 (mol/L), so that you can do the calculations by hand.

【H⁺】／K1 +1 + K2／【H⁺】

b =  10^-8／10^-6 +1 + 10^-9 ／10^-8  = 10^-2 +1 + 10^-1 = 1.11  ,    1/b = 0.9009009,,,

1/a =   10^-8/( b・10^-6 ) = 10^-2／1.11 =  0.009009009,,, 

1/c =  10^-9／(b・10^-8) = 10^-1／b  = 0.09009009,,,

In seawater at pH 8, we see that 【HCO₃^-】 accounts for 90%, 【CO₃^2-】 for 9%, and 【H₂CO₃】 for 1% of 【DIC】.

Note 1) Although we substituted the equilibrium constant and the value of 【H⁺】 at the end of this calculation, it is easier to summarize the equation by substituting it at the beginning of the series of calculations.

Note 2) I have conveniently changed the equilibrium constant, but you can use a function calculator or Excel to calculate the constant as it is.