# How to derive composite acidity constant for H2CO3*?

I see the equation (19) and (20) on Stumm and Morgan's Aquatic Chemistry book (Fig. 1 here). They said "a combination of equations 19 and 20" gives Equation 21. But I don't know how the Equation (21) is generated? Because if I make summation of (19) and (20), it will be:

$$\ce{2H2CO3 <=> H2O + CO2(aq) + H+ + HCO3-}$$ • @Zenix Yes, but here K_H2CO3* seems not the K times K_H2CO3. So I am confused about the way to derive from Equation (19&20) to Equation (21)
– T X
May 19, 2020 at 14:13

[OP] Because if I make summation of (19) and (20), it will be: [...]

They meant they combine the mathematical expressions, not the chemical equations.

To express

$$K_\ce{H2CO3^\Huge{*}} = \frac{K_\ce{H2CO3}}{1 + K}$$

in terms of concentrations, you substitute equations (1) and (2) into the expression:

$$K = \dfrac{\ce{[CO2(aq)]}}{\ce{[H2CO3]}}\tag{1}$$ $$K_\ce{H2CO3} = \dfrac{\ce{[H+][HCO3-]}}{\ce{[H2CO3]}}\tag{2}$$

So

$$\frac{K_\ce{H2CO3}}{1 + K} = \frac{\dfrac{\ce{[H+][HCO3-]}}{\ce{[H2CO3]}}}{1 + \dfrac{\ce{[CO2(aq)]}}{\ce{[H2CO3]}}}$$

Multiplying top and bottom of the fraction by $$\ce{[H2CO3]}$$ gives:

$$\frac{\ce{[H2CO3]}}{1 + K} = \frac{\ce{[H+][HCO3-]}}{\ce{[H2CO3]} + \ce{[CO2(aq)]}}$$

If you want to make the intention clearer, you define a total concentration of carbon dioxide as

$$[\ce{H2CO3^\huge{*}}] = \ce{[H2CO3]} + \ce{[CO2(aq)]}$$

and then write

$$K_\ce{H2CO3^\Huge{*}} = \frac{K_\ce{H2CO3}}{1 + K} = \frac{\ce{[H+][HCO3-]}}{[\ce{H2CO3^\huge{*}}]}$$

• +1 Ah... very nice. A much less convoluted derivation than mine.
– MaxW
May 19, 2020 at 23:01
• @AdnanAL-Amleh - I deleted my answer since Karsten's is so very much better. $K_\ce{H2CO3^*}$ is the fictional $K_{\alpha 1}$ for $\ce{H2CO3}$ assuming that the dissolved $\ce{CO2}$ is $\ce{H2CO3}$. In reality $\ce{CO2(aq) > \text{actual }H2CO3}$
– MaxW
May 19, 2020 at 23:12
• @MaxW Good point about the actual concentrations, so it should really be called $$\ce[\ce{CO2(aq)}^*]$$ May 20, 2020 at 0:05

I take perhaps a more intuitive view of the system, and in the case of, for example, Iron bicarbonate, it intuitively explains its formation in well water under pressure and its subsequent decomposition upon exiting a faucet at ground level.

I start with a modified cited equation (19) citing gaseous and not aqueous carbon dioxide in the equilibrium reaction:

$$\ce{ H2CO3 (aq) <-> CO2 (g) + H2O (l) }$$

I also cite Equation (20) and introduce dissociated CO2 as represented by carbonic acid:

$$\ce{ CO2 (d) = H2CO3 (aq) = H+ (aq) + HCO3- (aq)}$$

Now, at high pressure, examining the modified (19) above, per Le Chatelier's Principle, the position of equilibrium will move to produce more carbonic acid, and more bicarbonate per the second equation, resulting in the creation of Iron bicarbonate.

However, with the release of pressure, the bicarbonate equilibrium concentration shifts to create more carbonate acid, which can decompose releasing CO2 gas resulting in the ultimate removal of the bicarbonate ion and the decomposition of Iron bicarbonate, as reported.

Note, to preserve the total concentration of carbon dioxide in the system:

CO2 (d) + CO2 (g) = [CO2]