Equilibriums

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Ions and compounds are in equilibriums with each other. A typical equilibrium is the one between ammonia (NH₃) and ammonium (NH₄⁺) discussed in the section about pKa and Ka. These types of equilibriums are dependent on pH.

pH determines how much of a compounds or an ion in an equilibrium is dissociated. A simple example is already given in the pKa / Ka section, so here's a better one:

Consider the equilibrium between carbonic acid, bicarbonate and carbonic ion: H₂CO₃, HCO₃^- and CO₃^2-. First of all it should be noted that only approximately 1/400 of the carbonic acid H₂CO₃ is on the form H₂CO₃. The rest is on the form CO₂.

For this reason the carbonic acid in the "inorganic carbon equilibrium" is often denoted CO₂^* even though it at first sight seems unreasonable that CO₂^* can dissociate into HCO₃^-.

To calculate the concentration of individual species in the equilibrium one first have to introduce a variable denoting the sum of concentrations of CO₂^*, HCO₃^- and H₂CO₃. This variable should be called TIC - a shortcut for Total Inorganic Carbon.

TIC = [CO₂^*] + [HCO₃^-] + [CO₃^2-]

If we start by finding the [CO₂^*], CO₂^* should be isolated in the above equation. By rearranging the equation above to:

TIC = [CO₂^*]^. (1+ [HCO₃^-] / [CO₂^*] + [CO₃^2-] / [CO₂^*])

If both sides are divided by (1+ [HCO₃^-] / [CO₂^*] + [CO₃^2-] / [CO₂^*]) we get that:

[CO₂^*] = TIC / (1+ [HCO₃^-] / [CO₂^*] + [CO₃^2-] / [CO₂^*])

To move on with this equation and to see how it depend on [H⁺] we introduce [H⁺] in the denominator:

[CO₂^*] = TIC / (1+ [HCO₃^-]^.[H⁺] / [CO₂^*]^.[H⁺] + [CO₃^2-] / [CO₂^*])

At this point it is necessary to recall the definition of the dissociation constant. A more detailed explanation on this subject can be found on the site about pKa and Ka.

The ionisation constant of CO₂^* is [H⁺]^.[HCO₃^-]/[CO₂^*]

This ionisation constant can be substituted into denominator of the equation for TIC:

[CO₂^*] = TIC / (1 + Ka(CO₂^*) / [H⁺] + [CO₃^2-] / [CO₂^*])

The calculation is not finished yet. [CO₃^2-] and [CO₂^*] are not known.

They cannot be substituted the same way as before. However, the ionisation constant of HCO₃^- is [H⁺]^.[CO₃^2-]/[HCO₃^-].

If this, the ionisation constant of HCO₃^- is multiplied by the ionisation constant of CO₂^* we get that:

Ka(HCO₃^-)^.Ka(CO₂^*)=[H⁺]^.[HCO₃^-]/[CO₂^*] ^. [H⁺]^.[CO₃^2-]/[HCO₃^-] = [H⁺]²^.([CO₃^2-] / [CO₂^*]).

But this is the same as saying that:

Ka(CO₂^*)^.Ka(HCO₃^-) / [H⁺]² = [CO₃^2-] / [CO₂^*]

This can be substituted into the equation for the [CO₂^*] from before:

[CO₂^*] = TIC / (1 + Ka(CO₂^*) / [H⁺] + Ka(HCO₃^-)^.Ka(CO₂^*) / [H⁺]²)

Web resources

Theory about chemical equilibriums
Wikipedia equilibriums
A complete e-book about equilibriums (free)