The aim here is to revise the concept of pK and apply it to the dissociation of tooth mineral. The ion of primary interest in this is phosphate which, in its various states, is both an acid because it can donate a hydrogen ion and a base because it can accept one.

Strong electrolytes are almost completely ionized in aqueous solution. Weak electrolytes are only partially ionized and a solution contains the undissociated compound as well as ions.

Many acids (HA) are weak electrolytes and partially dissociate to produce hydrogen ions (H+) and the generalized anion (A-). It follows then that a measurement of the hydrogen ion concentration of a solution of a weak acid does not give the total hydrogen ion concentration because some of the hydrogen remains in the undissociated form. However, if the acid is titrated the undissociated compound progressively dissociates and eventually the total hydrogen ion concentration is found. This is why the "titratable acid" is distinguished from the actual acidity (pH). With strong acids the titratable acid and acidity are the same.

A weak acid (HA) dissociates in reversibly in water.

The dissociation constant (K) defines the ratio of the concentrations of the dissociated ions and the undissociated acid.

Square brackets denote concentration.

Rearranging the equation to isolate the hydrogen ion concentration

Convert both sides of the equation to log base 10

Multiply by -1 to create the term -log[H]

Substitute the term pH for -log[H] and pK for -logK

The end result are two terms. One is pH, the familiar measure of acidity or alkalinity, the other is pK.

Actually it is a very useful term. One way of looking at its value is to consider the phenomenon of pH buffering.

Buffer action is a property of a weak acid and its salt. Unlike the weak acid, the salt is fully dissociated. If we assume that the pK of the weak acid is 5 which is an entirely plausible value, then the [A] due to the presence of the acid is 0.00005M for each mole of acid. This means, in effect, that the acid's contribution to the total [A] is overwhelmed by the salt so it is possible to write the equation opposite as a good approximation which ignores the contribution to [A] by the weak acid.

If both the salt and the acid are present at 0.1M then from the equation opposite the pH of the solution is 5 since log1=0

If 0.01mole of a strong base, say BOH, is added it will react with the weak acid to produce an additional 0.01 mole of the salt BA and 0.01 mole of water which can be ignored because it is small compared with the total water present. The pH of the solution is now 5.09, an increase of only 0.09 pH unit. A similar addition of 0.01 mole of a strong acid would only decrease the pH by 0.09 pH unit.

Suppose we had done this using a strong acid instead of a weak acid. To produce a solution of pH 5 we would use a 0.00001 M solution of HCl. The addition of 0.01 moles of strong base would raise the pH of the solution to 12 which is a change of 7 pH units.

The titration curves of two weak acids are shown opposite. The pH at the midpoint of each curve, where 50% of the acid or base has been used up, gives the pK value since log [salt]/[acid] = zero.

The buffering action of any weak acid-salt mixture is at its greatest close to the pK of the weak acid.

Now that we have a description of pK it is possible to apply it to the dissociation of tooth mineral which, for the purpose of this explanation, we will consider as calcium hydroxyapatite rather than the more complex biological apatite.

The dissociation equilibrium equation for calcium hydroxyapatite is shown opposite. On dissociation, hydroxyapatite produces two ions, phosphate and hydroxyl, capable of accepting protons. Hydroxyapatite is, therefore, a weak base.

Phosphate is capable of accepting 3 protons and has 3 pK values shown as 12.3, 7.2 and 2.1. As the pH of the aqueous phase drops (becomes more acid) the ratio of the concentrations of the various forms of phosphate changes as more and more phosphate is progressively protonated.

Similarly, as the pH drops, hydroxyl ions are protonated and the [OH] reduces.

Knowledge of the relationship between pH and pK allows us to predict the behaviour of phosphate and relate this to mineral dissolution.

At near neutral pH, say pH 7.2, the ratio of mono- and di-hydrogen phosphate can be calculated.

Therefore

Which means that when the pH of the solution is equal to the pK, the concentration of mono- and DI-hydrogen phosphate is the same. This is precisely the ratio predicted by the titration curves of weak acids shown above.

Similar calculations can be applied to the other pK values but these are a long way outside the physiological values of pH which are experienced in vivo. It is, however, interesting to calculate the ratio of phosphate ion to mono-hydrogen phosphate at near neutral pH because it is the concentration of the unprotonated phosphate, together with hydroxyl ion, which affects the dissociation of hydroxyapatite.

The weak acid in this case is mono-hydrogen phosphate, pK = 12.3. At pH 7.2

Therefore

This means that at near neutral pH the concentration of mono-hydrogen phosphate is 7.9 million times greater than the unprotonated form and the same as DI-hydrogen phosphate.