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# Alpha Fractions

## Introduction

An fraction is the ratio of the equilibrium concentration of one specific form of a solute divided by the total concentration of all forms of that solute in an equilibrium mixture. It is thus a number between 0 and 1.

For the example of phosphoric acid, H3PO4:

H3PO4 (aq) H+(aq) + H2PO4-(aq) H+(aq) + HPO42-(aq) H+(aq) + PO43-(aq)

pKa1 = 2.15
pKa2 = 7.20
pKa3 = 12.38

Ctotal = [H3PO4] + [H2PO4-] + [HPO42-] + [PO43-]

0 = [H3PO4] / Ctotal

1 = [H2PO4-] / Ctotal

2 = [HPO42-] / Ctotal

3 = [PO43-] / Ctotal

Alpha fractions for a triprotic acid are calculated from the following equation:

[H+]3
0 = --------------------------------------
[H+]3 + Ka1[H+]2 + Ka1Ka2[H+] + Ka1Ka2Ka3

Ka1[H+]2
1 = --------------------------------------
[H+]3 + Ka1[H+]2 + Ka1Ka2[H+] + Ka1Ka2Ka3

Ka1Ka2[H+]
2 = --------------------------------------
[H+]3 + Ka1[H+]2 + Ka1Ka2[H+] + Ka1Ka2Ka3

Ka1Ka2Ka3
3 = --------------------------------------
[H+]3 + Ka1[H+]2 + Ka1Ka2[H+] + Ka1Ka2Ka3

Note that the denominators are the same in all of these equations. Plots of these four equations are shown below for phosphoric acid. Figure 6.2 on page 186 of Rubinson & Rubinson shows this plot on a log scale, which better shows the low concentrations.

See Figure 6.2 on page 186 of Rubinson & Rubinson to see this plot on a log scale.

Note that when the pH equals one of the pKa values, the acid to conjugate base ratio equals 1.

Why would we want to know the fraction of one particular species as a function of pH?

ADP(aq) + HPO42-(aq) + 2 H+(aq) ATP(aq) + H2O

This reaction has a positive G of 52.4 kJ/mol. Producing ATP is the body's way of storing energy, e.g., doughnuts ATP. When your body needs energy for important activities like thinking, ATP converts back to ADP to supply the energy.

Intracellular pH varies from 6.1 in muscle cells to approximately 7 in most other cells. Looking at the phosphoric acid alpha plot, you can see that 2 varies from 0.07 to 0.4 over this range.

For the body to store energy it needs a reliable supply of HPO42-, which is one of many reasons why buffer systems are so important in biology. Large changes in pH would convert HPO42- to H2PO4- or PO43- and you would not be able to convert the chemical energy of doughnuts to a usable form.

You might have noticed that Rubinson & Rubinson (Section 6.6, pg. 180) states that the carbonate/bicarbonate equilibrium is the main determinant of blood pH, which is 7.4. The carbonate system is the primary blood buffer, but look at the pKa values and the following alpha plot to decide if a carbonate/bicarbonate equilibrium would create a buffer system at a pH of 7.4.

 H2CO3 (aq) H+(aq) + HCO3-(aq) Ka = 4.45x10-7 pKa1 = 6.38 HCO3-(aq) + H2O CO32-(aq) + H3O+(aq) Ka2 = 4.7x10-11 pKa2 = 10.32

To achieve a pH of 7.4 requires bicarbonate ion and carbonic acid in a ratio of approximately 11. Carbonate ion, CO32-, is present but in relatively low concentrations: 3 = 1.08x10-3. The HCO3- to CO32- ratio might be important for other reasons, but it affects blood pH only indirectly.

As a side note, a blood pH of 7.4 is not in equilibrium with the atmospheric concentration of CO2. The body maintains blood pH at 7.4 by the kidneys excreting excess acid and pumping bicarbonate ion back into the blood.

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