# Calculating Equilibrium Concentrations of Polyprotic Acids

## Polyprotic acids have complex equilibriums due to the dissociation of hydrogen species at different pHs and the presence of multiple species in solution.

#### Key Points

• Polyprotic acids contain multiple acidic protons that can dissociate from the compound sequentially with unique acid dissociation constants for each proton.

• Because of the increased variety of possible ionic species in solution for each acid, the calculations to precisely determine the concentrations of different species at equilibrium can be very complicated.

• Certain simplifications can be made to make the calculations easier. What the simplifications are vary with the specific acid and the solution conditions.

#### Terms

• the state of a reaction in which the rates of the forward and reverse reactions are the same

• Referring to an acid with multiple protons capable of dissociating from the compound.

• Referring to chemical species with a residual charge; in acid-base equilibria, this concept refers to the charge arising from the loss or addition of electrons from chemical compounds.

#### Figures

1. ##### Fractional Ion Calculations for Polyprotic Acids

The above complex equations can be used to determine the fractional concentration of various ions from polyprotic acids.

2. ##### Phosphoric Acid

The chemical structure of phosphoric acid indicates it has three acidic protons.

Polyprotic acids are acids that can lose more than one proton. The dissociation constant of the first proton may be denoted as Ka1 and the constants for dissociation of successive protons as Ka2, etc. Common polyprotic acids include sulfuric acid, H2SO4 and phosphoric acid, H3PO4.

To determine equilibrium concentrations of different ions produced by polyprotic acids, the equations can get very complicated to account for all of the different components. For instance, one can calculate the fractional concentration (alpha) of different ions using complex equation that account for hydrogen concentration (pH) and equilibrium constants, as seen in Figure 1.

However, one can simplify the problem, depending on the polyprotic acid. The following examples indicate the mathematics and simplifications for a few polyprotic acids under specific conditions.

## Sulfuric Acid

If water is the solvent, sulfuric acid, H2SO4, loses one proton as a strong acid with an immeasurably large dissociation constant.

H2SO4 → H+ + HSO4-

It also can lose a second proton as a weak acid with a measurable dissociation constant.

$HSO_4^- \rightleftharpoons H^+ + SO_4^{2-}$  Ka2 = 1.20x10-2   pKa2 = 1.92

Therefore, one can assume that there is no measurable H2SO4 in the solution. Instead, it has completely dissociated to H+ and HSO4-, which in turn has dissociated to more protons and SO42-.

## Phosphoric Acid

Phosphoric acid, H3PO4, has three dissociations, as can be viewed in Figure 2:

$H_3PO_4 \rightleftharpoons H^+ + H_2PO^-$  pKa1 = 2.12

$H_2PO^- \rightleftharpoons H^+ + HPO^{2-}$   pKa2 = 7.21

$HPO^{2-} \rightleftharpoons H^+ + PO^{3-}$  pKa3 = 12.67

Thus, in an aqueous solution of phosphoric acid, there will theoretically be seven ionic and molecular species present: H3PO4, H2PO-, HPO2-, PO3-, H2O, H+, and OH-. Life might appear impossibly complicated, were we not able to make some approximations.

At a pH equal to the pKa for a particular dissociation, the two forms of the dissociating species are present in equal concentrations, due to the following mathematical observation. For the second dissociation of phosphoric acid, for which pKa2 = 7.21:

• pKa2 = -log(Ka2) = -log([H+]*[HPO42-]/[H2PO4-])
• pH = -log[H+]

Therefore, pH - pKa2 = log([HPO42-]/[H2PO4-])

When pH = pKa2, we have the ratio [HPO42-]/[H2PO4-] = 1.00. Hence, in a neutral solution, H2PO4- and HPO42- are present in about the same concentrations. Very little undissociated H3PO4 and fully dissociated PO43- will be found as can be determined through simliar equation with their given Ka's.

The only phosphate species that we have to consider near pH = 7 are H2PO4- and HPO42-. Similarly, in strong acid solutions near pH = 3, only H3PO4 and H2PO4- are important. As long as the pKa's of successive dissociations are separated by three or four units (as they almost always are), matters are simplified.

## Carbonic Acid

There is still another simplification. When a weak polyprotic acid, such as carbonic acid, H2CO3, dissociates, most of the protons present come from the first dissociation:

$H_2CO_3 \rightleftharpoons H^+ + HCO_3^-$  pKa1 = 6.37

Since the second dissociation constant is smaller by four orders of magnitude (pKa2 = 10.25 is larger by four units), the contribution of hydrogen ions from the second dissociation will be only one ten-thousandth as large. Correspondingly, the second dissociation has a negligible effect on the concentration of the product of the first dissociation, HCO3-.

#### Key Term Glossary

acid
an electron pair acceptor; generally capable of donating hydrogen ions
##### Appears in these related concepts:
aqueous
Consisting mostly of water.
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concentration
the proportion of a substance in a mixture
##### Appears in these related concepts:
constant
Consistently recurring over time; persistent
##### Appears in these related concepts:
dissociation
Referring to the process by which compounds split into smaller constituent molecules, usually in a reversible manner.
##### Appears in these related concepts:
equilibrium
the state of a reaction in which the rates of the forward and reverse reactions are the same
##### Appears in these related concepts:
equilibrium constant
referring to a numerical value derived from the ratio of concentrations of products to reactants of a reversible reaction
##### Appears in these related concepts:
ion
An atom or group of atoms bearing an electrical charge, such as the sodium and chlorine atoms in a salt solution.
##### Appears in these related concepts:
ionic
of, relating to, or containing ions
##### Appears in these related concepts:
matter
The basic structural component of the universe. Matter usually has mass and volume.
##### Appears in these related concepts:
multiple
one of a set of the same thing; a duplicate
##### Appears in these related concepts:
pH
the negative of the logarithm to base 10 of the concentration of hydrogen ions, measured in moles per liter; a measure of acidity or alkalinity of a substance, which takes numerical values from 0 (maximum acidity) through 7 (neutral) to 14 (maximum alkalinity)
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phosphate
any salt or ester of phosphoric acid
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pKa
A quantitative measure of the strength of an acid in solution; a weak acid has a pKa value in the approximate range −2 to 12 in water and a strong acid has a pKa value of less than about −2.
##### Appears in these related concepts:
polyprotic
of an acid (or a base) that can donate (or accept) more than one proton; polybasic
##### Appears in these related concepts:
polyprotic acids
Referring to an acid with multiple protons capable of dissociating from the compound.
##### Appears in these related concepts:
product
a chemical substance formed as a result of a chemical reaction
##### Appears in these related concepts:
proton
A positively charged subatomic particle forming part of the nucleus of an atom and determining the atomic number of an element; the nucleus of the most common isotope of hydrogen; composed of two up quarks and a down quark
##### Appears in these related concepts:
ratio
The relative magnitudes of two quantities (usually expressed as a quotient).
##### Appears in these related concepts:
solution
A homogeneous mixture, which may be liquid, gas or solid, formed by dissolving one or more substances.
##### Appears in these related concepts:
Solution
A homogeneous mixture, which may be liquid, gas or solid, formed by dissolving one or more substances.
##### Appears in these related concepts:
solvent
a substance that dissolves a solute, resulting in a solution
##### Appears in these related concepts:
strong acid
A strong acid is one that completely ionizes (dissociates) in water; in other words, one mole of a strong acid HA dissolves in water yielding one mole of H+ and one mole of the conjugate base, A−.
##### Appears in these related concepts:
weak acid
one that dissociates incompletely, releasing only some of its hydrogen atoms into solution