Fractional Ionization of a Monoprotic Weak Acid

Weak acids are those that are not leveled, or completely ionized in the solvent. One common weak acid in aqueous solution is acetic acid, CH₃CO₂H. Solutions of this acid are found in the household under the name vinegar. Chemists commonly abbreviate acetic acid using the symbolic formula, HOAc. The "H" is the acidic hydrogen, and the "OAc" is the acetate, typically found as the anion in solution.

Acetic acid is a relatively weak acid. The degree of ionization in aqueous solution depends on the formal concentration of HOAc, as well as the existence of other acid or base species that may be in solution. Ionization of HOAc produces OAc^-, the acetate anion, and H₃O⁺

Since HOAc ionizes to some extent, the molar concentration is different than the number of moles we add to a liter of solution. However, the formal concentration does not change since we have a fixed number of formula weights HOAc per liter of solution.

If we denote molar concentrations using square brackets, the molar concentration of the acetic acid molecule is [HOAc], while that of the acetate ion is [OAc^-]. These are the only two forms of acetic acid we have to consider for moderately weak pure aqueous solutions. Applying the mass conservation principle, the total moles of acetic and acetate species when HOAc is added to water is the sum of the molar concentrations, multiplied by the volume, V

If we divide both sides by volume, and use the definition of formal

which is a mathematical statement of mass balance, i.e., the sum of the molar concentrations of all acetic species equals the formula weights we put into solution.

The chemical equilibrium equation for acetic acid is

where K_a is the acid dissociation constant, which for HOAc is 1.8x10^-5. This equation is solved for [HOAc]

and substituted into the formal mass balance equation to yield

The fraction of acid in acetate form is [OAc^-]/F_HOAc. Solving the above for this fraction results in the amount of HOAc in the form of the acetate

The ratio [OAc^-]/F_HOAc is often called the "alpha" of the acetate. An expression for the fraction of acetic acid in the acid form, found using a similar derivation, is

Notice that these equations give the fractional amounts of acid and conjugate base as a function of proton or H₃O⁺ concentration, and the acid dissociation constant. Calculation of the fraction of species in a particular form is easy if we know the pH or similarly, the proton concentration, and the dissociation constant.

If we do not know the pH, then we first calculate the proton concentration using the standard equilibrium methods, then we find the species fractions using the above formulas.