Weak acids are acids that do not completely dissociate into their ions when dissolved in water. Instead, they partially ionize, meaning only a fraction of the acid molecules release hydrogen ions (H⁺) into the solution. The rest of the molecules remain in their undissociated form. This creates an equilibrium between the undissociated acid and its dissociation products (H⁺ and the conjugate base). Common examples of weak acids include acetic acid (CH₃COOH), formic acid (HCOOH), and carbonic acid (H₂CO₃). The ionization of a weak acids (acetic acid) in water can be written in the Bronsted–Lowry manner as:

$$ \underset{Acid_1}{HAc}+\underset{Base_2}{H_2O}⇌\underset{Acid_2}{H_3O^+}+\underset{Base_1}{Ac^-} $$

The arrows pointing in the forward and reverse directions indicate that the reaction is proceeding both to the right and to the left simultaneously. According to the law of mass action, the velocity or rate of the forward reaction, R_f, is proportional to the concentration of the reactants:

$$ R_f=k_1\times\left[HAc\right]^1\times\left[H_2O\right]^1 $$

The speed of the reaction is usually expressed in terms of the decrease in the concentration of the reactants per unit time. In this context, the terms rate, speed, and velocity have the same meaning. The reverse reaction:

$$ R_r=k_2\times\left[H_3O^+\right]^1\times\left[Ac^-\right]^1 $$

The rate, R_r, expresses the re-formation of un-ionized acetic acid. Since only one mole of each constituent appears in the reaction, each term is raised to the first power, and the exponents are not required in subsequent expressions for the dissociation of acetic acid and similar acids and bases. The symbols k₁ and k₂ are proportionality constants, commonly known as specific reaction rates for the forward and reverse reactions, respectively, and the brackets indicate concentrations.

Ionization Constant of Weak Acids (K_a)

According to the concept of equilibrium, the rate of the forward reaction decreases over time as acetic acid is depleted, while the rate of the reverse reaction starts at zero and increases as larger quantities of hydrogen ions and acetate ions are formed. Eventually, a balance is reached when the two rates are equal, that is, when:

$$ R_f=R_r $$

The concentrations of products and reactants are not necessarily equal at equilibrium; it is the speeds of the forward and reverse reactions that are equal. After substituting R_f and R_r, the equation becomes:

$$ k_1\left[HAc\right]\left[H_2O\right]=k_2\left[H_3O^+\right]\left[Ac^-\right] $$

And solving for the ratio k₁/k₂ , one obtains:

$$ k=\frac{k_1}{k_2}=\frac{\left[H_3O^+\right]\left[Ac^-\right]}{\left[HAc\right]\left[H_2O\right]} $$

In dilute solutions of acetic acid, water is in sufficient excess to be regarded as constant at about 55.3 moles per liter (1 liter of H₂O at 25°C weighs 997.07 g, and 997.07/18.02 = 55.3). It is thus combined with k₁/k₂ to yield a new constant K_a, the ionization constant or dissociation constant of acetic acid.

$$ K_a=55.3k=\frac{\left[H_3O^+\right]\left[Ac^-\right]}{\left[HAc\right]} $$

This equation is the equilibrium expression for the dissociation of acetic acid, and the dissociation constant K_a is an equilibrium constant in which the essentially constant concentration of the solvent is incorporated. In the discussion of equilibria involving charged as well as uncharged acids, according to the Brønsted–Lowry nomenclature, the term ionization constant of weak acids K_a is not satisfactory and is replaced by the term acidity constant.

We can write the acidity constant for an uncharged weak acid, HB, by the general expression:

$$ HB+H_2O⇌H_3O^++B^- $$

$$ K_a=\frac{\left[H_3O^+\right]\left[B^-\right]}{\left[HB\right]} $$

Using the symbol (c) to represent the initial molar concentration of acetic acid and x to represent the concentration [H₃O⁺]. The latter quantity is also equal to [Ac^–] because both ions are formed in equimolar concentration. The concentration of acetic acid remaining at equilibrium [HAc] can be expressed as (c – x). The reaction is:

$$ \underset{(c-x)}{HAc}+H_2O⇌\underset x{H_3O^+}+\underset x{Ac^-} $$

And the equilibrium expression becomes:

$$ K_a=\frac{x^2}{c-x} $$

Where (c) is large in comparison with x, the term (c – x) can be replaced by (c) without appreciable error, giving the equation:

$$ K_a≅\frac{x^2}{c} $$

which can be rearranged as follows for the calculation of the hydrogen ion concentration of weak acids:

$$ x^2=K_ac $$

$$ x=\left[H_3O^+\right]=\sqrt{K_ac} $$

Reference:

• Sinko, P. (2011). Martin’s Physical Pharmacy and Pharmaceutical Sciences. Baltimore, : Lippincott Williams & Wilkins, a Wolters Kluwer business.