'The ionization constant of dimethyl amine is \(5.4 \times 10^{-4} .\) Calculate its degree of ionization in its \(0.02 \mathrm{M}\) solution. What percentage of dimethyl amine is ionized if the solution is also \(0.1 \mathrm{M}\) in \(\mathrm{NaOH}\).

Ionization of weak bases, such as dimethyl amine, is a critical concept in understanding how such substances behave in solution. Weak bases do not completely disassociate in water, which results in an equilibrium between the base, its ionized form, and the hydroxide ions. The ionization can be represented by a balanced chemical equation:

DMA + H2O ⇌ DMAH+ + OH-.

In this equation, DMA represents dimethyl amine, and DMAH+ is the conjugate acid of DMA. During this ionization process, only a fraction of the base converts into hydroxide ions (OH-) and the conjugate acid. This fraction is quantified by the degree of ionization, 'x', meaning that only 'x' moles of OH- are produced per mole of the weak base in solution. To understand the extent of ionization, the base ionization constant (Kb) is used, which offers a measure of the equilibrium between the reactants and products.

The common ion effect plays a pivotal role in controlling the ionization equilibrium of weak acids and bases. It occurs when a solution containing a weak base (or acid) also contains another compound that provides an ion identical to the one produced by the ionization of the weak base. For instance, when NaOH is added to a dimethyl amine solution, it releases hydroxide ions (OH-) into the solution, which are common to the ionization product of dimethyl amine.

This influx of common ions shifts the equilibrium, according to Le Chatelier's principle, towards the left, reducing the ionization of the weak base. Consequently, the presence of a strong base like NaOH significantly suppresses the ionization of dimethyl amine, as the system seeks to reduce the concentration of the common ion, OH-, by reducing the extent of DMA ionization.

Calculating the percentage ionization provides insight into how much of a weak base has ionized in a solution. To determine this, the degree of ionization must first be established by using the ionization constant and the initial concentration of the base. In mathematical terms:

Percentage Ionization = \(\frac{x}{[DMA]_{initial}} \times 100%\)

In this formula, 'x' stands for the degree of ionization (the concentration of ionized base at equilibrium), and [DMA]initial is the initial concentration of the weak base before ionization. After calculating 'x' (as detailed in the step-by-step solution), it can be converted into a percentage to reflect the proportion of the base that has been ionized. However, when other sources provide a common ion, such as NaOH in the exercise, the percentage ionization can be significantly affected or even rendered negligible due to the suppression of ionization by the common ion effect.