For which of the following are we permitted to make the assumption that the equilibrium concentration of the acid or base is the same as the initial concentration when we calculate the \(\mathrm{pH}\) of the solution specified? (a) \(0.020 \mathrm{M} \mathrm{HC}_{2} \mathrm{H}_{3} \mathrm{O}_{2}\) (c) \(0.002 \mathrm{M} \mathrm{N}_{2} \mathrm{H}_{4}\) (b) \(0.10 \mathrm{M} \mathrm{CH}_{3} \mathrm{NH}_{2}\) (d) \(0.050 \mathrm{M} \mathrm{HCHO}_{2}\)

Understanding the equilibrium concentration of a substance is crucial when discussing reactions that involve weak acids and bases. Equilibrium concentration is the amount of reactant or product present when a reversible chemical reaction reaches a state where the rate of the forward reaction equals the rate of the reverse reaction, leading to no noticeable change in concentration over time.

In the context of weak acids or bases in an aqueous solution, determining the equilibrium concentrations of the various species involved (the weak acid/base itself, hydronium ions, and conjugate base/acid) is essential for calculating the pH. With weak acids and bases, it is often possible, for simplicity, to assume that the equilibrium concentration of the acid or base is nearly equal to its initial concentration, especially if it is low and the acid or base is weak -- this is because they only partially dissociate in solution.

However, this assumption is not without its limitations and should be carefully considered against factors such as the substance's dissociation extent and concentration. When the concentration is too high, even a weak acid or base could shift the equilibrium enough that the assumption no longer holds true. In the context of the exercise, this assumption is particularly useful for simplifying pH calculations.

Weak acids and bases are substances that partially dissociate into ions when dissolved in water. Unlike their strong counterparts, they do not completely ionize in solution, meaning that not all the molecules of the acid or base break apart. Hence, the presence of weak acids and bases in the solution sets up an equilibrium condition between the undissociated molecules and the ions produced.

For instance, acetic acid (HC2H3O2) is a classic example of a weak acid. In an aqueous solution, it partially dissociates into acetate ions (C2H3O2-) and hydrogen ions (H+), which combine with water to form hydronium ions (H3O+). The ability of a weak acid or base to resist changes in pH—a concept known as 'buffering'—makes understanding them crucial, especially in biological and environmental systems.

Weak acids generally have a pH greater than 3 but less than 7, while weak bases usually have a pH greater than 7 but less than 11. The exercise presents examples of weak acids and bases used to highlight the application of theoretical concepts in practical pH calculations.

In acid-base reactions, the acid-base equilibrium is the point at which the forward reaction (acid dissociation) and the reverse reaction (reassociation) occur at the same rate. This dynamic process leads to a constant concentration of reactants and products when viewed over time, yet on the molecular level, the exchange of particles continues unceasingly.

An understanding of acid-base equilibrium is essential for predicting the behavior of acids and bases in various chemical environments. For weak acids and bases, the equilibrium can be disturbed by changes in concentration, temperature, and the presence of other substances in the solution. The position of the equilibrium can give insight into the relative strengths of the acids and bases present, which is fundamental to various applications, from industrial processes to biological systems.

Using the acid dissociation constant and the initial concentrations of acids and bases, it is possible to calculate the equilibrium concentrations and, subsequently, the pH of the solution, which is the primary goal of the exercise we are examining.

The acid dissociation constant, represented by the symbol Ka, is an essential quantitative measure of the strength of an acid in solution. It is the equilibrium constant for the dissociation reaction of the acid into its conjugate base and a hydrogen ion. The higher the value of Ka, the stronger the acid, because it indicates a greater tendency for the acid to lose a proton.

A larger Ka value corresponds to a larger concentration of hydrogen ions, which leads to a lower pH, indicating a stronger acid. Conversely, a smaller Ka value signifies a weaker acid, as it implies that less dissociation occurs, and thus, fewer hydrogen ions are formed. Bases have a similar constant called the base dissociation constant, Kb.

In practice, the Ka value is critical for calculating the pH of solutions containing weak acids. It is used in conjunction with the initial concentration and the acid-base equilibrium expression to determine what may initially seem like an intractable concentration of H3O+. The step-by-step solution provided in the exercise primarily intends to help students understand how these various elements interact within the realm of acid-base chemistry.