Calculate the concentration of each species present in a \(0.010-M\) solution of phthalic acid, \(\mathrm{C}_{6} \mathrm{H}_{4}\left(\mathrm{CO}_{2} \mathrm{H}\right)_{2}\). $$\mathrm{C}_{6} \mathrm{H}_{4}\left(\mathrm{CO}_{2} \mathrm{H}\right)_{2}(a q)+\mathrm{H}_{2} \mathrm{O}(l) \rightleftharpoons \mathrm{H}_{3} \mathrm{O}^{+}(a q)+\mathrm{C}_{6} \mathrm{H}_{4}\left(\mathrm{CO}_{2} \mathrm{H}\right)\left(\mathrm{CO}_{2}\right)^{-}(a q) \quad K_{\mathrm{a}}=1.1 \times 10^{-3}$$, $$\mathrm{C}_{6} \mathrm{H}_{4}\left(\mathrm{CO}_{2} \mathrm{H}\right)\left(\mathrm{CO}_{2}\right)(a q)+\mathrm{H}_{2} \mathrm{O}(l) \rightleftharpoons \mathrm{H}_{3} \mathrm{O}^{+}(a q)+\mathrm{C}_{6} \mathrm{H}_{4}\left(\mathrm{CO}_{2}\right)_{2}^{2-}(a q) \quad K_{\mathrm{a}}=3.9 \times 10^{-6}$$.

Understanding the equilibrium constant expression is crucial in tackling problems related to the dissociation of weak acids, such as phthalic acid. An equilibrium constant expression, commonly referred to as the law of mass action, relates the concentrations of products and reactants at equilibrium. For a general reversible reaction:

\[ aA + bB \rightleftharpoons cC + dD \]
the equilibrium constant expression (K) would be:

\[ K = \frac{[C]^c \times [D]^d}{[A]^a \times [B]^b} \]
The concentrations of the reactants and products are raised to the power of their respective coefficients in the balanced chemical equation. For the dissociation of phthalic acid, we have two equilibrium expressions corresponding to its two dissociation steps. The equilibrium constants for these reactions, denoted as \(K_{a1}\) and \(K_{a2}\), provide a quantitative measure of the acid's tendency to lose \(H^+\) ions in solution.

The acid dissociation constant (Ka) is specific to the equilibrium constant expression for the ionization of an acid in water. It is a measure of the acid's strength, where a larger Ka value indicates a stronger acid capable of losing a proton more readily. Weak acids, such as phthalic acid, dissociate partially in solution leaving an equilibrium mixture of the undissociated acid, the corresponding anions, and hydronium ions (\(H_3O^+\)).

For the first dissociation step of phthalic acid, the acid dissociation constant \(K_{a1}\) is determined experimentally and used to predict the extent of dissociation at a given concentration of the acid. In the second step, a separate \(K_{a2}\) describes the further dissociation of the monohydrogen phthalate ion produced in the first step.

To calculate the concentration of all species in a weak acid equilibrium problem, one must first establish the initial concentrations of the reactants and products. Often, it is assumed that the concentration of products is initially zero, as the acid has not yet dissociated. As the reaction proceeds towards equilibrium, concentration changes are accounted for using an 'ICE' table, which stands for Initial, Change, and Equilibrium concentrations.
The changes in concentration are related to the stoichiometry of the reaction, expressed in terms of the variable 'x', which represents the amount of acid dissociated at equilibrium. These values are then used in the equilibrium constant expression to solve for 'x', and ultimately, to determine the equilibrium concentrations of each species in the solution.

For weak acids like phthalic acid, the equilibrium favors the undissociated form of the acid. That is, in an aqueous solution, only a small fraction of the acid molecules donate their protons to water, leading to a relatively low concentration of hydronium ions compared to the concentration of the undissociated acid. The extent to which a weak acid dissociates is governed by its Ka value and the initial concentration of the acid.

Navigating through weak acid equilibrium calculations involves approximations that simplify the mathematical process, such as assuming that the change in concentration of the weak acid is negligible compared to its initial concentration. These rational simplifications speed up the calculation process without significantly affecting the accuracy of the final concentrations, as done in the step-by-step solutions for phthalic acid dissociation.