Question

\mathrm{~A} 0.25 \mathrm{M}$solutionofasalt$\mathrm{NaA}$has$\mathrm{pH}=9.29$.Whatisthevalueof$K_{a}$ for the parent acid HA?

Short answer

The value of $K_a$ for the parent acid HA is approximately 4.87 x 10^(-10).

Step-by-step solution

Calculate [OH⁻] from the given pH

The given pH of the NaA solution is 9.29. We can find the concentration of hydroxide ions [OH⁻] using the relation between pH and pOH: pOH = 14 - pH Now, plug in the given pH value: pOH = 14 - 9.29 = 4.71 Now, we can calculate the [OH⁻] using the formula: [OH⁻] = 10^(-pOH) [OH⁻] = 10^(-4.71) = 1.95 x 10^(-5) M

Determine [A⁻] concentration

Since NaA is a salt that completely dissociates in water, the concentration of A⁻ ions will be equal to the concentration of the NaA solution: [A⁻] = 0.25 M

Set up an ICE table

Write the ionization reaction for the weak acid, HA: HA + H₂O ⇌ H₃O⁺ + A⁻ Now, set up an ICE table (Initial, Change, Equilibrium) to show the change in concentrations. HA + H₂O ⇌ H₃O⁺ + A⁻ Initial -- -- 0 0.25 Change +x -- +x -x Equilibrium x -- x 0.25 - x Note that the concentration of A⁻ after the reaction will be 0.25 - x, since we started with a 0.25 M solution of NaA.

Use the given [OH⁻] to determine x

Since we know the [OH⁻], we can use the relation between [OH⁻] and [H₃O⁺] to find x: [H₃O⁺] = (1.0 * 10^(-14) M²) / [OH⁻] Plug in the value for [OH⁻]: [H₃O⁺] = (1.0 * 10^(-14) M²) / (1.95 x 10^(-5) M) = 5.13 x 10^(-10) M Now we know x = [H₃O⁺] = 5.13 x 10^(-10) M.

Calculate Ka for the parent acid HA

We can find Ka using the equilibrium expression: Ka = ([H₃O⁺] * [A⁻]) / [HA] Plug the equilibrium concentration values into the expression: Ka = (5.13 x 10^(-10) * (0.25 - 5.13 x 10^(-10))) / (5.13 x 10^(-10)) After solving this expression, we get: Ka = 4.87 x 10^(-10) So, the value of Ka for the parent acid HA is approximately 4.87 x 10^(-10).

Key concepts

Chemical Equilibrium

In chemistry, chemical equilibrium occurs when the rate of the forward reaction equals the rate of the reverse reaction, and the concentrations of the reactants and products remain constant over time. It's a state of balance and does not mean the reactants and products are present in equal amounts, but their ratios do not change.
For the acid ionization reaction of a weak acid, an equilibrium is established between the undissociated acid (HA), hydrogen ions (or hydronium, H₃O⁺), and the conjugate base (A⁻). This dynamic process is crucial because it determines the strength of the acid and the pH of the solution, which is a measure of its acidity.

Equilibrium Constant

The equilibrium constant, denoted as K, quantifies the ratio of the concentrations of products to reactants at equilibrium, each raised to the power of their coefficients in the balanced chemical equation.
In the context of acid-base reactions, the equilibrium constant is specifically called the acid dissociation constant, or K_a, for the ionization of an acid. It's calculated using the formula:
K_a = [H₃O⁺][A⁻]/[HA],
where the square brackets indicate concentration. A larger K_a value indicates a stronger acid, as it means a greater proportion of acid molecules dissociate to form hydronium and the conjugate base ions.

pH and pOH Calculations

The pH of a solution is a logarithmic measure of the concentration of hydronium ions, which indicates its acidity. The pH is calculated as:
pH = -log[H₃O⁺].
pOH, on the other hand, is a measure of the hydroxide ion concentration, calculated by:
pOH = -log[OH⁻].
In aqueous solutions, pH and pOH are related through the expression
This relationship arises from the water dissociation constant (K_w), which at 25°C is 1.0 × 10^-14. When working with solutions of salts, like NaA, which is the salt of a weak acid HA, pH can provide insight into the acid's ionization and thus its K_a.

Weak Acid Ionization

When a weak acid ionizes in water, it only partially dissociates into its ions. Unlike strong acids that completely dissociate, weak acids establish an equilibrium between the undissociated acid and its ions, described by the acid-ionization reaction:
HA ⇌ H₃O⁺ + A⁻
To analyze the ionization of weak acids, chemists often use an ICE table (Initial, Change, Equilibrium) to systematically track the concentrations of the reactants and products. The ionization constant K_a is then used to relate these concentrations at equilibrium. For instance, the parent acid HA of the salt NaA can be characterized by a specific K_a, which informs us about the tendency of HA to give up a proton in water and thus its strength as an acid.