Question

What is the hydronium ion concentration of a \(0.25 \mathrm{M} \mathrm{HA}\) solution? \(\left(K_{a}=4 \times 10^{-8}\right)\) (a) \(10^{-4}\) (b) \(10^{-5}\) (c) \(10^{-7}\) (d) \(10^{-10}\)

Short answer

The hydronium ion concentration of a 0.25 M HA solution is 10^{-4} M.

Step-by-step solution

Write the Dissociation Equation

Write down the equation representing the dissociation of the acid HA in water. HA(aq) ⇌ H₃O⁺(aq) + A⁻(aq).

Set Up the Ka Expression

Write the expression for the acid dissociation constant (Ka) for HA. Ka = [H₃O⁺][A⁻] / [HA]

Assume Equal Concentrations

Since HA partially dissociates, we can make the assumption that [H₃O⁺] and [A⁻] will be equal and much smaller than the initial concentration of HA. Thus, [H₃O⁺] ≈ [A⁻] and [HA] remains approximately 0.25 M.

Write the Ionization Equation

Express Ka using the assumption that [H₃O⁺] = [A⁻] = x, where x is the hydronium ion concentration. Ka = x^2 / (0.25 - x)

Solve for x assuming (0.25 - x) ≈ 0.25

Given that Ka is much smaller than the initial concentration of HA, x will be small compared to 0.25 M, hence (0.25 - x) ≈ 0.25. We get Ka = x^2 / 0.25. Solve for x: x = √(Ka * 0.25) = √(4 × 10^(-8) * 0.25)

Calculate the Hydronium Ion Concentration

x = √(4 × 10^(-8) * 0.25) = √(10^(-8)) = 10^(-4). Therefore, the concentration of hydronium ions is 10^(-4) M.

Key concepts

Acid Dissociation Constant (Ka)

The acid dissociation constant, denoted as Ka, is a quantitative measure of an acid's strength. It represents the equilibrium constant for the dissociation of an acid into its conjugate base and a hydronium ion in aqueous solution. For a generic acid HA, dissociating into H₃O⁺ and A⁻, the dissociation equation is HA(aq) ⇌ H₃O⁺(aq) + A⁻(aq), and the Ka expression is Ka = [H₃O⁺][A⁻] / [HA].

In simpler terms, the larger the Ka value, the stronger the acid, because it implies a greater concentration of hydronium ions. When solving problems involving weak acids, like the provided exercise, we often assume that the ionization is small compared to the initial concentration of the acid. This simplifies calculations and allows us to estimate the hydronium ion concentration without complicated equations.

Chemical Equilibrium

Chemical equilibrium occurs when the rate of the forward reaction equals the rate of the reverse reaction in a chemical process, resulting in no net change in the concentration of reactants and products over time. It is a dynamic state where the concentrations remain constant, but both reactions continue to occur. For our discussion on acid-base chemistry, equilibrium is represented by the acid dissociation constant—when the HA dissociates into H₃O⁺ and A⁻ at the same rate they recombine.

In the exercise, establishing equilibrium allowed us to use the constant Ka to find the unknown concentration of hydronium ions, assuming that the system had reached equilibrium where the forward and reverse reactions were occurring at the same rate.

Ionic Product of Water (Kw)

The ionic product of water, Kw, is the equilibrium constant for water's self-ionization, where water acts as both an acid and a base. The equation for this is 2H2O(l) ⇌ H₃O⁺(aq) + OH⁻(aq), and the Kw expression is Kw = [H₃O⁺][OH⁻]. At 25°C, Kw is always 1.0 x 10⁻¹⁴. This is important because it links the concentrations of hydronium and hydroxide ions in water, providing a basis for understanding the balance of acid and base properties in a solution.

Even though our exercise focuses on calculating the hydronium ion concentration directly, it's useful to know that in any aqueous solution at 25°C, the product of the hydronium and hydroxide ion concentrations will always be 10⁻¹⁴.

pH Calculations

The pH is a measure of the acidity or basicity of a solution, defined as the negative logarithm (base 10) of the hydronium ion concentration. Mathematically, pH = -log[H₃O⁺]. Because the pH scale is logarithmic, each whole pH value below 7 (which is neutral) is ten times more acidic than the next higher value. In our exercise, after calculating the hydronium ion concentration to be 10⁻⁴ M, we can directly determine the pH of the solution to be 4.

Understanding pH calculations is vital for determining the acidity of a solution and for predicting the direction of acid-base reactions. It is also an essential concept for many practical applications, such as adjusting the pH in pools, soils for agriculture, and the human body's tightly regulated pH balance.