Question

You have 75.0 \(\mathrm{mL}\) of 0.10 \(\mathrm{M}\) HA. After adding 30.0 \(\mathrm{mL}\) of 0.10 \(\mathrm{M} \mathrm{NaOH}\) , the \(\mathrm{pH}\) is \(5.50 .\) What is the \(K_{\mathrm{a}}\) value of \(\mathrm{HA}\) ?

Short answer

The $K_{a}$ value of HA is approximately \(2.06 \times 10^{-5}\).

Step-by-step solution

Calculate the moles of HA and NaOH before the reaction

first, we will calculate the moles of HA and NaOH initially present using their concentrations and volumes: Moles of HA = Concentration of HA × Volume of HA = \(0.10\,\mathrm{M} \times 75.0\,\mathrm{mL}\) = \(0.0075\,\mathrm{mol}\) Moles of NaOH = Concentration of NaOH × Volume of NaOH = \(0.10\,\mathrm{M} \times 30.0\,\mathrm{mL}\) = \(0.0030\,\mathrm{mol}\)

Determine the limiting reagent, moles and concentrations of species after the reaction

Since NaOH is a strong base, it will completely react with the weak acid HA. So, the limiting reagent will be NaOH, and after the reaction: Moles of HA = \(0.0075\,\mathrm{mol} - 0.0030\,\mathrm{mol} = 0.0045\,\mathrm{mol}\) The reaction consumes all the moles of NaOH, forming an equal amount of the conjugate base A⁻. Moles of A⁻ = Moles of NaOH reacted = \(0.0030\,\mathrm{mol}\)

Calculate the concentrations of species after the reaction

Now, we calculate the new concentration of HA and A⁻ in the solution: Total volume of the solution = Initial volume of HA + Initial volume of NaOH = \(75.0\,\mathrm{mL} + 30.0\,\mathrm{mL}\) = \(105.0\,\mathrm{mL}\) Concentration of HA = Moles of HA / Total volume of the solution = \(0.0045\,\mathrm{mol} / 105.0\,\mathrm{mL}\) = \(0.0429\,\mathrm{M}\) Concentration of A⁻ = Moles of A⁻ / Total volume of the solution = \(0.0030\,\mathrm{mol} / 105.0\,\mathrm{mL}\) = \(0.0286\,\mathrm{M}\)

Use the pH value to find the concentration of hydrogen ions [H+]

The pH of the solution is 5.50, so we can find the concentration of H⁺ ions using: pH = -log\(_{10}\)([H⁺]) 5.50 = -log\(_{10}\)([H⁺]) [H⁺] = 10\(^{-5.50}\) = \(3.16 \times 10^{-6}\,\mathrm{M}\)

Use the equilibrium expression to calculate Ka

Using the equilibrium expression for the dissociation of HA: Ka = \(\frac{[\mathrm{H}^+][\mathrm{A}^-]}{[\mathrm{HA}]}\) Ka = \(\frac{(3.16 \times 10^{-6})(0.0286)}{0.0429}\) Ka = \(2.06 \times 10^{-5}\) So, the Ka value of HA is about \(2.06 \times 10^{-5}\).

Key concepts

Buffer Solution

A buffer solution is a unique chemical arrangement that helps maintain a stable pH within a certain range, even when small amounts of acids or bases are added. This stability is crucial in numerous biochemical and chemical applications. To create such a buffer, a weak acid and its conjugate base are usually combined.

In the provided problem, the buffer solution is formed by adding a strong base, NaOH, to the weak acid, HA. As NaOH neutralizes some HA, the conjugate base A⁻ forms, creating a mixture that resists drastic pH changes.

• **Components of a Buffer Solution**: Typically involves a weak acid (HA) and its conjugate base (A⁻).
• **Function**: Helps to stabilize the pH by neutralizing added acids or bases.
• **Mechanism**: The weak acid can donate protons (H⁺), while the conjugate base can accept protons to counter pH changes.

Understanding buffer solutions is essential for manipulating pH in lab settings and biological systems.

Equilibrium Expression

An equilibrium expression represents the balance of concentrations between reactants and products in a dynamic chemical equilibrium. It describes how the concentrations relate to each other when a reaction is at equilibrium.

For the weak acid HA, its dissociation in water can be expressed as an equilibrium involving HA, H⁺, and A⁻:- **HA ⇌ H⁺ + A⁻**The equilibrium constant, known as the acid dissociation constant ( K_a ), quantifies the extent of this dissociation. It is given by the expression:
\[K_a = \frac{[ \mathrm{H}^+ ][ \mathrm{A}^- ]}{[ \mathrm{HA} ]}\]

• **Weak Acid Dissociation**: Only partially dissociates to form H⁺ and A⁻.
• **Equilibrium Constant (K_a)**: Measures the acidity; a higher K_a signifies a stronger acid.
• **Importance**: K_a helps predict the degree of dissociation and the resulting pH.

By calculating K_a, we gain insights into the acid strength and buffering capacity of the solution.

pH Calculation

Accurately calculating pH is fundamental in chemistry as it allows us to determine the acidity or basicity of a solution. The pH is calculated from the concentration of hydrogen ions ( [H^+] ).

For the problem at hand, by using the equilibrium expression and given pH, we extract the concentration of [H^+] . This is achieved as follows: The pH is defined by the formula:\[ pH = -\log_{10}([\mathrm{H}^+])\]Given pH = 5.50:\[ [\mathrm{H}^+] = 10^{-\mathrm{pH}} = 10^{-5.50} = 3.16 \times 10^{-6} \mathrm{M}\]

• **pH Definition**: A measure of how acidic or basic a solution is; ranges from 0 to 14.
• **Cation Concentration Impact**: Directly affects the pH value.
• **Application**: Helps in determining the ionization of weak acids and bases.

Calculating the pH is crucial for understanding the chemical nature and behavior of the solution, helping in designing experiments and applications where specific pH levels are necessary.