Question

You have \(75.0 \mathrm{~mL}\) of \(0.10 \mathrm{M} \mathrm{HA}\). 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 acidic dissociation constant Ka of the weak acid HA is \(2 \times 10^{-5}\).

Step-by-step solution

Calculate moles of the weak acid (HA) and strong base (NaOH) in the solution

To find the moles of HA and NaOH, multiply the initial volume (L) of each solution with their respective concentration (mol/L): Moles of HA = Initial volume of HA (L) × Concentration of HA (mol/L) Moles of HA = 0.075 L × 0.10 mol/L = 0.0075 mol Moles of NaOH = Initial volume of NaOH (L) × Concentration of NaOH (mol/L) Moles of NaOH = 0.030 L × 0.10 mol/L = 0.0030 mol

Determine the moles of HA and A- after the reaction with NaOH

In the reaction, NaOH neutralizes the weak acid, forming its conjugate base A- and water. Since we are given the amount of moles of HA and NaOH, we can determine the moles of HA and A- after the reaction: Moles of HA (after reaction) = Initial moles of HA - moles of NaOH Moles of HA (after reaction) = 0.0075 mol - 0.0030 mol = 0.0045 mol Moles of A- (after reaction) = Initial moles of A- + moles of NaOH Moles of A- (after reaction) = 0 + 0.0030 mol = 0.0030 mol

Calculate the final concentrations of HA and A- in the solution

The total volume of the solution after adding NaOH to HA is 75.0 mL + 30.0 mL = 105.0 mL (or 0.105 L). The concentration of each species can be determined by dividing the moles by the total volume of the solution: [HA] = Moles of HA / Total volume of the solution [HA] = 0.0045 mol / 0.105 L = 0.0429 M [A-] = Moles of A- / Total volume of the solution [A-] = 0.0030 mol / 0.105 L = 0.0286 M

Use the given pH to find the Hydrogen ion concentration [H+]

Recall that pH = -log[H+], where [H+] is the concentration of hydrogen ions. Rearranging this equation, we can find [H+]: [H+] = 10^(-pH) [H+] = 10^(-5.5) = 3.16 × 10^(-6) M

Calculate the Ka value for the weak acid HA

We can now use the weak acid equilibrium expression: Ka = ([H+][A-])/[HA] to find the Ka value of HA: Ka = (3.16 × 10^(-6) × 0.0286) / 0.0429 Calculate Ka: Ka = 2 × 10^(-5) The acidic dissociation constant Ka of the weak acid HA is 2 × 10^(-5).

Key concepts

Weak Acid and Base Reaction

Understanding the interaction between weak acids and bases is fundamental in chemistry, especially when calculating the acid dissociation constant (Ka). Weak acids, such as HA in this problem, only partially dissociate in water, creating a dynamic equilibrium between the undissociated acid (HA) and its ions (H+ and A-).

When a strong base like NaOH is added to the weak acid solution, it reacts completely with HA, neutralizing the acid and forming water (H2O) and the conjugate base (A-). This process shifts the equilibrium, and depending on the amounts of acid and base involved, can significantly affect the pH of the solution. Our goal is to calculate the remaining concentration of the weak acid and its conjugate base after the neutralization reaction to determine the acid's Ka.

pH Calculation

The pH is a measure of the acidity or alkalinity of a solution, expressed on a logarithmic scale. Mathematically, it is described as pH = -log[H+], where [H+] is the concentration of hydrogen ions in the solution. In this exercise, after adding a strong base to the weak acid, the pH was measured as 5.50.

To connect pH with the acid dissociation constant (Ka), we need to calculate the hydrogen ion concentration [H+] using the inverse of the pH definition: \( [H+] = 10^{-\text{pH}} \). The hydrogen ion concentration is directly proportional to the strength of the acid: the more H+ ions present, the lower the pH and the stronger the acid. This relationship is pivotal because it links the measured pH to the equilibrium concentrations needed for the Ka calculation.

Chemical Equilibrium

Chemical equilibrium occurs when the rates of the forward and reverse reactions are equal, resulting in no net change in the concentrations of reactants and products over time. It is vital in the context of weak acid-base reactions because the dissociation of weak acids establishes an equilibrium condition.

The Ka value is a quantitative measure of the acid's strength and represents the equilibrium constant for the dissociation of a weak acid into hydrogen ions and its conjugate base. By using the classroom example, we explored how to solve for Ka using the equilibrium concentrations of HA and A-, which were obtained through stoichiometric calculations after the neutralization reaction took place. The balanced equation for the dissociation of HA into H+ and A- provides the framework for developing and solving the expression for Ka, which is \( Ka = \frac{[H^+][A^-]}{[HA]} \). Ultimately, understanding chemical equilibrium not only allows scientists to predict the direction of a reaction but also enables them to calculate fundamental properties such as the acid dissociation constant.