Question

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

Short answer

The value of \(K_u\) for the acid HA is approximately \(4.20 \times 10^{-6}\).

Step-by-step solution

Calculate initial moles of weak acid HA and strong base NaOH

Using the given volume and molarity information, we can determine the initial moles of HA and NaOH. moles_HA = volume_HA × Molarity_HA = 75.0 mL × 0.10 M = 7.50 mmol moles_NaOH = volume_NaOH × Molarity_NaOH = 30.0 mL × 0.10 M = 3.00 mmol

Calculate the moles of acid and base remaining after the reaction

Since NaOH is a strong base, it will react completely with the weak acid HA. The moles of the remaining weak acid and the strong base can be calculated as: moles_HA_remaining = moles_HA - moles_NaOH = 7.50 mmol - 3.00 mmol = 4.50 mmol moles_NaOH_remaining = 0 mmol (All NaOH reacts with HA)

Calculate the concentrations of the remaining acid and base in the final solution

To find the concentration of the remaining acid, we need to take into account the total volume of the solution after mixing. Final volume = volume_HA + volume_NaOH = 75.0 mL + 30.0 mL = 105.0 mL Now, we can find the concentration of the remaining acid: \[HA_{concentration} = \frac{moles_{HA}}{Final~volume} = \frac{4.50~mmol}{105.0~mL} = 0.04286~M\]

Determine pOH and concentration of OH- ions

Since we have the pH of the final solution, we can determine the pOH using the following formula: pOH = 14 - pH = 14 - 5.50 = 8.50 Now we can calculate the concentration of OH- ions: \[OH^-_{concentration} = 10^{-pOH} = 10^{-8.5} \approx 3.16 \times 10^{-9}~M\]

Calculate the concentration of the acid's conjugate base (A-)

Now we need to determine the concentration of A- in the final solution by considering the dissociation of HA: \[HA \rightleftharpoons H^+ + A^-\] From the given pH, we can calculate H+ concentration: \[H^+_{concentration}=10^{-5.50}\approx 3.16\times10^{-6}~M\] As HA is the only source of A- in the solution, we can write the following equation: \[A^-_{concentration} = HA_{initial~concentration} - HA_{concentration} + OH^-_{concentration}\] Substitute the values: \[A^-_{concentration} = 0.10 M - 0.04286 M + 3.16 \times 10^{-9} M\] \[A^-_{concentration} \approx 0.05714 M\]

Calculate the acid dissociation constant (K_u)

Using the equilibrium concentrations of HA, A-, and H+, we can calculate the acid dissociation constant (K_u) using the following formula: \[K_u = \frac{[H^+][A^-]}{[HA]}\] Substituting the values: \[K_u = \frac{(3.16 \times 10^{-6})(0.05714)}{(0.04286)} \approx 4.20 \times 10^{-6}\] The value of K_u for the acid HA is approximately \(4.20 \times 10^{-6}\).

Key concepts

Weak Acid and Strong Base Titration

When a weak acid such as HA is titrated with a strong base like NaOH, the reaction proceeds as the base neutralizes the acid. This process, called titration, involves adding one solution to another until the chemical reaction between the two solutes is complete. In the case of a weak acid and strong base, the titration forms water and the conjugate base of the weak acid (A-).
In the exercise, the titration is not taken to completion; instead we're examining the solution after adding 30.0 mL of 0.10 M NaOH to 75.0 mL of 0.10 M HA. At this point, the pH is no longer representative of the weak acid alone but is now influenced by the remaining weak acid and the conjugate base.
To understand the titration's impact on pH, it is essential to appreciate that the weak acid partially dissociates in solution, forming its conjugate base and hydrogen ions, while the strong base fully dissociates to produce hydroxide ions. Knowing the volumes and concentrations of the reactants allows us to calculate the remaining amount of weak acid after titration and helps us in determining the acid's dissociation constant (Ka) value.

pH Calculation

The pH of a solution is a measure of its acidity or alkalinity, which is directly related to the concentration of hydrogen ions (H+) in the solution. Calculating pH is a crucial step in understanding the acid-base properties of a system. The pH scale ranges from 0 to 14, with lower values being more acidic, higher values being more alkaline, and 7 being neutral.
In the provided exercise, the pH after partial titration is given as 5.50. However, pH alone does not fully describe the acid-base chemistry occurring in the solution. Therefore, to calculate the pH accurately during any stage of the titration, it is necessary to know the concentrations of all species present: acid, conjugate base, and hydroxide ions from the strong base. In this case, after establishing concentrations by following the steps provided from the exercise, we would use the concentration of hydrogen ions to calculate the pH, utilizing the formula pH = -log(H+). The resulting pH helps to reflect the new equilibrium state after adding a precise volume of strong base to the weak acid.

Equilibrium Concentration

The equilibrium concentration refers to the concentrations of reactants and products in a chemical reaction that has reached a state where their concentrations no longer change with time. This does not necessarily mean that the reactants and products are in equal concentrations, but rather that the rates of the forward and reverse reactions are equal.
In the context of the exercise, we encountered an equilibrium system where the weak acid HA is in equilibrium with its conjugate base A- and hydrogen ions. By performing a series of calculations using the initial conditions of the reaction and the known pH, we can determine the equilibrium concentrations of these species. Importantly, the equilibrium concentration of the hydroxide ions introduced by the strong base contributes to the formation of the conjugate base, affecting the overall equilibrium.
Using the calculated equilibrium concentrations of HA, A-, and H+, we can then determine the acid dissociation constant (Ka), which is a quantifiable measure of the acid's strength. The Ka value characterizes the acid's tendency to donate a proton to the solution and is crucial for understanding the behavior of the weak acid in various chemical contexts.