Question

Two students were asked to determine the \(K_{b}\) of an unknown base. They were given a bottle with a solution in it. The bottle was labeled "aqueous solution of a monoprotic strong acid." They were also given a pH meter, a buret, and an appropriate indicator. They reported the following data: volume of acid required for neutralization \(=21.0 \mathrm{~mL}\) \(\mathrm{pH}\) after \(7.00 \mathrm{~mL}\) of strong acid added \(=8.95\) Use the students' data to determine the \(K_{\mathrm{b}}\) of the unknown base.

Short answer

Question: An unknown base is titrated with a strong acid. After adding 7.00 mL of the strong acid, the resulting solution's pH is measured to be 8.95. This was repeated until 21.0 mL of the strong acid was required to neutralize the base. Using these experimental data, find the \(K_b\) of the unknown base. Answer: The \(K_b\) of the unknown base is \(1.19 \times 10^{-14}\).

Step-by-step solution

Find the initial concentrations of the base and the strong acid

As we know, 21.0 mL of strong acid is required for the neutralization of the base. We also have the initial pH value after adding 7.00 mL of the strong acid to be 8.95. From this pH value, we can calculate the \([OH^-]\) concentration and, consequently, the \([H^+]\) concentration of the strong acid. pOH = 14 - pH = 14 - 8.95 = 5.05 \([OH^-] = 10^{-pOH} = 10^{-5.05} = 8.91 \times 10^{-6} \mathrm{M}\) Now, we know the acid neutralized 7.00 mL of the base: \([H^+] \times 7.00 = [OH^-] \times 21.0\) \( [H^+] = \frac{8.91 \times 10^{-6}\times 21.0}{7.00} = 2.67 \times 10^{-5} \mathrm{M}\)

Write the equilibrium expression for the reaction

Since the solution is a mixture of a strong acid and a weak base, we can write the equilibrium expression for the reaction between them as: \(K_b = \frac{[BH^+][OH^-]}{[B]}\)

Find the change in concentrations during the reaction

Initially, we have 21.0 mL of the unknown base, B. Then, we add 7.00 mL of the strong acid with \([H^+] = 2.67 \times 10^{-5}\). Let's find the change in concentrations during the reaction: \([OH^-] = [H^+] - \frac{[H^+] \times 7.00}{21.0} = 2.67 \times 10^{-5} - \frac{2.67 \times 10^{-5} \times 7.00}{21.0} =1.78\times 10^{-5}\) M

Find the initial concentration of the unknown base

Now we can determine the initial concentration of the unknown base by dividing the number of moles by the total volume of the solution. Total volume = 21.0 mL + 7.00 mL = 28.0 mL \([B] = \frac{[OH^-] \times 21.0}{28.0} = \frac{1.78\times 10^{-5} \times 21.0}{28.0} = 1.34 \times 10^{-5} \mathrm{M}\)

Calculate the \(K_b\) of the unknown base

Finally, we can now calculate the \(K_b\) for the reaction using the calculated initial concentrations: \(K_b = \frac{([H^+] - [OH^-])([OH^-])}{[B]} = \frac{(2.67 \times 10^{-5} - 1.78 \times 10^{-5})(1.78 \times 10^{-5})}{1.34 \times 10^{-5}} = 1.19 \times 10^{-14}\) Therefore, the \(K_b\) of the unknown base is \(1.19 \times 10^{-14}\).

Key concepts

Acid-Base Equilibrium

In acid-base chemistry, understanding equilibrium is key to grasping how acids and bases behave in solution. Acid-base equilibrium refers to the state where the rate of the forward reaction (the acid donating a proton) equals the rate of the reverse reaction (the base accepting a proton). In this dynamic state, the concentrations of reactants and products remain constant over time. This concept is central to predicting the pH of a solution and understanding how titration works. During titration, a strong acid reacts with a weak base, or vice versa, and we move towards the establishment of an acid-base equilibrium, where the amount of acid equals the amount of base, resulting in a neutralization reaction.

Understanding the equilibrium constant ((K_{eq})) for acid-base reactions is crucial as it indicates the extent of the reaction. The equilibrium constant changes with the strength of acids and bases involved. Strong acids and bases disassociate completely, moving the equilibrium far to the right, while weak acids and bases do not disassociate fully, establishing a more central equilibrium point.

pH Calculation

Calculating the pH is a common task in chemistry that helps determine the acidity or basicity of a solution. The pH scale ranges from 0 to 14, with 7 being neutral. Values below 7 denote acidity, while values above 7 indicate basicity. To calculate pH, you need to know the concentration of hydrogen ions, ([H^+]), which can be found using the formula pH = -log([H^+]). In the case of the acid-base titration, once you have determined the hydroxide ion concentration ([OH^-]), you can find the corresponding pOH (pOH = -log([OH^-])), after which pH is easily obtained by subtracting the pOH from 14 (pH = 14 - pOH).

When doing pH calculations, it's important to use the correct formulas and remember that pH is a logarithmic scale, meaning each whole pH value below 7 is ten times more acidic than the next higher value. For example, a pH of 3 is ten times more acidic than a pH of 4.

Hydroxide Ion Concentration

The hydroxide ion concentration ([OH^-]) plays a fundamental role in understanding the basicity of a solution. It's inversely related to the hydrogen ion concentration, as implied by the water dissociation equilibrium ([H^+][OH^-] = 10^{-14}). In an acid-base titration, once you add a known amount of acid to a base, you need to account for the change in hydroxide ion concentration. As the strong acid reacts with the weak base, hydroxide ions are consumed, leading to a decrease in their concentration, which can be calculated based on the volumes and concentrations of the reactants involved in the titration.

Knowing the hydroxide ion concentration is essential not just for finding pH but also for calculating the base's equilibrium constant (K_b), as it shows how much of the base has reacted and therefore how much remains in the basic form.

Kb Determination

The equilibrium constant for the dissociation of a base (K_b) is a quantitative measure of the base's strength. It is defined by the concentration of the hydroxide ions and the protonated base formed, divided by the concentration of the unprotonated base at equilibrium. The K_b can be calculated once you've established the concentrations of the base ([B]), the hydroxide ion ([OH^-]), and the protonated base ([BH^+]) in your solution.

In a titration problem, where an acid neutralizes a base, you can indirectly calculate K_b by first, as in our exercise, calculating the [OH^-] concentration at a specific pH. This information, combined with the initial concentrations and the volume ratios of the acid and base, allows you to find out the concentrations needed to determine K_b. The process involves comprehensive understanding and application of acid-base equilibrium and correct pH, and hydroxide ion concentration calculations.