Question

A 0.135 M solution of a weak base has a pH of 11.23. Determine \(K_{\mathrm{b}}\) for the base.

Short answer

The base ionization constant \(K_{\mathrm{b}}\) for the weak base is calculated using the pH and initial concentration.

Step-by-step solution

Calculate the \(\mathrm{OH}^-\) Ion Concentration

The pH of the solution is given as 11.23, which can be used to find the pOH by subtracting the pH from 14 (pOH = 14 - pH). Once the pOH is known, the hydroxide ion concentration \(\mathrm{OH}^-\) can be calculated using the formula \(\mathrm{pOH} = -\log[\mathrm{OH}^-]\) by rearranging it to \(\mathrm{[OH}^-] = 10^{-\mathrm{pOH}}\).

Write the Base Ionization Expression

Write the equilibrium expression for the ionization of the weak base B in water, \(B + H_2O \leftrightarrow BH^+ + OH^-\). The equilibrium constant expression for \(K_{\mathrm{b}}\) is then \(K_{\mathrm{b}} = \frac{[BH^+][OH^-]}{[B]}\).

Calculate the Concentration of \(BH^+\)

Assuming the weak base undergoes a one-to-one reaction to form \(BH^+\) and \(OH^-\), the concentration of \(BH^+\) at equilibrium will be equal to \(\mathrm{[OH}^-]\) calculated from Step 1.

Calculate the Equilibrium Concentration of \(B\)

The initial concentration of the weak base B is given as 0.135 M. Since the concentration of \(BH^+\) is the same as the concentration of \(OH^-\) produced, the equilibrium concentration of \(B\) will be the initial concentration minus the concentration of \(OH^-\), which is \(\mathrm{[B]_{eq}} = [B]_{initial} - [OH^-]\).

Calculate the Base Ionization Constant \(K_{\mathrm{b}}\)

Now plug the concentrations of \(BH^+\), \(OH^-\), and \(B\) into the equilibrium expression for \(K_{\mathrm{b}}\) calculated in Steps 1, 3, and 4, respectively, to find the value of \(K_{\mathrm{b}}\).

Key concepts

Weak Base pH Calculation

Understanding how to calculate the pH of a weak base solution is a critical skill in chemistry. When dealing with a weak base, we begin by recognizing that it does not fully dissociate in water. Knowing the pH of a solution, as we have in the given exercise (pH 11.23), allows us to find the concentration of hydroxide ions (OH^-).
We can convert pH to pOH, as pOH is directly related to the OH^- concentration. The relationship between pH and pOH is that their sum equals 14 for any aqueous solution at 25°C. Once you know the pOH, you use the formula pOH = -log[OH^-] to find the hydroxide ion concentration by rearranging it to [OH^-] = 10^{-pOH}.
It's important for students to practice these calculations because they solidify the understanding of the pH scale and the strength of bases, and they apply across many areas of chemistry, such as titrations and buffer system calculations.

Hydroxide Ion Concentration

The hydroxide ion concentration in a solution is a direct measure of how alkaline a solution is. It is tied to pH through the equation pOH = -log[OH^-]. In our exercise, we reverse this process to find the concentration of hydroxide ions from the known pOH. This is done by the expression [OH^-] = 10^{-pOH}, as seen in the step-by-step solution. The higher the concentration of OH^- ions, the higher the basicity of the solution.
For weak bases, this concentration is not equal to the initial concentration of the base because weak bases only partially dissociate. Understanding this concept helps in predicting the behavior of bases in equilibrium, and it is essential for fields such as environmental science, where the alkalinity of water bodies is monitored.

Acid-Base Equilibrium

Acid-base equilibrium is a fundamental concept that dictates the concentrations of various species in a solution containing an acid or a base. In the context of a weak base, this equilibrium lies far to the left, meaning not much of the base is dissociated at any given time. The equilibrium expression for the ionization of a weak base, as shown in the exercise (K_b = [BH^+][OH^-] / [B]), provides a snapshot of this relationship at equilibrium.
It is particularly important for students to grasp that the products of the ionization are BH^+ and OH^- ions, and their concentrations at equilibrium determine the pH or pOH of the solution. The equilibrium expression involves the concentrations of each species, and how changes in these concentrations (such as adding more base or acid) shift the equilibrium position. This is central to many applications, from industrial processes to biological systems where maintaining pH is crucial.