Question

Find the \(\mathrm{pH}\) of a \(0.1 \mathrm{M}\) solution of ammonia, \(\mathrm{NH}_{3} \cdot \mathrm{pK}_{\mathrm{b}}=\) \(1.76 \times 10^{-5}\)

Short answer

The pH of the \(0.1\text{ M}\) ammonia solution is calculated by the following steps: \(1.\) Find the Kb value using \(\text{Kb} = 10^{-\text{pKb}}\); \(2.\) Calculate the OH⁻ concentration using the Kb expression \(\text{Kb} = \frac{x^2}{0.1-x}\) and solve for x; \(3.\) Determine the H⁺ concentration using the Kw expression \(\text{Kw} = [\text{H}^{+}] [\text{OH}^{-}]\); and \(4.\) Calculate the pH using the H⁺ concentration with the pH formula \(\text{pH} = -\log [\text{H}^{+}]\).

Step-by-step solution

Relationship between Kb and pKb

The relationship between the Kb value and the pKb value is given by the equation: \[ \text{Kb} = 10^{-\text{pKb}} \] We are given the pKb value, so we can find the Kb value.

Calculate the Hydroxide Ion Concentration (OH⁻)

The Kb expression for the reaction of ammonia with water can be written as: \[ \text{Kb} = \frac{[\text{NH}_4^{+}][\text{OH}^-]}{[\text{NH}_3]} \] Let x be the concentration of \(\text{NH}_4^{+}\) and \(\text{OH}^-\) at equilibrium. The initial concentration of \(\text{NH}_3\) is 0.1 M, and at equilibrium, it will be \((0.1-x)\) M. The Kb expression becomes: \[ \text{Kb} = \frac{x^2}{0.1-x} \] Now, we substitute the Kb value from Step 1 and solve for x.

Determine the Hydrogen Ion Concentration (H⁺)

To find the H⁺ concentration, we can use the relationship between H⁺ and OH⁻ given by the Kw expression: \[ \text{Kw} = [\text{H}^{+}] [\text{OH}^{-}] \] where Kw is the ion-product constant of water, \(1.0 \times 10^{-14}\) at 25°C. Using the OH⁻ concentration calculated in Step 2, we can find the H⁺ concentration.

Calculate the pH

The pH is defined as the negative logarithm of the H⁺ concentration: \[ \text{pH} = -\log [\text{H}^{+}] \] Now, we can use the H⁺ concentration from Step 3 to find the pH of the ammonia solution.

Key concepts

Relationship Between Kb and pKb

In the study of acid-base chemistry, one often discusses the base dissociation constant (Kb) and its logarithmic counterpart, the pKb. The relationship between Kb and pKb is critical in understanding how bases interact with water to produce hydroxide ions. Kb represents the equilibrium constant for the reaction in which a base accepts a proton from water, forming its conjugate acid and hydroxide ions. The pKb, on the other hand, is simply the negative logarithm to the base 10 of the Kb:\[\begin{equation}\text{pKb} = -\log(\text{Kb})\end{equation}\]To go from pKb to Kb, you would perform the inverse logarithmic operation, raising 10 to the power of negative pKb:\[\begin{equation}\text{Kb} = 10^{-\text{pKb}}\end{equation}\]This relationship is used to determine the strength of a base. A lower pKb indicates a stronger base, as it corresponds to a higher value of Kb. Understanding this concept enables us to predict the extent to which a base will ionize in solution, which is crucial for calculating pH.

Hydroxide Ion Concentration Calculation

Calculating the hydroxide ion concentration in a solution is an essential step in determining the pH. For bases, such as ammonia in an aqueous solution, the concentration of hydroxide ions (OH-) can be found using the base dissociation constant (Kb). The equation representing the ionization of ammonia in water is:\[\begin{equation}\text{NH}_3 + \text{H}_{2}\text{O} \rightleftharpoons \text{NH}_4^{+} + \text{OH}^-\end{equation}\]In equilibrium, the concentration of products over the concentration of reactants is given by Kb:\[\begin{equation}\text{Kb} = \frac{[\text{NH}_4^{+}][\text{OH}^-]}{[\text{NH}_3]}\end{equation}\]By assuming that the degree of ionization is small compared to the initial concentration of ammonia, we can use the approximation that the equilibrium concentration of ammonia is approximately equal to its initial concentration. This simplification allows us to calculate the concentration of hydroxide ions, which then can be used to find pH.

Hydrogen Ion Concentration Calculation

When working with bases, finding the concentration of hydrogen ions (H+) can seem less intuitive but is just as critical when calculating pH. The connection between hydroxide ions (OH-) and hydrogen ions is governed by the ion-product constant of water (Kw). At 25°C, Kw is always \[\begin{equation}\text{Kw} = [\text{H}^{+}][\text{OH}^{-}] = 1.0 \times 10^{-14}\end{equation}\]Due to this constant relationship, we can calculate the concentration of hydrogen ions in the solution once we have the concentration of hydroxide ions. Knowing the OH- concentration, we rearrange the ion-product water constant equation to solve for the hydrogen ion concentration:\[\begin{equation}[\text{H}^{+}] = \frac{\text{Kw}}{[\text{OH}^{-}]}\end{equation}\]This step is pivotal for calculating the pH of not only acidic solutions but also basic solutions, by impacting the balance between hydroxide and hydrogen ions.

Ion-Product Constant of Water (Kw)

The ion-product constant of water, Kw, has a central role in acid-base chemistry. It is the equilibrium constant for the self-ionization of water and is defined at 25°C as:\[\begin{equation}\text{Kw} = [\text{H}^{+}][\text{OH}^{-}] = 1.0 \times 10^{-14}\end{equation}\]Kw is a constant at a given temperature and does not change regardless of the presence of acids or bases. This invariance provides a bridge between the hydrogen ion and hydroxide ion concentrations in any aqueous solution, enabling us to determine pH. It's important to recognize that an increase in [OH-] results in a decrease in [H+], and vice versa, to maintain the constant value of Kw. Whether you’re working with acidic or basic solutions, the ion-product constant of water is the key to unlocking the pH puzzle. It's important to remember that while Kw remains constant at a specific temperature, it does vary with temperature, which could be relevant in different environmental conditions or industrial processes.