Problem 89 - Chemistry: The Central Science

Question

Sorbic acid \(\left(\mathrm{C}_{5} \mathrm{H}_{7} \mathrm{COOH}\right)\) is a weak monoprotic acid with \(K_{a}=1.7 \times 10^{-5} .\) Its salt (potassium sorbate) is added to cheese to inhibit the formation of mold. What is the \(\mathrm{p} \mathrm{H}\) of a solution containing \(11.25 \mathrm{~g}\) of potassium sorbate in \(1.75 \mathrm{~L}\) of solution?

Step-by-Step Solution

Answer

The pH of a solution containing 11.25 g of potassium sorbate in 1.75 liters of solution is approximately 2.99.

Step 1: Determine the concentration of potassium sorbate in the solution

First, we need to calculate the concentration of potassium sorbate in the solution. To do this, we need to find the moles of potassium sorbate and divide it by the volume of the solution. Moles of potassium sorbate = \(\dfrac{\text{mass of potassium sorbate}}{\text{molecular weight of potassium sorbate}}\) The molecular weight of potassium sorbate, \(\mathrm{C}_{5} \mathrm{H}_{7} \mathrm{COOK}\), is approximately \(112.2\ \mathrm{g/mol}\). Moles of potassium sorbate = \(\dfrac{11.25\ \mathrm{g}}{112.2\ \mathrm{g/mol}}\) = \(0.10023\) moles Concentration of potassium sorbate = \(\dfrac{\text{moles of potassium sorbate}}{\text{volume of solution}} = \dfrac{0.10023\ \text{moles}}{1.75\ \text{L}} = 0.05727\ \text{M}\).

Step 2: Convert the concentration to the concentration of sorbic acid

Since potassium sorbate is the salt of sorbic acid, the concentration of sorbic acid in the solution is equal to the concentration of potassium sorbate. Therefore, \([\mathrm{C}_{5} \mathrm{H}_{7} \mathrm{COO}^-] = 0.05727\ \text{M}\).

Step 3: Set up an equilibrium expression for the dissociation of sorbic acid and solve for \([\mathrm{H}^{+}]\)

The equilibrium expression for the dissociation of sorbic acid is: \(K_a = \dfrac{[\mathrm{H}^+][\mathrm{C}_{5} \mathrm{H}_{7} \mathrm{COO}^-]}{[\mathrm{C}_{5} \mathrm{H}_{7} \mathrm{COOH}]}\) Let \(x\) represent the concentration of \([\mathrm{H}^+]\) and \([\mathrm{C}_{5} \mathrm{H}_{7} \mathrm{COOH}]\) produced by the dissociation of sorbic acid. \(K_a = \dfrac{x^2}{0.05727-x}\) Solving for \(x\): \(1.7 \times 10^{-5} = \dfrac{x^2}{0.05727-x}\) Since \(K_a\) is small, the value of \(x\) will be much smaller than \(0.05727\), so we can approximate: \(1.7 \times 10^{-5} \approx \dfrac{x^2}{0.05727}\) Solving for \(x\), we get: \(x \approx 1.025 \times 10^{-3}\ \text{M}\) So, the concentration of \([\mathrm{H}^{+}] \approx 1.025 \times 10^{-3}\ \text{M}\).

Step 4: Calculate the pH

Now that we have the concentration of \([\mathrm{H}^{+}]\), we can calculate the pH using the following formula: \(\mathrm{pH} = -\log{[\mathrm{H}^{+}]}\) \(\mathrm{pH} = -\log{(1.025 \times 10^{-3})} = 2.989\) Thus, the pH of the solution is approximately 2.99.

Key Concepts

Weak Acids

Weak acids are a fascinating topic in chemistry. Unlike strong acids, which completely dissociate in water, weak acids only partially dissociate. This means that only a portion of the acid molecules release hydrogen ions ( H^+ ) into the solution. Because of this partial dissociation, weak acids yield an equilibrium between the undissociated acid and the ions formed.

Some key characteristics of weak acids include:

• Lower dissociation compared to strong acids
• Proton ( H^+ ) donors in reactions, though not as readily
• Require equilibrium constants to describe their ionization

Sorbic acid, a weak monoprotic acid, is often found in food preservation due to its ability to inhibit mold growth. Understanding its weak nature is crucial for calculating the pH of solutions containing its salts.

Acid Dissociation Constant (Ka)

The acid dissociation constant, K_a, is a valuable tool in chemistry as it quantifies the strength of an acid in solution. Specifically, K_a measures the extent to which a weak acid ionizes in water.

The general formula for K_a is:\[K_a = \frac{[\text{H}^+][\text{A}^-]}{[\text{HA}]}\]
Where:

• [H⁺] is the concentration of hydrogen ions
• [A^-] is the concentration of the conjugate base
• [HA] is the concentration of the undissociated acid

A small K_a value indicates that the acid only sparingly dissociates, pointing to its status as a weak acid. For instance, in the sorbic acid example, K_a = 1.7 \times 10^{-5}, which classifies it as a weak acid. This K_a helps us set up the equilibrium expression we need to determine [H^+] and ultimately the pH of the solution.

Acid-Base Equilibrium

Acid-base equilibrium is pivotal in understanding how solutions reach a state of balance. In the context of weak acids and their dissociation, equilibrium describes the point at which the forward reaction (dissociation of the acid) and the reverse reaction (reformation of the acid) occur at the same rate.

For weak acids like sorbic acid, this balance is represented mathematically in its dissociation constant equation:\[K_a = \frac{x^2}{C - x}\]
In this formula:

• "x" is the concentration of H^+ ions produced
• "C" is the initial concentration of the acid or its salt
• The equation considers the negligible change due to tiny K_a, simplifying to K_a = \frac{x^2}{C}

Achieving equilibrium allows chemists to make approximations, simplifying calculations to find crucial measurements like pH. Understanding the interplay of these reactions is key to mastering the behavior of solutions containing weak acids and their salts.