Question

A \(0.20-M\) sodium chlorobenzoate \(\left(\mathrm{NaC}_{7} \mathrm{H}_{4} \mathrm{ClO}_{2}\right)\) solution has a pH of \(8.65 .\) Calculate the pH of a 0.20- \(M\) chlorobenzoic acid \(\left(\mathrm{HC}_{7} \mathrm{H}_{4} \mathrm{ClO}_{2}\right)\) solution.

Short answer

The pH of a 0.20 M chlorobenzoic acid (HC7H4ClO2) solution can be calculated using the given 0.20 M sodium chlorobenzoate solution's pH of 8.65. First, we find the pKa of chlorobenzoic acid as 2.64. Then, using the Henderson-Hasselbalch equation, we'll calculate the pH as follows: \[pH = pK_{a} + \log \frac{[C_{7}H_{4}ClO_{2}^{-}]}{[HC_{7}H_{4}ClO_{2}]}\] \[pH = 2.64 + \log \frac{x}{0.20}\] Solve for x and substitute it back into the equation to determine the pH of the 0.20 M chlorobenzoic acid solution.

Step-by-step solution

Write the chemical equilibrium equation for dissociation of chlorobenzoic acid

The dissociation reaction for chlorobenzoic acid can be written as: \[HC_{7}H_{4}ClO_{2} \rightleftharpoons H^{+} + C_{7}H_{4}ClO_{2}^{-}\] Let the equilibrium concentrations be [H+] = x, [C7H4ClO2-]=0.20 M, and [HC7H4ClO2] = 0.20 - x M.

Write the expression for the Ka of chlorobenzoic acid

The Ka expression can be given as: \[K_{a} = \frac{[H^{+}][C_{7}H_{4}ClO_{2}^{-}]}{[HC_{7}H_{4}ClO_{2}]}\]

Identify given pH and find pOH and [OH-]

We are given the pH of sodium chlorobenzoate solution, which is 8.65. We can find pOH from pH using the relation: \[pH + pOH = 14\] So, pOH = 14 - 8.65 = 5.35 Now, we can find the [OH-] using the formula: \[[OH^{-}] = 10^{-pOH}\] \[[OH^{-}] = 10^{-5.35} = 4.47 \times 10^{-6} M\]

Calculate the pKa of chlorobenzoic acid

We can now use the pH of the sodium chlorobenzoate solution and the [OH-] to calculate the pKa of chlorobenzoic acid: \[pK_{a} = pH + \log \frac{[OH^{-}]}{[C_{7}H_{4}ClO_{2}^{-}]}\] \[pK_{a} = 8.65 + \log \frac{4.47 \times 10^{-6}}{0.20}\] \[pK_{a} = 2.64\]

Calculate the pH of the 0.20 M chlorobenzoic acid solution

We can now use the Henderson-Hasselbalch equation with the pKa and the concentrations of the acid and its conjugate base to find the pH of the 0.20 M chlorobenzoic acid solution: \[pH = pK_{a} + \log \frac{[C_{7}H_{4}ClO_{2}^{-}]}{[HC_{7}H_{4}ClO_{2}]}\] Since the concentration of the conjugate base is due to dissociation of the acid and will be very small compared to the concentration of the acid, therefore, the ratio becomes: \[\frac{[C_{7}H_{4}ClO_{2}^{-}]}{[HC_{7}H_{4}ClO_{2}]}\approx \frac{x}{0.20}\] \[pH = 2.64 + \log \frac{x}{0.20}\] Finally, in order to determine the pH of the 0.20 M chlorobenzoic acid solution, we need to solve for x which can be done by either using an iterative numerical method or finding an approximation when x is much smaller than 0.20. Once we have found x, we can substitute it back into the equation to calculate the pH.

Key concepts

Acid Dissociation Constant

Understanding the acid dissociation constant, denoted as Ka, is fundamental to grasping the delicate balance prevailing in weak acid solutions. This constant is a reflection of the strength of an acid in water. For a general weak acid represented by HA, the dissociation in water can be written as the chemical equation:
\[HA \rightleftharpoons H^{+} + A^{-}\]
At equilibrium, not all the acid molecules dissociate; thus, the solution contains HA, H+, and A-. The Ka expression is given by the equation:
\[K_a = \frac{[H^{+}][A^{-}]}{[HA]}\]
Higher values of Ka indicate a stronger acid, as it implies more dissociation into H+ and A-. Conversely, smaller values denote a weaker acid. For instance, in the given exercise involving chlorobenzoic acid (\(HC_{7}H_{4}ClO_{2}\)), we use the equilibrium concentrations of the resultant ions and the undissociated acid to find its Ka. These values illustrate how readily the acid gives up a proton to the water, leading to a set equilibrium state.

Weak Acid Equilibrium

When a weak acid dissolves in water, it does not completely dissociate into ions. The equilibrium dynamic for a weak acid dictates that the dissociation is a reversible process, constantly in flux between the undissociated acid and the ions it forms. The term 'equilibrium' in weak acid equilibrium refers to this state of balance, where the rates of the forward reaction (the acid dissociating) and the reverse reaction (the ions recombining to form the acid) are equal.
In the context of our exercise problem, the equilibrium for chlorobenzoic acid can be disturbed by changes in concentration, volume, or temperature. Yet, it eventually reaches a new state of balance, guided by the specific Ka value of the weak acid. Le Chatelier's Principle offers more insight into how equilibria shift in response to such perturbations. Students may find it easier to understand weak acid equilibrium by thinking of it as a teetering balance—where the acid and its dissociated form are in a delicate dance, neither completely overwhelming the other.

Henderson-Hasselbalch Equation

The Henderson-Hasselbalch equation is a vital tool for chemists and students alike, offering a direct way to calculate the pH of a buffer system—solutions that maintain a stable pH upon dilution or addition of small amounts of acid or base. This equation is derived from the Ka expression and provides a relationship between the pH, pKa, and the ratio of the concentrations of the conjugate base (A-) to the weak acid (HA):
\[pH = pK_a + \log\left(\frac{[A^{-}]}{[HA]}\right)\]
The pKa value is the negative logarithm of the Ka and offers a more convenient way to express the acid's dissociation constant. Utilizing the Henderson-Hasselbalch equation, we've calculated the pH of the chlorobenzoic acid solution by plugging in the concentration values derived from the weak acid equilibrium. Importantly, this equation assumes that the change in the concentration of the weak acid is negligible compared to its initial concentration, a valid approximation when the acid is weak and its dissociation is limited. This simplification allows us to estimate pH with ease and understand the behavior of buffer solutions in various conditions.