Question

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

Short answer

The pH of the 0.20 M chlorobenzoic acid solution can be calculated using the Henderson-Hasselbalch equation, given the \(K_a\) value obtained from the sodium chlorobenzoate solution with a pH of 8.65. By calculating the \(K_a\), then using the Henderson-Hasselbalch equation, the pH of the 0.20 M chlorobenzoic acid solution is found to be equal to the negative logarithm of the acid dissociation constant.

Step-by-step solution

1. Write the equations for the reaction and equilibrium constants for the given substances

The given substances are sodium chlorobenzoate (NaC₇H₄ClO₂) and chlorobenzoic acid (HC₇H₄ClO₂). Sodium chlorobenzoate is a salt where chlorobenzoate is the conjugate base of chlorobenzoic acid. The dissociation of chlorobenzoic acid can be represented with the following equation: \( HC_7H_4ClO_2 \rightleftharpoons H^+ + C_7H_4ClO_2^- \) And the dissociation constant, \(K_a\) can be written as follows: \( K_a = \frac{[H^+][C_7H_4ClO_2^-]}{[HC_7H_4ClO_2]} \)

2. Calculate the \( K_a \) for chlorobenzoic acid

To calculate the \(K_a\) for chlorobenzoic acid, we can use the given pH of the sodium chlorobenzoate solution, which is 8.65. Because sodium chlorobenzoate is a salt of a weak acid and a strong base, it will increase the pH of the solution above 7. Therefore, the concentration of hydroxide ions in the sodium chlorobenzoate solution is larger than the concentration of hydrogen ions. To find the concentration of hydrogen ions [H⁺] for the sodium chlorobenzoate solution, we can calculate the concentration of hydroxide ions and then use the relation \(K_w = [H^+][OH^-]\) \[ pOH = 14 - pH = 14 - 8.65 = 5.35 \] \[ [OH^-] = 10^{-pOH} = 10^{-5.35} \] Now, we can calculate the concentration of hydrogen ions using \(K_w = [H^+][OH^-]\) The ion product of water, \(K_w\) is \(1 \times 10^{-14}\) at room temperature. \[ [H^+] = \frac{K_w}{[OH^-]} = \frac{1 \times 10^{-14}}{10^{-5.35}} \] With the concentration of hydrogen ions [H⁺], we can find the concentration of the chlorobenzoate ion [C₇H₄ClO₂⁻] which is equal to the concentration of the sodium chlorobenzoate solution because sodium is a monovalent ion. \[ [C_7H_4ClO_2^-]= 0.20 \mathrm{M} \] Now we can calculate the \( K_a \) for chlorobenzoic acid using the following relationship: \[ K_a = \frac{[H^+][C_7H_4ClO_2^-]}{[HC_7H_4ClO_2]} \] Due to the low ionization of weak acid, the concentration of weak acid \( [HC_7H_4ClO_2] \approx 0.20 \mathrm{M} \) \[ K_a = \frac{[H^+][C_7H_4ClO_2^-]}{0.20} \]

3. Calculate the pH of the chlorobenzoic acid solution

Now that we have \( K_a \) for the chlorobenzoic acid, we can find the pH of the 0.20 M chlorobenzoic acid solution using the Henderson-Hasselbalch equation: \[ pH = pK_a + log \frac{[C_7H_4ClO_2^-]}{[HC_7H_4ClO_2]} \] Since both the concentration of chlorobenzoic acid and its conjugate base are equal, the pH will be equal to the \(pK_a\) of the solution. \[ pH = pK_a \] And, \[ pH = -log (K_a) \] So, the pH of the 0.20 M chlorobenzoic acid solution is equal to the negative logarithm of the acid dissociation constant.

Key concepts

pH calculation

Understanding pH is crucial in chemistry, especially in the context of acid-base equilibrium. The term "pH" refers to the potential of hydrogen and is a measure that evaluates the acidity or basicity of a solution. The pH scale ranges from 0 to 14; a pH less than 7 indicates an acidic solution, whereas a pH greater than 7 indicates a basic or alkaline solution.
pH is calculated using the formula: \[ pH = -\log [H^+] \] where \( [H^+ ] \) is the concentration of hydrogen ions in the solution. For example, if a solution has a hydrogen ion concentration of \(1 \times 10^{-3} \, M\), its pH would be 3. Furthermore, calculating pH is a critical step when analyzing solutions involving weak acids or bases, as it indicates how the said acids or bases affect the surrounding environment.

equilibrium constant

In chemical reactions, an equilibrium constant \( K \) helps determine the extent to which reactions occur. For acids and bases, the equilibrium constant is known as the acid dissociation constant, \( K_a \). This value illustrates how well an acid can donate a proton in a chemical reaction.
For the dissociation of a weak acid such as chlorobenzoic acid \( HC_7H_4ClO_2 \), it dissociates into \( H^+ \) and \( C_7H_4ClO_2^- \), and the equilibrium constant is expressed by the equation:\[ K_a = \frac{[H^+][C_7H_4ClO_2^-]}{[HC_7H_4ClO_2]} \] Calculating \( K_a \) is an essential step in predicting the behavior of weak acids under varying conditions. It also provides fundamental information required for further calculations, such as determining the pH or utilizing the Henderson-Hasselbalch equation.

Henderson-Hasselbalch equation

The Henderson-Hasselbalch equation is a useful tool in the realm of acid-base chemistry, offering a simplified way to determine the pH of buffer solutions. This equation is expressed as:\[ pH = pK_a + \log \frac{[A^-]}{[HA]} \] where \( [A^-] \) is the concentration of the conjugate base and \( [HA] \) is the concentration of the weak acid. The \( pK_a \) is the negative logarithm of the \( K_a \).This equation is especially handy when dealing with buffer solutions, as it relates pH directly to the pKa and the ratio of the concentrations of the conjugate base and the weak acid. In the case of equal concentrations of \( [A^-] \) and \( [HA] \), the pH equals the \( pK_a \), simplifying the calculation even further.
extul> ### Key Usage Tips:- Ideal for systems where both acid and conjugate base are present.- Much straightforward when concentrations are not drastically different from one another.

weak acid dissociation

Weak acid dissociation is a fundamental concept when dealing with solutions involving acids that do not completely ionize in water. Weak acids partially dissociate, releasing fewer hydrogen ions than strong acids.
The general expression for the dissociation of a weak acid \( HA \) can be written as:\[ HA \rightleftharpoons H^+ + A^- \] This reversible reaction indicates that the acid exists in equilibrium with its ions in the solution. The degree of dissociation depends on the acid dissociation constant \( K_a \).
extul> ### Important Considerations:- Only a fraction of weak acid molecules donate their protons.- Knowing the concentration of hydrogen ions helps predict the pH.- The concept is key for understanding buffer systems.