Question

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

Short answer

The pH of the 0.20M chlorobenzoic acid solution is 1.55.

Step-by-step solution

Write the chemical equations for the sodium chlorobenzoate and chlorobenzoic acid solutions

For sodium chlorobenzoate solution, the chemical equation for the dissociation of C7H4ClO2- ion is: \(C7H4ClO2^- + H2O \rightleftharpoons HC7H4ClO2 + OH^-\) For chlorobenzoic acid solution, the chemical equation for the dissociation of HC7H4ClO2 is: \(HC7H4ClO2 + H2O \rightleftharpoons C7H4ClO2^- + H3O^+\)

Calculate the pOH of the sodium chlorobenzoate solution

The given pH of the sodium chlorobenzoate solution is 8.65. To find the pOH, we can use the relationship: \(pH + pOH = 14\) Therefore, pOH = 14 - pH = 14 - 8.65 = 5.35.

Calculate the concentration of OH- ions in the sodium chlorobenzoate solution

To calculate the concentration of OH- ions, we can use the formula: \([OH^-] = 10^{-pOH}\) Therefore, [OH-] = \(10^{-5.35}\) = \(2.24 x 10^{-6}M\).

Calculate the Ka using the Kb expression for sodium chlorobenzoate solution

The Kb expression for sodium chlorobenzoate solution is as follows: \(Kb = \frac{[OH^-][HC7H4ClO2]}{[C7H4ClO2^-]}\) Since the initial concentration of sodium chlorobenzoate is 0.20M and it is almost completely dissociated, \([C7H4ClO2^-]\) = 0.20M, and \([HC7H4ClO2]\) = [OH-]. Now we can calculate the Kb value using the expression: \(Kb = \frac{(2.24 x 10^{-6})(2.24 x 10^{-6})}{0.20}\) = \(2.504 x 10^{-11}\).

Determine the Ka of chlorobenzoic acid from Kb of sodium chlorobenzoate

To find the Ka of chlorobenzoic acid, we use the equation \(Kw = Ka \times Kb\), where Kw is the ion product of water, which equals \(1.0 x 10^{-14}\). Then, Ka = \(Kw / Kb\) = \(\frac{1.0 x 10^{-14}}{2.504 x 10^{-11}}\) = \(3.99 x 10^{-4}\).

Write the acid dissociation expression and solve for [H3O+]

The acid dissociation expression for chlorobenzoic acid is: \(Ka = \frac{[H3O^+][C7H4ClO2^-]}{[HC7H4ClO2]}\) Since the initial concentration of chlorobenzoic acid is 0.20M, for a small value of x, the concentrations at equilibrium are: \([H3O^+]\) = x and \([C7H4ClO2^-]\) = x \([HC7H4ClO2]\) = (0.20 - x) Now we can substitute these values and solve for x, which represents the [H3O+] in the chlorobenzoic acid solution: \(3.99 x 10^{-4} = \frac{x^2}{(0.20-x)}\) We can approximate x << 0.20, then: \(3.99 x 10^{-4}\) = \(\frac{x^2}{0.20}\) x = \(H3O^+\) = \(\sqrt{3.99 x 10^{-4}\times 0.20}\) = \(2.82 x 10^{-2}\)

Calculate the pH of the chlorobenzoic acid solution

Finally, we can calculate the pH of the chlorobenzoic acid solution using the formula: \(pH = -log_{10} [H3O^+]\) pH = -log(2.82 x 10^{-2}) = 1.55 Therefore, the pH of the 0.20M chlorobenzoic acid solution is 1.55.

Key concepts

Acid Dissociation Constant (Ka)

The acid dissociation constant, abbreviated as \(K_a\), measures the strength of an acid in solution. It reflects how well an acid can donate protons, or hydrogen ions \((H^+)\), to the solution. To understand \(K_a\), consider a generic weak acid \(HA\), which dissociates in water according to the following equilibrium equation:
\[ HA + H_2O \rightleftharpoons H_3O^+ + A^- \] In this expression, \(H_3O^+\) represents the hydronium ion, and \(A^-\) is the conjugate base. The equation for \(K_a\) is: \[ K_a = \frac{[H_3O^+][A^-]}{[HA]} \] This formula shows the ratio of the products (hydronium ion and the conjugate base concentration) to the reactants (the concentration of undissociated acid). A larger \(K_a\) value implies a stronger acid, meaning it dissociates to a greater extent, producing more \(H_3O^+\) ions.
In practical terms, if you encounter a compound like chlorobenzoic acid, you calculate its \(K_a\) to predict how acidic it will be in solution, using measured \([H_3O^+]\) and its concentration at equilibrium. Understanding \(K_a\) is crucial for predicting the pH of solutions.

Base Dissociation Constant (Kb)

The base dissociation constant, denoted as \(K_b\), helps us understand how bases dissociate in solution. This constant tells us about a base's strength by showing its ability to accept protons. Consider a typical base \(B\) that reacts with water as:\[ B + H_2O \rightleftharpoons BH^+ + OH^- \]Here, \(BH^+\) is the conjugate acid, and \(OH^-\) is the hydroxide ion. The mathematical expression for \(K_b\) is:\[ K_b = \frac{[BH^+][OH^-]}{[B]} \]This ratio indicates the concentration of products \((BH^+\) and \([OH^-])\) to the reactant \((B)\). The value of \(K_b\) gives insight into how well a base ionizes in water. A higher \(K_b\) value means a stronger base that is more effective at generating hydroxide ions \((OH^-)\), which leads to a higher pH.In the sodium chlorobenzoate solution, \(K_b\) is used to evaluate the solution's basic nature. We calculate \(K_b\) from the given \([OH^-]\) to find how much sodium chlorobenzoate dissociates. By understanding \(K_b\), we can make informed predictions regarding the basicity of the solution.

Chemical Equilibrium

Chemical equilibrium is a state where the concentrations of reactants and products remain constant over time. This occurs because the forward and reverse reactions proceed at the same rate. In a chemical system at equilibrium, such as the dissociation of weak acids or bases, the ratio of the concentration of products to reactants becomes a constant value.For acids and bases, the equilibrium is represented by the equations we use to calculate \(K_a\) and \(K_b\). These equilibrium constants provide a snapshot of the reaction's balance point, where neither reactants nor products dominate significantly.The concept of equilibrium is particularly integral in calculating pH, as seen with chlorobenzoic acid and sodium chlorobenzoate solutions. For each dissociation process, equilibrium expressions help us derive the \(K_a\) or \(K_b\), reflecting how the substances dissociate in solution.When dealing with chemical equilibrium,

• Recognize that changes in concentration, temperature, or pressure can shift the equilibrium position.
• Use Le Chatelier's Principle to predict shifts in equilibrium.

Equilibrium calculations often involve approximations, assuming that changes in concentration (like a negligible \(x\) in quadratic approximations) remain small compared to initial values. Understanding equilibrium enables accurate pH calculations and predictions of reaction behavior.