Question

Calculate the \(\mathrm{pH}\) of an aqueous solution at \(25^{\circ} \mathrm{C}\) that is \(0.34 \mathrm{M}\) in phenol \(\left(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{OH}\right) .\left(K_{\mathrm{a}}\right.\) for phenol \(=1.3 \times 10^{-10}\).)

Short answer

The pH of the solution is approximately 5.18.

Step-by-step solution

Understanding the Acid Dissociation

Phenol (\(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{OH}\)) is a weak acid, which partially dissociates in water according to the equation: \[\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{OH} \rightleftharpoons \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}^{-} + \mathrm{H}^{+}\]The dissociation constant \(K_a\) is the equilibrium constant for this reaction.

Writing the Expression for Ka

The expression for the acid dissociation constant \(K_a\) is: \[K_a = \frac{[\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}^{-}][\mathrm{H}^+]}{[\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{OH}]}\]Given: \(K_a = 1.3 \times 10^{-10}\) and the initial concentration of phenol is \(0.34 \mathrm{M}\).

Setting Up the Ice Table

Set up an ICE (Initial, Change, Equilibrium) table to determine the concentrations at equilibrium. - **Initial:** - \([\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{OH}] = 0.34\ \mathrm{M}\) - \([\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}^{-}] = 0\ \mathrm{M}\) - \([\mathrm{H}^+] = 0\ \mathrm{M}\)- **Change:** - \([\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{OH}] = -x\ \mathrm{M}\) - \([\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}^{-}] = x\ \mathrm{M}\) - \([\mathrm{H}^+] = x\ \mathrm{M}\)- **Equilibrium:** - \([\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{OH}] = 0.34 - x\ \mathrm{M}\) - \([\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}^{-}] = x\ \mathrm{M}\) - \([\mathrm{H}^+] = x\ \mathrm{M}\)\

Solving for x

Substitute the equilibrium concentrations into the \(K_a\) expression:\[1.3 \times 10^{-10} = \frac{x^2}{0.34 - x}\]Assume \(x\) is small compared to \(0.34\), so \(0.34 - x \approx 0.34\):\[1.3 \times 10^{-10} = \frac{x^2}{0.34}\]Solving for \(x\):\[x^2 = (1.3 \times 10^{-10}) \times 0.34\]\[x^2 = 4.42 \times 10^{-11}\]\[x = \sqrt{4.42 \times 10^{-11}}\]\[x \approx 6.64 \times 10^{-6}\]

Calculating pH

The \([\mathrm{H}^+]\) concentration at equilibrium is \(x\), which is approximately \(6.64 \times 10^{-6} \mathrm{M}\).Calculate the \(\mathrm{pH}\):\[\mathrm{pH} = -\log(6.64 \times 10^{-6})\]\[\mathrm{pH} \approx 5.18\]

Verifying the Assumption

Verify that \(x\) is small compared to \(0.34\). Since \(6.64 \times 10^{-6}\) is indeed much smaller than \(0.34\), the assumption is valid.

Key concepts

Weak Acid Dissociation

Weak acids are unique because they do not completely dissociate in water. Instead, they only partially break apart into their ions. For instance, phenol, a weak acid, dissociates to form phenoxide ions ( \( \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}^{-}\ \)), and hydrogen ions ( \( \mathrm{H}^{+}\ \)) in a solution. This means that at any given time, there is an equilibrium between the intact phenol molecules and the ions it dissociates into.
Understanding this partial dissociation is crucial in chemistry, as it affects both the solution's pH and its overall reactivity. Compared to strong acids, weak acids like phenol have a less drastic effect on their environment, which can be advantageous in certain chemical reactions.
This equilibrium behavior is expressed using the dissociation constant, \(K_a\), which indicates how far the dissociation proceeds under equilibrium conditions.

Acid Dissociation Constant

The acid dissociation constant, often symbolized as \(K_a\), is a fundamental concept in determining the strength of a weak acid. It provides a quantitative measure of the extent to which an acid dissociates in solution. For weak acids like phenol, a low \(K_a\) value (e.g., \(1.3 \times 10^{-10}\)) signifies that the acid doesn’t dissociate much. The lower the \(K_a\), the weaker the acid.
The formula for \(K_a\) is given by:- \[K_a = \frac{[\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{O}^{-}][\mathrm{H}^+]}{[\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{OH}]}\]This equation conveys the relationship between the concentrations of the undissociated acid and its dissociation products. Understanding \(K_a\) helps predict how an acid behaves in different chemical environments, which is essential for tasks like calculating solutions' pH or manipulating chemical reactions.

Equilibrium Concentration

In a chemical equilibrium involving acids like phenol, determining the final concentrations of all substances at equilibrium is essential to understanding the system's behavior. The equilibrium concentration refers to the amounts of the acid, its conjugate base, and hydrogen ions present once the system has reached a state of balance.
For weak acids, equilibrium concentration is critical because these values dictate the acid's reactivity and its effect on the solution's pH. The concentration of hydrogen ions, directly connected to the concept of pH, is also determined at this stage. This calculation involves using the initial concentrations and changes that occur due to the acid's partial dissociation.
The equilibrium concentration is typically found using tools like the ICE table and tested assumptions to simplify calculations, which are pivotal for deriving wise predictions about the solution's characteristics.

ICE Table

The ICE table is an invaluable tool in chemistry for systematically determining the changes in concentration and the equilibrium state of chemical reactions. ICE stands for Initial, Change, and Equilibrium, and it helps organize these transitions for transparent calculations.
1. **Initial** - This section records the starting concentrations of each component before any reaction occurs.2. **Change** - Here, you account for the changes that occur as the reaction moves toward equilibrium. This is depicted through variables like \(x\).3. **Equilibrium** - This section calculates the final concentrations of substances at equilibrium.By filling in these sections, the ICE table enables simplified calculation of unknowns, such as the concentration of hydrogen ions in weak acid solutions. It supports efficient problem-solving, thus, making pH calculation and analysis of weak acid reactions more accessible for students.