Problem 23 - General Chemistry: Principles and Modern Applications

Question

For the ionization of phenylacetic acid, $$\begin{array}{r} \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{CH}_{2} \mathrm{CO}_{2} \mathrm{H}+\mathrm{H}_{2} \mathrm{O} \rightleftharpoons \mathrm{H}_{3} \mathrm{O}^{+}+\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{CH}_{2} \mathrm{CO}_{2} \\\ K_{\mathrm{a}}=4.9 \times 10^{-5} \end{array}$$ (a) What is \(\left[\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{CH}_{2} \mathrm{CO}_{2}^{-}\right]\) in \(0.186 \mathrm{M} \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{CH}_{2} \mathrm{CO}_{2} \mathrm{H} ?\) (b) What is the \(\mathrm{pH}\) of \(0.121 \mathrm{M} \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{CH}_{2} \mathrm{CO}_{2} \mathrm{H} ?\)

Step-by-Step Solution

Answer

The concentration of \([C_6H_5CH_2CO_{2}^-]\) ion in \(0.186M\, C_6H_5CH_2CO_2H\) solution and the pH of \(0.121M\, C_6H_5CH_2CO_2H\) solution can be calculated by following the above steps. Since the solutions for 'x' provide both the ion concentration and \([H_3O^+]\) needed for pH calculation, a clear understanding of weak acid ionization and solving equations is critical for solving this exercise.

Step 1: Set up ICE table

To solve this problem, we use an ICE (Initial, Change, Equilibrium) table. The initial concentration of the acid is given as 0.186 M and the ion and hydronium ion have concentrations of 0 initially. During the reaction the acid loses some amount of H+, which will be represented as ‘x’, hence decreasing by ‘x’. Hence adding ‘x’ to the ion (product) and hydronium ion (product).

Step 2: Write the equation for Ka and solve for ‘x’

The ionization constant or acid dissociation constant Ka is given by the equation: \[ Ka = \frac{[H_3O^{+}][C_6H_5CH_2CO_{2}^{-}]}{[C_6H_5CH_2CO_2H]} \]Substitute the equilibrium concentrations in terms of ‘x’ into the equation and solve for ‘x’, which represents the concentration of the ion. We now have: \[4.9 * 10 ^{-5} = \frac{x * x}{0.186 - x}\]. Solve this quadratic equation to get the value of 'x'.

Step 3: Finding ion concentration of the phenylacetic acid

After solving the quadratic equation we get two solutions for 'x' \([-H_3O^{+}]\) and \([C_6H_5CH_2CO_{2}^-]\). Of the two solutions, the negative one must be discarded (as concentration cannot be negative). Hence, the positive solution for 'x' represents the equilibrium concentration of \([C_6H_5CH_2CO_{2}^-]\) ion.

Step 4: Calculation of pH

pH is calculated using the equation: \[ pH = -log[H_3O^{+}] \]Since \([H_3O^{+}]\) is equal to 'x', substitute the positive ‘x’ value obtained in step 3 and compute to find the pH.

Key Concepts

Ionization Constants

Ionization constants, represented as \( K_a \) for acids, are essential in understanding how strongly an acid donates protons in water. Phenylacetic acid is a weak acid, which means it doesn't completely ionize in solution. The \( K_a \) value helps us measure this tendency.

For the ionization reaction:

• \[ \text{C}_6\text{H}_5\text{CH}_2\text{CO}_2\text{H} + \text{H}_2\text{O} \rightleftharpoons \text{H}_3\text{O}^+ + \text{C}_6\text{H}_5\text{CH}_2\text{CO}_2^- \]
• \( K_a = \frac{[H_3O^+][C_6H_5CH_2CO_2^-]}{[C_6H_5CH_2CO_2H]} \)

A higher \( K_a \) implies a stronger acid. For phenylacetic acid, \( K_a = 4.9 \times 10^{-5} \), indicating it's a weak acid. By knowing \( K_a \), we can use it in calculations to understand the acid's behavior in solution.

ICE Tables

ICE tables, which stand for Initial, Change, Equilibrium, are a systematic way to keep track of concentrations during a chemical reaction. In the context of weak acids, they help us visualize what's happening as the acid ionizes.

For the given problem:

• **Initial:** We start with an initial concentration of 0.186 M for phenylacetic acid, with no ions initially present.
• **Change:** As the reaction progresses, 'x' amount of acid ionizes, increasing hydronium and ion concentration by 'x' and decreasing the acid concentration by 'x'.
• **Equilibrium:** We end with \( (0.186 - x) \) for the acid and 'x' for both the ion and the hydronium ions.

Using an ICE table helps to set up the mathematical equation based on the \( K_a \) in order to solve for 'x'. This process provides clarity on how much of each species is present at equilibrium.

pH Calculation

Calculating pH gives us insight into the acidity of a solution. It is calculated using the concentration of hydronium ions. In this case, hydronium ions form as the acid ionizes.

Using the relation:

• \[ pH = -\log[H_3O^+] \]

Since \([H_3O^+] = x\), once we solve the quadratic equation from the ionization step, we find 'x'. This value is inserted into the logarithmic equation above to calculate the pH.
For weak acids like phenylacetic acid, the pH is typically greater than 2 but less than 7, indicating a slightly acidic solution. A calculated pH value helps us determine the strength and behavior of the acid in the given concentration.