Calculate the pH of `0.05M` sodium acetate solution, if the `pK_(a)` of acetic acid is `4.74`.

Options

A. 3.37
B. 4.37
C. 7.74
D. 0.474

Answer

To calculate the pH of a 0.05 M sodium acetate solution, we can follow these steps:

### Step 1: Understand the Reaction
Sodium acetate (CH₃COONa) dissociates in water to form acetate ions (CH₃COO⁻) and sodium ions (Na⁺). The acetate ions can react with water to form acetic acid (CH₃COOH) and hydroxide ions (OH⁻):

\[ \text{CH}_3\text{COO}^- + \text{H}_2\text{O} \rightleftharpoons \text{CH}_3\text{COOH} + \text{OH}^- \]

### Step 2: Determine the Concentration of Acetate Ion
The concentration of acetate ions in the solution is given as 0.05 M.

### Step 3: Calculate \(K_a\) from \(pK_a\)
The \(pK_a\) of acetic acid is given as 4.74. We can calculate \(K_a\) using the formula:

\[ K_a = 10^{-pK_a} = 10^{-4.74} \approx 1.82 \times 10^{-5} \]

### Step 4: Use the Hydrolysis Equation
The hydrolysis of acetate can be described by the equilibrium expression:

\[ K_b = \frac{K_w}{K_a} \]

Where:
- \(K_w\) is the ion product of water, \(1.0 \times 10^{-14}\).
- \(K_b\) is the base dissociation constant for acetate.

Calculating \(K_b\):

\[ K_b = \frac{1.0 \times 10^{-14}}{1.82 \times 10^{-5}} \approx 5.49 \times 10^{-10} \]

### Step 5: Set Up the Hydrolysis Equilibrium Expression
Let \(x\) be the concentration of OH⁻ produced at equilibrium:

\[ K_b = \frac{[CH_3COOH][OH^-]}{[CH_3COO^-]} = \frac{x^2}{0.05 - x} \approx \frac{x^2}{0.05} \quad (\text{assuming } x \text{ is small}) \]

Thus:

\[ 5.49 \times 10^{-10} = \frac{x^2}{0.05} \]

### Step 6: Solve for \(x\)
Rearranging gives:

\[ x^2 = 5.49 \times 10^{-10} \times 0.05 \]

Calculating:

\[ x^2 = 2.745 \times 10^{-11} \]
\[ x = \sqrt{2.745 \times 10^{-11}} \approx 5.24 \times 10^{-6} \text{ M (OH⁻ concentration)} \]

### Step 7: Calculate pOH
Now, we can find pOH:

\[ pOH = -\log[OH^-] = -\log(5.24 \times 10^{-6}) \approx 5.28 \]

### Step 8: Calculate pH
Finally, we can find the pH:

\[ pH = 14 - pOH = 14 - 5.28 = 8.72 \]

### Final Answer
The pH of the 0.05 M sodium acetate solution is approximately **8.72**.

---