Certain weak acid has `P^(ka)` = 4. Find pH of 0.01M NaA(ag)

Answer

To find the pH of a 0.01 M solution of sodium acetate (NaA), where the weak acid has a pKa of 4, we can follow these steps:

### Step 1: Understand the Nature of the Salt
Sodium acetate (NaA) is a salt formed from a weak acid (acetic acid) and a strong base (sodium hydroxide). In solution, it dissociates into sodium ions (Na⁺) and acetate ions (A⁻). The acetate ion can undergo hydrolysis, which affects the pH of the solution.

### Step 2: Determine the pKa and pKw
Given:
- pKa = 4
- pKw = 14 (at 25°C)

### Step 3: Calculate pKb
Using the relationship between pKa and pKb:
\[ pKb = pKw - pKa \]
\[ pKb = 14 - 4 = 10 \]

### Step 4: Calculate the concentration of A⁻
The concentration of acetate ions (A⁻) in the solution is the same as the concentration of the salt:
\[ [A^-] = 0.01 \, M \]

### Step 5: Use the Hydrolysis Equation
The hydrolysis of the acetate ion can be represented as:
\[ A^- + H_2O \rightleftharpoons HA + OH^- \]

### Step 6: Set Up the Equilibrium Expression
Let \( x \) be the concentration of OH⁻ produced at equilibrium. The equilibrium concentrations will be:
- [A⁻] = 0.01 - x (approximately 0.01 for weak hydrolysis)
- [HA] = x
- [OH⁻] = x

The base dissociation constant (Kb) can be calculated using:
\[ Kb = \frac{Kw}{Ka} \]
Where \( Ka = 10^{-pKa} = 10^{-4} \).

Thus,
\[ Kb = \frac{1.0 \times 10^{-14}}{1.0 \times 10^{-4}} = 1.0 \times 10^{-10} \]

### Step 7: Set Up the Kb Expression
Using the Kb expression:
\[ Kb = \frac{[HA][OH^-]}{[A^-]} \]
Substituting the equilibrium concentrations:
\[ 1.0 \times 10^{-10} = \frac{x \cdot x}{0.01} \]
\[ 1.0 \times 10^{-10} = \frac{x^2}{0.01} \]
\[ x^2 = 1.0 \times 10^{-10} \times 0.01 = 1.0 \times 10^{-12} \]
\[ x = \sqrt{1.0 \times 10^{-12}} = 1.0 \times 10^{-6} \]

### Step 8: Calculate pOH
The concentration of OH⁻ is \( 1.0 \times 10^{-6} \), so:
\[ pOH = -\log[OH^-] = -\log(1.0 \times 10^{-6}) = 6 \]

### Step 9: Calculate pH
Using the relationship:
\[ pH + pOH = 14 \]
\[ pH = 14 - pOH = 14 - 6 = 8 \]

### Final Answer
The pH of a 0.01 M solution of sodium acetate (NaA) is **8**.

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