pH of a salt of a strong base with weak acid

Options

A. `pH=1/2pK_w+1/2pK_a+1/2logC`
B. `pH=1/2pK_w-1/2pK_a-1/2logC`
C. `pH=1/2pK_w+1/2pK_a-1/2logC`
D. None of these

Answer

To find the pH of a salt formed from a strong base and a weak acid, we can follow these steps:

### Step 1: Identify the components of the salt
The salt in question is formed from a strong base (e.g., NaOH) and a weak acid (e.g., CH₃COOH). The salt can be represented as CH₃COONa, where CH₃COO⁻ is the conjugate base of the weak acid.

**Hint:** Remember that the pH of a salt depends on the nature of its constituent acid and base.

### Step 2: Understand the behavior of the salt in water
When the salt CH₃COONa is dissolved in water, it dissociates into CH₃COO⁻ and Na⁺ ions. The Na⁺ ion does not affect the pH since it comes from a strong base. However, the CH₃COO⁻ ion can hydrolyze in water to form OH⁻ ions, leading to a basic solution.

**Hint:** Hydrolysis of the conjugate base will determine if the solution is acidic or basic.

### Step 3: Write the hydrolysis reaction
The hydrolysis of the acetate ion can be represented as:
\[ \text{CH}_3\text{COO}^- + \text{H}_2\text{O} \rightleftharpoons \text{CH}_3\text{COOH} + \text{OH}^- \]

**Hint:** This reaction shows that the acetate ion can accept a proton from water, generating hydroxide ions.

### Step 4: Set up the equilibrium expression
The equilibrium constant for the hydrolysis can be expressed using the dissociation constant \( K_a \) of the weak acid:
\[ K_a = \frac{[\text{CH}_3\text{COO}^-][\text{H}^+]}{[\text{CH}_3\text{COOH}]} \]

Since \( K_w = K_a \cdot K_b \), we can find \( K_b \) for the acetate ion:
\[ K_b = \frac{K_w}{K_a} \]

**Hint:** Use the relationship between \( K_a \), \( K_b \), and \( K_w \) to find the base dissociation constant.

### Step 5: Calculate the pH
Using the concentration of the salt (C), we can express the concentration of OH⁻ produced from the hydrolysis. The pH can be calculated using the formula:
\[ \text{pH} = \frac{1}{2} pK_w + \frac{1}{2} pK_a + \log C \]

Where:
- \( pK_w = 14 \)
- \( pK_a \) is the negative logarithm of the acid dissociation constant of the weak acid.

Since \( pK_w = 14 \), we can simplify:
\[ \text{pH} = 7 + \frac{1}{2} pK_a + \log C \]

**Hint:** The pH is greater than 7 for a salt of a strong base and a weak acid.

### Conclusion
Thus, the pH of a salt formed from a strong base and a weak acid can be expressed as:
\[ \text{pH} = \frac{1}{2} pK_w + \frac{1}{2} pK_a + \log C \]

The correct answer from the options provided is option A.