Calculate the pH of a solution of ammonium acetate
(Given ` : K_a= 1.78 xx 10 ^(-5) ,K_b= 1.8 xx 10 ^(-5) and K_w = 1.8 xx 10^(-14))`

Answer

To calculate the pH of a solution of ammonium acetate (CH₃COONH₄), we can follow these steps:

### Step 1: Identify the components
Ammonium acetate is a salt formed from a weak acid (acetic acid, CH₃COOH) and a weak base (ammonium hydroxide, NH₄OH). The relevant constants are given as:
- \( K_a \) (for acetic acid) = \( 1.78 \times 10^{-5} \)
- \( K_b \) (for ammonium hydroxide) = \( 1.8 \times 10^{-5} \)
- \( K_w \) (ion product of water) = \( 1.8 \times 10^{-14} \)

### Step 2: Calculate \( pK_a \) and \( pK_b \)
The \( pK_a \) and \( pK_b \) can be calculated using the formulas:
\[ pK_a = -\log(K_a) \]
\[ pK_b = -\log(K_b) \]

Calculating \( pK_a \):
\[ pK_a = -\log(1.78 \times 10^{-5}) \approx 4.75 \]

Calculating \( pK_b \):
\[ pK_b = -\log(1.8 \times 10^{-5}) \approx 4.74 \]

### Step 3: Use the formula for pH of a salt from weak acid and weak base
For a salt formed from a weak acid and a weak base, the pH can be calculated using the formula:
\[ pH = 7 + \frac{1}{2}(pK_a - pK_b) \]

### Step 4: Substitute the values into the formula
Substituting the calculated \( pK_a \) and \( pK_b \) into the pH formula:
\[ pH = 7 + \frac{1}{2}(4.75 - 4.74) \]
\[ pH = 7 + \frac{1}{2}(0.01) \]
\[ pH = 7 + 0.005 \]
\[ pH = 7.005 \]

### Step 5: Finalize the answer
Thus, the pH of the ammonium acetate solution is approximately:
\[ \text{pH} \approx 7.005 \]