40 ml of `0.1` M ammonia is mixed with 20 ml of `0.1 M HCI`. What is the pH of the mixture ? (`pK_(b)` of ammonia solution is `4.74.`)

Options

A. `4.74`
B. `2.26`
C. `9.26`
D. `5.00`

Explanation

To solve the problem, we need to find the pH of the mixture of ammonia and hydrochloric acid. Here’s a step-by-step solution:

### Step 1: Calculate the moles of ammonia and hydrochloric acid
- **Ammonia (NH₃)**:
- Volume = 40 mL = 0.040 L
- Molarity = 0.1 M
- Moles of NH₃ = Molarity × Volume = 0.1 mol/L × 0.040 L = 0.004 moles

- **Hydrochloric Acid (HCl)**:
- Volume = 20 mL = 0.020 L
- Molarity = 0.1 M
- Moles of HCl = Molarity × Volume = 0.1 mol/L × 0.020 L = 0.002 moles

### Step 2: Determine the limiting reactant
- Ammonia reacts with hydrochloric acid in a 1:1 ratio.
- We have 0.004 moles of NH₃ and 0.002 moles of HCl.
- Since HCl is the limiting reactant, it will completely react with an equal amount of NH₃.

### Step 3: Calculate the remaining moles after the reaction
- Moles of NH₃ remaining after reaction = Initial moles of NH₃ - Moles of HCl reacted
- Moles of NH₃ remaining = 0.004 moles - 0.002 moles = 0.002 moles

- Moles of NH₄Cl formed = Moles of HCl reacted = 0.002 moles

### Step 4: Calculate the total volume of the mixture
- Total volume = Volume of NH₃ + Volume of HCl = 40 mL + 20 mL = 60 mL = 0.060 L

### Step 5: Calculate the concentrations of NH₃ and NH₄Cl
- Concentration of NH₃ = Moles of NH₃ remaining / Total volume = 0.002 moles / 0.060 L = 0.0333 M

- Concentration of NH₄Cl = Moles of NH₄Cl formed / Total volume = 0.002 moles / 0.060 L = 0.0333 M

### Step 6: Use the Henderson-Hasselbalch equation to find pH
- Since we have a buffer solution (NH₃ and NH₄Cl), we can use the formula:
\[ pOH = pK_b + \log\left(\frac{[\text{Salt}]}{[\text{Base}]}\right) \]
- Here, \( pK_b = 4.74 \), and since the concentrations of NH₃ and NH₄Cl are equal:
\[ pOH = 4.74 + \log\left(\frac{0.0333}{0.0333}\right) = 4.74 + \log(1) = 4.74 \]

### Step 7: Calculate pH from pOH
- pH = 14 - pOH = 14 - 4.74 = 9.26

### Final Answer
The pH of the mixture is **9.26**.

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