Calculate the pH of a solution made by adding 0.01 mole of HCl in 100 mL of a solution which is 0.2 M in `NH_(3)(pK_(b)=4.74)` and 0.3 M in `NH_(4)^(+)` :
(Assuming no change in volume )

Options

A. 5.34
B. 8.66
C. 7.46
D. None of these

Answer

To calculate the pH of the solution formed by adding 0.01 mole of HCl to a solution that is 0.2 M in ammonia (NH₃) and 0.3 M in ammonium ion (NH₄⁺), we can follow these steps:

### Step 1: Understand the System
We have a buffer solution consisting of ammonia (NH₃) and its conjugate acid, ammonium ion (NH₄⁺). When we add HCl, it will react with the ammonia to form more ammonium ions.

### Step 2: Calculate the Molarity of HCl
We need to find the molarity of the HCl solution:
- Moles of HCl = 0.01 moles
- Volume of solution = 100 mL = 0.1 L

\[ \text{Molarity of HCl} = \frac{\text{moles of HCl}}{\text{volume in L}} = \frac{0.01 \text{ moles}}{0.1 \text{ L}} = 0.1 \text{ M} \]

### Step 3: Determine the Change in Concentrations
When HCl is added:
- HCl will donate H⁺ ions, which will react with NH₃ to form NH₄⁺.
- Initial concentration of NH₄⁺ = 0.3 M
- Initial concentration of NH₃ = 0.2 M

After the reaction:
- NH₄⁺ concentration will increase by 0.1 M (due to the reaction with H⁺).
- NH₃ concentration will decrease by 0.1 M.

New concentrations:
- \([NH₄^+] = 0.3 + 0.1 = 0.4 \text{ M}\)
- \([NH₃] = 0.2 - 0.1 = 0.1 \text{ M}\)

### Step 4: Use the Henderson-Hasselbalch Equation
Since we have a buffer solution, we can use the Henderson-Hasselbalch equation to find the pOH:
\[ \text{pOH} = pK_b + \log\left(\frac{[NH₄^+]}{[NH₃]}\right) \]
Given \(pK_b = 4.74\):
\[ \text{pOH} = 4.74 + \log\left(\frac{0.4}{0.1}\right) \]

### Step 5: Calculate the Logarithm
Calculate the logarithm:
\[ \log\left(\frac{0.4}{0.1}\right) = \log(4) = 0.6 \]
Thus,
\[ \text{pOH} = 4.74 + 0.6 = 5.34 \]

### Step 6: Calculate the pH
Now, we can find the pH using the relationship:
\[ \text{pH} + \text{pOH} = 14 \]
So,
\[ \text{pH} = 14 - \text{pOH} = 14 - 5.34 = 8.66 \]

### Final Answer
The pH of the solution is **8.66**.

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