Calculate the number of moles of `NH_(4)Cl` that should be added to one litre of `"1.0 M "NH_(4)OH` to prepare buffer solution with
`pH=9[K_(b)=2xx10^(-5)," take log"2=0.3]`

Options

A. 3.4
B. 2.6
C. 1.5
D. 2

Answer

To solve the problem, we need to calculate the number of moles of \( NH_4Cl \) that should be added to 1 liter of \( 1.0 M \) \( NH_4OH \) to prepare a buffer solution with a pH of 9. We will use the Henderson-Hasselbalch equation to find the required concentration of \( NH_4Cl \).

### Step-by-Step Solution:

1. **Identify the components of the buffer:**
- We have \( NH_4OH \) (ammonium hydroxide), which is a weak base.
- We will add \( NH_4Cl \), which is the salt of the weak base \( NH_4OH \) and a strong acid (HCl). This provides the conjugate acid \( NH_4^+ \).

2. **Write the Henderson-Hasselbalch equation:**
\[ pH = pK_b + \log \left( \frac{[A^-]}{[HA]} \right) \]
Here, \( [A^-] \) is the concentration of the base (\( NH_4OH \)) and \( [HA] \) is the concentration of the conjugate acid (\( NH_4Cl \)).

3. **Convert pH to pOH:**
Since we are given pH, we can find pOH:
\[ pOH = 14 - pH = 14 - 9 = 5 \]

4. **Find \( pK_b \):**
We are given \( K_b = 2 \times 10^{-5} \). We can find \( pK_b \):
\[ pK_b = -\log(K_b) = -\log(2 \times 10^{-5}) = 4.699 \]
(Using the hint that \( \log(2) \approx 0.3 \))

5. **Set up the Henderson-Hasselbalch equation:**
\[ pOH = pK_b + \log \left( \frac{[NH_4OH]}{[NH_4Cl]} \right) \]
Substituting the known values:
\[ 5 = 4.699 + \log \left( \frac{1}{[NH_4Cl]} \right) \]

6. **Rearranging the equation:**
\[ 5 - 4.699 = \log \left( \frac{1}{[NH_4Cl]} \right) \]
\[ 0.301 = \log \left( \frac{1}{[NH_4Cl]} \right) \]

7. **Convert from logarithmic to exponential form:**
\[ \frac{1}{[NH_4Cl]} = 10^{0.301} \approx 2 \]
Therefore,
\[ [NH_4Cl] \approx \frac{1}{2} = 0.5 \, M \]

8. **Calculate the number of moles of \( NH_4Cl \):**
Since we have a volume of 1 liter:
\[ \text{Number of moles of } NH_4Cl = [NH_4Cl] \times \text{Volume} = 2 \, \text{mol/L} \times 1 \, \text{L} = 2 \, \text{moles} \]

### Final Answer:
The number of moles of \( NH_4Cl \) that should be added is **2 moles**.