A buffer solution is prepared in which the concentration of `NH_(3)` is `0.30 M` and the concentration of `NH_(4)^(+)` is `0.20 M`. If the equilibrium constant, `K_(b)` for `NH_(3)` equals `1.8xx10^(-5)`, what is the `pH` of this solution? (`log 2.7=0.43`)

Options

A. `9.43`
B. `11.72`
C. `8.73`
D. `9.08`

Explanation

To find the pH of the buffer solution containing ammonia (NH₃) and ammonium ion (NH₄⁺), we can use the Henderson-Hasselbalch equation for basic buffers:

\[ \text{pH} = 14 - \text{pK}_b + \log\left(\frac{[\text{Salt}]}{[\text{Base}]}\right) \]

### Step 1: Calculate pK_b
Given the equilibrium constant \( K_b \) for ammonia (NH₃) is \( 1.8 \times 10^{-5} \), we first calculate \( pK_b \):

\[ pK_b = -\log(K_b) = -\log(1.8 \times 10^{-5}) \]

Using the logarithmic property:

\[ pK_b = -\log(1.8) - \log(10^{-5}) = -\log(1.8) + 5 \]

From the problem, we have \( \log(2.7) = 0.43 \). Since \( 1.8 \) is approximately \( 2.7/1.5 \), we can estimate:

\[ \log(1.8) \approx \log(2.7) - \log(1.5) \approx 0.43 - 0.176 = 0.254 \]

Thus,

\[ pK_b \approx 5 - 0.254 = 4.746 \approx 4.76 \]

### Step 2: Use the Henderson-Hasselbalch equation
Now we can substitute the values into the Henderson-Hasselbalch equation. The concentrations given are:

- \([NH₃] = 0.30 \, M\)
- \([NH₄^+] = 0.20 \, M\)

Substituting these values into the equation:

\[ \text{pH} = 14 - 4.76 + \log\left(\frac{0.20}{0.30}\right) \]

### Step 3: Calculate the log ratio
Now we calculate the log ratio:

\[ \frac{0.20}{0.30} = \frac{2}{3} \]

Using logarithmic properties:

\[ \log\left(\frac{2}{3}\right) = \log(2) - \log(3) \]

From standard logarithm values:

- \( \log(2) \approx 0.3010 \)
- \( \log(3) \approx 0.4771 \)

Thus,

\[ \log\left(\frac{2}{3}\right) \approx 0.3010 - 0.4771 = -0.1761 \]

### Step 4: Final calculation of pH
Now substituting back into the pH equation:

\[ \text{pH} = 14 - 4.76 - 0.1761 \approx 14 - 4.76 + 0.1761 = 9.43 \]

### Conclusion
Therefore, the pH of the buffer solution is approximately:

\[ \text{pH} \approx 9.43 \]