`34.05 mL` of phosphorus vapours weighs `0.0625 g` at `546^(@)C` and `0.1` bar pressure. What is the molar mass of phossphorus ?

Options

A. `124 . 77 g mol^(-1)`
B. `1247 . 74 g mol ^(-1)`
C. `12 . 47 g mol^(-1)`
D. `30 g mol^(-1)`

Explanation

To find the molar mass of phosphorus vapors given the volume, mass, temperature, and pressure, we can use the Ideal Gas Law equation, which is:

\[ PV = nRT \]

Where:
- \( P \) = pressure (in bar)
- \( V \) = volume (in liters)
- \( n \) = number of moles
- \( R \) = ideal gas constant
- \( T \) = temperature (in Kelvin)

### Step-by-Step Solution:

1. **Identify the Given Values:**
- Volume \( V = 34.05 \, \text{mL} \)
- Mass \( m = 0.0625 \, \text{g} \)
- Temperature \( T = 546 \, ^\circ C \)
- Pressure \( P = 0.1 \, \text{bar} \)

2. **Convert Volume from mL to Liters:**
\[ V = \frac{34.05 \, \text{mL}}{1000} = 0.03405 \, \text{L} \]

3. **Convert Temperature from Celsius to Kelvin:**
\[ T = 546 + 273 = 819 \, \text{K} \]

4. **Use the Ideal Gas Constant \( R \):**
- The value of \( R \) in appropriate units is \( 0.0821 \, \text{L} \cdot \text{bar} \cdot \text{K}^{-1} \cdot \text{mol}^{-1} \).

5. **Rearrange the Ideal Gas Law to Find Molar Mass:**
The number of moles \( n \) can be expressed as:
\[ n = \frac{m}{M} \]
Substituting this into the Ideal Gas Law gives:
\[ PV = \frac{m}{M}RT \]
Rearranging to find molar mass \( M \):
\[ M = \frac{mRT}{PV} \]

6. **Substitute the Values into the Molar Mass Formula:**
\[ M = \frac{0.0625 \, \text{g} \times 0.0821 \, \text{L} \cdot \text{bar} \cdot \text{K}^{-1} \cdot \text{mol}^{-1} \times 819 \, \text{K}}{0.1 \, \text{bar} \times 0.03405 \, \text{L}} \]

7. **Calculate the Molar Mass:**
- First, calculate the numerator:
\[ 0.0625 \times 0.0821 \times 819 = 4.1835 \, \text{g} \cdot \text{L} \cdot \text{bar} \cdot \text{K}^{-1} \cdot \text{mol}^{-1} \]
- Then calculate the denominator:
\[ 0.1 \times 0.03405 = 0.003405 \, \text{bar} \cdot \text{L} \]
- Now, divide the two results:
\[ M = \frac{4.1835}{0.003405} \approx 1228.44 \, \text{g/mol} \]

8. **Final Result:**
The molar mass of phosphorus is approximately \( 1228.44 \, \text{g/mol} \).

### Conclusion:
Since the calculated value does not exactly match any of the options, we can conclude that the closest option is likely the correct answer.