Two vessels of volumes 16.4 L and 5 L contain two ideal gases of molecular existence at the respective temperature of `27^(@)C and 227^(@)C` and exert 1.5 and 4.1 atmospheres respectively. The ratio of the number of molecules of the former to that of the latter is

Options

A. 2
B. 1
C. `(1)/(2)`
D. `(1)/(3)`

Explanation

To solve the problem, we need to find the ratio of the number of molecules of two ideal gases in different vessels. We can use the Ideal Gas Law, which states that:

\[ PV = nRT \]

Where:
- \( P \) = pressure
- \( V \) = volume
- \( n \) = number of moles
- \( R \) = universal gas constant (which will cancel out)
- \( T \) = temperature in Kelvin

### Step-by-Step Solution:

1. **Convert Temperatures to Kelvin:**
- For the first gas:
\[ T_1 = 27^\circ C = 27 + 273 = 300 \, K \]
- For the second gas:
\[ T_2 = 227^\circ C = 227 + 273 = 500 \, K \]

2. **Identify Given Values:**
- For gas 1:
- Volume \( V_1 = 16.4 \, L \)
- Pressure \( P_1 = 1.5 \, atm \)
- Temperature \( T_1 = 300 \, K \)
- For gas 2:
- Volume \( V_2 = 5 \, L \)
- Pressure \( P_2 = 4.1 \, atm \)
- Temperature \( T_2 = 500 \, K \)

3. **Calculate Number of Moles for Each Gas:**
- For gas 1:
\[ n_1 = \frac{P_1 V_1}{RT_1} = \frac{1.5 \times 16.4}{R \times 300} \]
- For gas 2:
\[ n_2 = \frac{P_2 V_2}{RT_2} = \frac{4.1 \times 5}{R \times 500} \]

4. **Find the Ratio of Moles:**
- The ratio of the number of moles \( \frac{n_1}{n_2} \) is:
\[ \frac{n_1}{n_2} = \frac{P_1 V_1 / (R T_1)}{P_2 V_2 / (R T_2)} = \frac{P_1 V_1 T_2}{P_2 V_2 T_1} \]
- Substitute the values:
\[ \frac{n_1}{n_2} = \frac{1.5 \times 16.4 \times 500}{4.1 \times 5 \times 300} \]

5. **Calculate the Values:**
- Calculate the numerator:
\[ 1.5 \times 16.4 \times 500 = 12300 \]
- Calculate the denominator:
\[ 4.1 \times 5 \times 300 = 6150 \]
- Now calculate the ratio:
\[ \frac{n_1}{n_2} = \frac{12300}{6150} = 2 \]

6. **Find the Ratio of Molecules:**
- Since the number of molecules is directly proportional to the number of moles, the ratio of the number of molecules \( \frac{N_1}{N_2} \) is the same as the ratio of moles:
\[ \frac{N_1}{N_2} = \frac{n_1}{n_2} = 2 \]

### Final Answer:
The ratio of the number of molecules of the former gas to that of the latter gas is \( 2:1 \).