If 1 litre of N2 at `27^(@)C` and 760 mm Hg contains N molecules, 4 litres of `O_(2)` under the same conditions of temperature and pressure, shall contain

Options

A. N molecules
B. 2N molecules
C. `(N)/(4)` molecules
D. 4N molecules

Explanation

To solve the problem, we will use the ideal gas law and the relationship between volume and the number of molecules at constant temperature and pressure.

### Step-by-Step Solution:

1. **Understand the Given Information:**
- We have 1 liter of nitrogen gas (N₂) at 27°C and 760 mm Hg, which contains N molecules.
- We need to find out how many molecules are present in 4 liters of oxygen gas (O₂) under the same conditions of temperature and pressure.

2. **Use the Ideal Gas Law:**
- The ideal gas law states that at constant temperature and pressure, the volume of a gas is directly proportional to the number of moles (or molecules) of that gas.
- Mathematically, this can be expressed as:
\[ V \propto n \]
- Where \( V \) is the volume and \( n \) is the number of moles (or molecules).

3. **Set Up the Proportionality:**
- Since the conditions (temperature and pressure) are constant, we can set up the following relationship:
\[ \frac{V_1}{V_2} = \frac{n_1}{n_2} \]
- Here, \( V_1 = 1 \) liter (for N₂), \( V_2 = 4 \) liters (for O₂), \( n_1 = N \) (for N₂), and \( n_2 \) is what we need to find (for O₂).

4. **Substitute the Values:**
- Substitute the known values into the equation:
\[ \frac{1 \text{ L}}{4 \text{ L}} = \frac{N}{n_2} \]

5. **Cross-Multiply to Solve for \( n_2 \):**
- Cross-multiplying gives:
\[ n_2 = 4N \]

6. **Conclusion:**
- Therefore, 4 liters of O₂ under the same conditions of temperature and pressure will contain \( 4N \) molecules.

### Final Answer:
4 liters of O₂ will contain \( 4N \) molecules.

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