The capacity of a vessel is 3 litres. It contains 6 gm oxygen, 8 gm nitrogen and 5 gm CO₂ mixture at 27°C. If R = 8.31 J/mole x kelvin, then the pressure in the vessel in 2 N /m² will be (approx.)

The correct option is 5x10⁵.

To find the pressure in the vessel, we can use Dalton's Law of Partial Pressures or calculate the total number of moles of the gaseous mixture and apply the ideal gas equation.

Step 1: Calculate the number of moles of each gas component
The number of moles of a gas is given by the formula:
$$n=\frac{\text{Mass of gas (m)}}{\text{Molar mass of gas (M)}}$$

Using the molar masses: Oxygen (O₂) = 32 g/mol, Nitrogen (N₂) = 28 g/mol, and Carbon Dioxide (CO₂) = 44 g/mol:
For Oxygen:
$$n_{O_2}=\frac{6\text{g}}{32\text{g/mol}}$$
For Nitrogen:
$$n_{N_2}=\frac{8\text{g}}{28\text{g/mol}}$$
For Carbon Dioxide:
$$n_{C{O}_2}=\frac{5\text{g}}{44\text{g/mol}}$$

Step 2: Calculate the total number of moles in the mixture
$$n_{\text{total}}=n_{O_2}+n_{N_2}+n_{C{O}_2}$$
$$n_{\text{total}}=0.1875+0.2857+0.1136\approx 0.5868\text{mol}$$

Step 3: Convert variables to SI units
Volume ($V$) = 3 litres = $3\times {10}^{-3}{\text{m}}^3$
Temperature ($T$) = 27°C = 27 + 273.15 = 300.15 K (approximately 300 K)
Universal gas constant ($R$) = 8.31 J/(mol·K)

Step 4: Use the ideal gas equation to find the total pressure
The ideal gas law is:
$$PV=nRT$$
Rearranging for pressure ($P$):
$$P=\frac{n_{\text{total}}RT}{V}$$
Substitute the values into the equation:
$$P=\frac{0.5868\times 8.31\times 300}{3\times {10}^{-3}}$$
$$P=0.5868\times 8.31\times {10}^5$$
$$P\approx 4.88\times {10}^5{\text{N/m}}^2$$

Rounding to the nearest option, we get:
$$P\approx 5\times {10}^5{\text{N/m}}^2$$