A mixture of Nitrogen, Oxygen and Helium contains 0.25, 0.15 and 0.4 mole of these

Explanation

To solve this problem, we need to use the concept of partial pressures and Dalton's Law of Partial Pressures. The law states that in a mixture of non-reacting gases, the total pressure exerted by the mixture is equal to the sum of the partial pressures exerted by each individual gas. Mathematically, it can be represented as:

\(P_{total} = P_{N_2} + P_{O_2} + P_{He}\)

From the question, we know the moles of each gas and the pressure contribution due to oxygen, which is 2.5 atm. We can use the mole fraction of each gas to find the corresponding partial pressures. The mole fraction of a gas is the ratio of the moles of that gas to the total moles of all gases in the mixture. So, first, we need to find the total moles:

\(n_{total} = n_{N_2} + n_{O_2} + n_{He} = 0.25 + 0.15 + 0.4 = 0.8\)

Now we can find the mole fraction of helium:

\(\chi_{He} = \frac{n_{He}}{n_{total}} = \frac{0.4}{0.8} = 0.5\)

We also know the mole fraction of oxygen:

\(\chi_{O_2} = \frac{n_{O_2}}{n_{total}} = \frac{0.15}{0.8} = 0.1875\)

According to Dalton's Law, the ratio of partial pressures is equal to the ratio of mole fractions:

\(\frac{P_{He}}{P_{O_2}} = \frac{\chi_{He}}{\chi_{O_2}}\)

Plugging in the values, we get:

\(\frac{P_{He}}{2.5} = \frac{0.5}{0.1875}\)

Now, we can solve for the partial pressure of helium:

\(P_{He} = 2.5 \times \frac{0.5}{0.1875} = 6.67 \,\text{atm}\)

So, the partial pressure of helium in the mixture is 6.67 atm, which corresponds to Option D.