The ratio of velocities of diffusion of gases a and b is 1:4. If the ratio of their masses present in the mixture is2:3, calculate the ratio of their Mole fractions.

To solve the problem of finding the ratio of mole fractions of gases A and B given their diffusion velocities and mass ratio, we can use Graham's law of effusion and the concept of mole fraction. Let's break it down step by step.

Understanding the Given Information

We know two key pieces of information:

• The ratio of the velocities of diffusion of gases A and B is 1:4.
• The ratio of their masses in the mixture is 2:3.

Applying Graham's Law

Graham's law states that the rate of diffusion of a gas is inversely proportional to the square root of its molar mass. Mathematically, this can be expressed as:

Rate of diffusion ∝ 1/√M

Where M is the molar mass of the gas. Given the ratio of velocities:

v_a/v_b = 1/4

This implies:

√M_b/√M_a = 1/4

Squaring both sides gives:

M_b/M_a = 1/16

This means that the molar mass of gas B is 16 times that of gas A.

Relating Masses to Molar Masses

Let’s denote the molar mass of gas A as M and that of gas B as 16M. Now, we can express the masses of gases A and B in terms of their molar masses and the number of moles:

• Mass of gas A = n_a * M
• Mass of gas B = n_b * 16M

According to the problem, the mass ratio is given as:

Mass of A / Mass of B = 2 / 3

Substituting the expressions for mass, we have:

(n_a * M) / (n_b * 16M) = 2 / 3

The M cancels out, simplifying to:

n_a / (n_b * 16) = 2 / 3

Cross-multiplying gives:

3n_a = 32n_b

From this, we can express the number of moles of gas A in terms of gas B:

n_a = (32/3)n_b

Calculating Mole Fractions

The mole fraction of a component in a mixture is given by the formula:

X_A = n_a / (n_a + n_b)

X_B = n_b / (n_a + n_b)

Substituting for n_a:

X_A = (32/3)n_b / ((32/3)n_b + n_b)

This simplifies to:

X_A = (32/3)n_b / ((32/3 + 3/3)n_b) = (32/3) / (35/3) = 32/35

For gas B:

X_B = n_b / ((32/3)n_b + n_b) = n_b / ((32/3 + 3/3)n_b) = 1 / (35/3) = 3/35

Final Ratio of Mole Fractions

The ratio of the mole fractions of gases A and B is:

X_A : X_B = (32/35) : (3/35) = 32 : 3

Thus, the ratio of the mole fractions of gases A and B is 32:3. This shows how the diffusion rates and mass ratios influence the composition of the gas mixture in terms of mole fractions.