Question

The rates of diffusion of two gases A and B are in the ratio 1 : 4. If the ratio of their masses present in the mixture is 2 : 3. Find the ratio of their mole fraction.(${{9}^{1/3}}\text{ = 2}\text{.08}$)

Answer 1

Try to recall Graham's law of diffusion. We need to apply graham's law in order to find out the molecular mass of the hydrocarbon.
Formula: $\dfrac{{{r}_{A}}}{{{r}_{B}}}=\dfrac{\sqrt{{{M}_{B}}}}{\sqrt{{{M}_{A}}}}$
Where,
${{r}_{A}}$ is the rate of diffusion of first gas,
${{r}_{B}}$ is the rate of diffusion of second gas,
${{M}_{A}}$ is the molar mass of gas A,
${{M}_{B}}$ is the molar mass of gas B.

Complete step-by-step answer:
We will try to understand Graham's law of diffusion and then apply the formula to find the answer.
- Graham's law states that the rate of diffusion of a gas is inversely proportional to the square root of its molecular weight.
$r\propto \dfrac{1}{M}$
- The complete explanation of the law was given by the kinetic theory of gases a few years later. Graham's law is used in isolating isotopes of an element as they have different rates of diffusion.
- Rate of diffusion (r) =$\dfrac{volume\\_diffused}{time\\_taken}$
We will now calculate the mole fraction of gases A and B.
Let the total mass of mixture be m. Then,
Mass of A = $\dfrac{\text{2m}}{5}$
Mass of B = $\dfrac{\text{3m}}{5}$
According to Graham's law of diffusion,
$\dfrac{{{r}_{A}}}{{{r}_{B}}}=\dfrac{\sqrt{{{M}_{B}}}}{\sqrt{{{M}_{A}}}}$
$\dfrac{1}{4}=\dfrac{\sqrt{{{M}_{B}}}}{\sqrt{{{M}_{A}}}}$
$\dfrac{1}{16}=\dfrac{{{M}_{B}}}{{{M}_{A}}}$
${{M}_{A}}\text{ = 16}{{M}_{B}}$
Here, mole fraction of A becomes = $\dfrac{\dfrac{\text{2m}}{\text{5}{{\text{M}}_{\text{A}}}}}{\dfrac{\text{2m}}{\text{5}{{\text{M}}_{\text{A}}}}\text{+}\dfrac{\text{3m}}{\text{5}{{\text{M}}_{\text{B}}}}}$
= $\dfrac{\dfrac{\text{2m}}{\text{5}{{\text{M}}_{\text{A}}}}}{\dfrac{\text{2m}}{\text{5}{{\text{M}}_{\text{A}}}}\text{+}\dfrac{\text{3m}}{\text{5}\dfrac{{{\text{M}}_{A}}}{16}}}$
= 0.04
Mole fraction of B = 1 - mole fraction of A = 1-0.04 = 0.96

The ratio of mole fraction of A and mole fraction of B is = $\dfrac{0.96}{0.04}\text{ = 24}$.

Note: The formula for calculating the rate of diffusion i.e. Graham's law of diffusion is used when the diffusion of gases happens at the same conditions like temperature, pressure etc. Remember that the rate of diffusion is inversely proportional to molecular mass to avoid errors while calculating the ratio of rate of diffusions of two gases.