Answer:
The molar mass of the unknown gas is [tex]\mathbf{ 51.865 \ g/mol}[/tex]
Explanation:
Let assume that the gas is O2 gas
O2 gas is to effuse through a porous barrier in time t₁ = 4.98 minutes.
Under the same conditions;
the same number of moles of an unknown gas requires time t₂ = 6.34 minutes to effuse through the same barrier.
From Graham's Law of Diffusion;
Graham's Law of Diffusion states that, at a constant temperature and pressure; the rate of diffusion of a gas is inversely proportional to the square root of its density.
i.e
[tex]R \ \alpha \ \dfrac{1}{\sqrt{d}}[/tex]
[tex]R = \dfrac{k}{d}[/tex] where K = constant
If we compare the rate o diffusion of two gases;
[tex]\dfrac{R_1}{R_2}= {\sqrt{\dfrac{d_2}{d_1}}[/tex]
Since the density of a gas d is proportional to its relative molecular mass M. Then;
[tex]\dfrac{R_1}{R_2}= {\sqrt{\dfrac{M_2}{M_1}}[/tex]
Rate is the reciprocal of time ; i.e
[tex]R = \dfrac{1}{t}[/tex]
Thus; replacing the value of R into the above previous equation;we have:
[tex]\dfrac{R_1}{R_2}={\dfrac{t_2}{t_1}}[/tex]
We can equally say:
[tex]{\dfrac{t_2}{t_1}}= {\sqrt{\dfrac{M_2}{M_1}}[/tex]
[tex]{\dfrac{6.34}{4.98}}= {\sqrt{\dfrac{M_2}{32}}[/tex]
[tex]M_2 = 32 \times ( \dfrac{6.34}{4.98})^2[/tex]
[tex]M_2 = 32 \times ( 1.273092369)^2[/tex]
[tex]M_2 = 32 \times 1.62076418[/tex]
[tex]\mathbf{M_2 = 51.865 \ g/mol}[/tex]
