Question

The density of the vapour of a substance at $1$ atm pressure and $500K$ is $0.36\;kg\;{m^{ - 3}}$. The vapour effuses through a small hole at a rate times faster than oxygen under the same condition.
Determine
(i)mol.wt
(ii)molar volume
(iii)compression factor(z) of the vapour
(iv)which forces among the gas molecules are dominating, the attractive or repulsive

Answer 1

The rate of diffusion is inversely proportional to the square root of the molecular mass for the diffusion through the orifice. Also, the dominant forces among the gases are related to the compressibility factor named Z. This factor describes the deviation of a gas from the ideal gas behaviour is termed as compressibility factor.

Complete step-by-step answer: We have been given the density of substance at one atmospheric pressure as $0.36\;kg\;{m^{ - 3}}$
We know that rate of effusion of vapour by the rate of effusion of oxygen is equal to the square root of the molecular weight of oxygen divided by the molecular weight of vapour following the graham’s law.
Then according to the question,
$\dfrac{{{r_1}}}{{{r_{{o_2}}}}} = \sqrt {\dfrac{{{M_{{O_2}}}}}{{{M_1}}}}$
Given that $\dfrac{{{r_1}}}{{{r_{{o_2}}}}} = 1.33$
${M_{{O_2}}} = 32$
$\therefore 1.33 = \sqrt {\dfrac{{32}}{{{M_1}}}}$
Thus the molecular weight of the substance,
$1.33 = \sqrt {\dfrac{{32}}{{{M_1}}}}$
$\Rightarrow \dfrac{{32}}{{{M_1}}} = {(1.33)^2}$
$\Rightarrow {M_1} = \dfrac{{32}}{{1.33 \times 1.33}} = 18.09$
Now molecular volume is the volume occupied by one mole of a substance
$\Rightarrow M.V = \dfrac{{18.09 \times {{10}^{ - 3}}}}{{0.36}} = 50.25 \times {10^{ - 3}}$
Now compressibility factor is equal to the product of pressure and volume divided by RT
$z = \dfrac{{PV}}{{RT}}$
Where z is the compressibility factor
V is the volume
R is the gas constant
T is the temperature
$z = \dfrac{{1.1325 \times {{10}^5} \times 50.25 \times {{10}^{ - 3}}}}{{8.314 \times 500}} = 1.224$
$\Rightarrow z = 1.224$
Here the compressibility factor is greater than one so repulsive forces dominate.

Additional information: The deviation of real gas from ideal gas behaviour is explained in the terms of the compressibility factor, z

Note: When the compressibility factor is less than one, then the dominant forces are attractive and the gas is more expandable. Also when the compressibility factor is greater than one the repulsive forces are more dominant and it is more compressible. The compressibility factor for an ideal is however one.