PV = nRT

$$ \Rightarrow $$ PV = $${W \over M}RT$$

$$ \Rightarrow $$ $$P = {W \over M} \times {{RT} \over V}$$

$$ \Rightarrow $$$$P = {{dRT} \over M}$$   [Density = $${{Mass} \over {Volume}}$$]

$$ \Rightarrow d = {{PM} \over {RT}} = {{5 \times 28} \over {0.0821 \times 500}} = 3.41\,g/ml$$

C_p for monoatomic gas mixture of same volume = $${5 \over 2}R$$

$$ \therefore $$ C_V = $${3 \over 2}R$$

$$ \Rightarrow {{{C_P}} \over {{C_V}}} = {{{5 \over 2}R} \over {{3 \over 2}R}} = {5 \over 3} = 1.67$$

Van der Waal gas constant '$$a$$' represent intermolecular force of attraction of gaseous molecules and Van der Waal gas constant 'b' represent effective size of molecules . Therefore order should be

(I) H₂ < He < O₂ < CO₂   (II) CH₄ > O₂ > H₂

According to Graham's law of diffusion

$$r \propto {1 \over {\sqrt d }} \propto {1 \over {\sqrt M }}$$

$$ \Rightarrow {{{r_1}} \over {{r_2}}} = \sqrt {{{{M_2}} \over {{M_1}}}} $$

Rate of diffusion = $${{Volume\,of\,gas\,diffused\,(V)} \over {Times\,taken\,(t)}}$$

$$ \therefore $$ $${{{V_1}/{t_1}} \over {{V_2}/{t_2}}} = \sqrt {{{{M_2}} \over {{M_1}}}} $$

If same volume of two gases diffuse then V₁ = V₂

$$ \Rightarrow $$ $${{{t_1}} \over {{t_2}}} = \sqrt {{{{M_2}} \over {{M_1}}}} $$

Here t₂ = 3t, M₁ = 4 u, M₂ = ?

$$ \therefore {{3{t_1}} \over {{t_1}}} = \sqrt {{{{M_2}} \over 4}} \Rightarrow 3 = \sqrt {{{{M_2}} \over 4}} $$

$$ \Rightarrow 9 = {{{M_2}} \over 4} \Rightarrow {M_2} = 36\,u$$