Question

One mole of ${N_2}$ gas at $0.8atm$ takes 38 seconds to diffuse through a pinhole , whereas one mole of an unknown compound of Xenon with $F$ at $1.6atm$ takes 57 seconds to diffuse through the same hole . Calculate the molecules formula of the compound .
(Atomic weight $X = 136,F = 19$ )

Answer 1

Hint: The ability of a gas to mix spontaneously and to form a homogenous mixture is known as diffusion . The law which deals with this phenomenon is Graham's law .

Complete answer:
Graham's law of diffusion states that under similar conditions of temperature and pressures , the rates of diffusion or effusion of different gases are inversely proportional to the square root of their densities .
 Therefore , $\dfrac{{{r_1}}}{{{r_2}}} = \sqrt {\dfrac{{{d_1}}}{{{d_2}}}} $
Where , r denotes the rate of diffusion and d denotes the density of the gas .
If two gases are taken at different pressures , then greater the pressure , greater is the number of molecules hitting per unit area , greater is the rate of diffusion . In such cases , Graham's law of diffusion can be written as
$\dfrac{{{t_2}}}{{{t_1}}} = \dfrac{{{P_1}}}{{{P_2}}}\sqrt {\dfrac{{{d_1}}}{{{d_2}}}} = \dfrac{{{P_1}}}{{{P_2}}}\sqrt {\dfrac{{{M_2}}}{{{M_1}}}} $
Our aim is to find the molecular formula of the Xenon gas .
We will use the above formula for calculation
Now , in the question , it is given ${t_1} = 38\sec ,{t_2} = 57\sec $ , ${P_{nitrogen}} = 0.8atm,{P_{gas}} = 1.6atm$
(P denotes pressure )
Also , we know that molecular mass of ${N_2}$ is $28g/mol$ .
On substituting the values in the formula , we get ,
$\dfrac{{57}}{{38}} = \dfrac{{0.8}}{{1.6}}\sqrt {\dfrac{{{M_{gas}}}}{{28}}} $
On solving , we get
${M_{gas}} = 252g$
Let us assume the formula of Xenon gas to be $Xe{F_n}$
Given atomic weight of $X = 136,F = 19$
Therefore , ${M_{gas}} = 136 + (19 \times n) = 252$
$ \Rightarrow n = 6.1 \approx 6$
Therefore the molecular formula of the compound is $Xe{F_6}$ .

Note:
It should be noted that according to Graham's law the time taken by a gas to diffuse is inversely proportional to its Pressure and directly proportional to the square root of their molecular mass .