Question

A gas is found to have formula ${{\text{ }\!\![\!\!\text{ CO }\!\!]\!\!\text{ }}_{\text{x}}}$. If its vapour density is 70, the value of $x$is [DCE $2004$]
A.$2.5$
B.$3.0$
C.$5.0$
D.$6.0$

Answer 1

Hint: Vapour density is defined by the mass of a certain volume of a gas divided by the mass of the same volume of hydrogen. Also, the molecular mass of a gas can be expressed two times the vapour density of that particular gas. By using this relation we can calculate the value of $x$.

Formula Used:The molecular mass of a gas is related to the vapour density of the gas by the following relation:
The molecular mass of a gas $=2\times D$
$D=$Vapour density of the gas

Complete step by step solution:Simply vapour density can be expressed as the mass of a certain volume of a gas divided by the mass of the same volume of hydrogen.
Vapour density,$D$$=\dfrac{Mass\,of\,a\,\,given\,\,volume\,\,of\,\,a\,\,vapour\,\,at\,\,STP}{Mass\,\,of\,\,the\,\,same\,\,volume\,\,of\,\,hydrogen}$
According to Avogadro’s law, equal volumes of all gases or vapours contain an equal number of molecules.
Now let there are the number of molecules in a given volume$=1$
Thus, $D\,\,(\,STP)=\dfrac{Mass\,\,of\,\,1\,\,molecule\,\,of\,\,a\,\,vapour\,\,at\,\,STP}{Mass\,\,of\,\,1\,molecule\,\,of\,\,hydrogen}$
As we know Hydrogen is a diatomic molecule, then we can write
$D\,\,(\,STP)=\dfrac{Mass\,\,of\,\,1\,\,molecule\,\,of\,\,a\,\,vapour\,\,at\,\,STP}{2\times Mass\,\,of\,\,1\,atom\,\,of\,\,hydrogen}$
Or,$2\times D\,\,=\dfrac{Mass\,\,of\,\,1\,\,molecule\,\,of\,\,a\,\,vapour\,\,at\,\,STP}{Mass\,\,of\,\,1\,atom\,\,of\,\,hydrogen}$
Or,$2\times D\,\,=\dfrac{Molecular\,\,mass\,\,at\,\,STP}{\,\,1\,}$
Or, Molecular mass at $STP\,$$=2\times D$
Now the given vapour density,$D\,\,=\,\,70$
$\therefore $Molecular mass at STP$=2\times 70=140$
Also, the formula of the given gas is ${{(CO)}_{x}}$
The molecular mass of ${{(CO)}_{x}}$$=x$(Atomic weight of $C+$Atomic weight of $O$)$x(12+16)=28x$
$\therefore 28x=140$
Or,$x=\dfrac{140}{28}=5.0$
Hence the value $x$is $5.0$

Thus, option (C) is correct.

Note: To deal with this type of problem we should remember the formula of vapour density and also the atomic weight of the elements from the periodic table (at least Hydrogen to calcium) as sometimes the atomic weight of the elements is not provided in the questions. Also, remember that vapour density is a unitless quantity.