Question

The density of air at N.T.P. is $1.292\dfrac{gm}{lit}$. If the pressure is tripled keeping its temperature constant its density becomes

& A.~~~~~~~~~~~~~3.87\dfrac{gm}{ltr} \\\ & B.~~~~~~~~~~~~~1.293\dfrac{gm}{ltr} \\\ & C.~~~~~~~~~~~~~2.586\dfrac{gm}{ltr} \\\ & D.~~~~~~~~~~~~~0.431\dfrac{gm}{ltr} \\\ \end{aligned}$$

Answer 1

Hint: If the temperature of gas is kept constant the density of gas is directly proportional to its pressure. The relation between density and pressure for an ideal gas is given by
$d=\dfrac{P{{M}_{o}}}{RT}$, where
${{M}_{o}}$ is molar mass of gas, P is pressure of gas, T is temperature of gas, d is density of gas, and R is universal gas constant.

Formula Used: Ideal gas equation
$PV=nRT$, where P is pressure of gas, T is temperature of gas, V is volume of gas, R is universal gas constant and n is number of moles
Density of gas
$d=\dfrac{P{{M}_{o}}}{RT}$
Moles of gas is equal to
$n=\dfrac{M}{{{M}_{o}}}$

Complete Step By Step Solution:
Density of any substance is mass of substance per unit volume. Therefore
$d=\dfrac{M}{V}$
$M=dV$ … (1)
Where M is mass of gas, V is volume of gas and d is density of gas
We know that for all gases
$n=\dfrac{M}{{{M}_{o}}}$, where
n is moles of gas, ${{M}_{o}}$ is molar mass of gas, and M is mass of gas.
Putting value of M from equation 1
$n=\dfrac{dV}{{{M}_{o}}}$
$\dfrac{n}{V}=\dfrac{d}{{{M}_{o}}}$ … (2)
According to ideal gas equation
$PV=nRT$
$\dfrac{n}{V}=\dfrac{P}{RT}$
Putting value of $\dfrac{n}{V}$ from equation 2
$\dfrac{d}{{{M}_{o}}}=\dfrac{P}{RT}$
$d=\dfrac{P{{M}_{o}}}{RT}$
Now, according to the question the density of gas was $1.292\dfrac{gm}{lit}$ at some pressure. Let us assume that pressure to be$P{}_{o}$.
$1.292=\dfrac{P{}_{o}{{M}_{o}}}{RT}$ … (3)
Now if the pressure is tripled keeping its temperature constant, let density of gas be d
$d=\dfrac{3{{P}_{o}}{{M}_{o}}}{RT}$
Putting value $\dfrac{P{}_{o}{{M}_{o}}}{RT}$ from equation 3
$d=3\times 1.292=3.87\dfrac{gm}{ltr}$
Hence the correct option is A.

Note: It should always be kept in mind that density of gas is proportional to its pressure only when the temperature is kept constant. Also if pressure is kept constant density of gas is inversely proportional to its temperature. Also the formula $d=\dfrac{P{{M}_{o}}}{RT}$ should be remembered for future problems.