Question

A gas is filled in a cylinder, its temperature is increased by 20% on kelvin scale and volume is reduced by 90%. How much percentage of the gas will leak for pressure to remain constant?
(A) 20%
(B) 25%
(C) 30%
(D) 40%

Answer 1

Hint: The ideal gas law relating the thermodynamic state properties of the gas such as pressure, temperature, volume and the gas constant needs to be used to compute the conditions before and after the state change.

Complete step by step solution:
Using ideal gas constant, we have,
\[PV = nRT\]
\[{P_1}{V_1} = {n_1}R{T_1}\]and \[{P_2}{V_2} = {n_2}R{T_2}\]
\[\dfrac{{{P_1}{V_1}}}{{{P_2}{V_2}}} = \dfrac{{{n_1}R{T_1}}}{{{n_2}R{T_2}}}\]
We have been given that the pressure should be constant.
\[{P_1} = {P_2}\]
Also, it is given that the volume is reduced by 90% and temperature is increased by 20%.
\[{V_2} = 0.9{V_1}\]and \[{T_2} = 1.2{T_1}\]
\[\dfrac{1}{{0.9}} = \dfrac{{{n_1}}}{{{n_2}}} \times \dfrac{1}{{1.2}}\]
\[{n_2} = \dfrac{3}{{4{n_1}}}\]
Proportion of mass with respect to the initial mass to be leaked out will be,
=\[\dfrac{{({n_2} - {n_1})}}{{{n_1}}} = (0.75 - 1) = 0.25\]
=\[25\% \]

The correct option is B.

Note: The ideal gas law typically can be defined based on the number of moles of the gas or the mass of the gas. When using the gas law on the basis of the number of moles, the gas constant would be the universal gas constant given by 8.314 kJ/mol K. However, when using based on the mass, then we will be using the specific gas constant. In this scenario, since we deal with the same gas, we can use either the ideal gas law based on the mass or the number of moles as the molecular weights are the same before and after the process. But when dealing with the conditions where additional gas is involved, we have to be careful about the choice.