Question

An LPG cylinder weighs $14.8kg$ when empty. When full it weighs $29.0kg$ and after consumption, the weight of the full cylinder reduces to $23.2kg$. The volume of the gas consumed in cubic metres at the normal usage conditions is:
[Assume LPG to be $n -$butane with a normal boiling point of ${0^o}C$ .]
A. $V = 2.642{m^3}$
B. $V = 2.956{m^3}$
C. $V = 2.115{m^3}$
D. $V = 2.447{m^3}$

Answer 1

The ideal gas equation is the equation of state of a hypothetical ideal gas which follows all the gas laws at standard conditions of temperature, pressure and volume along with the number of moles of the gas. The inter-relation of Boyle’s law, Charle’s law, Gay Lussac’s law and Avogadro law gives an overall generalized ideal as equation.

Complete step by step answer:

According to the ideal gas equation, we know that:
$PV = nRT$
Where, $P =$ pressure of the ideal gas
$V =$ volume of the ideal gas
$n =$ number of moles of the gas
$R =$ universal gas constant$= 0.0821L - atm{K^{ - 1}}mo{l^{ - 1}}$
$T =$ temperature of the gas
As per the question, weight of the empty cylinder =$14.8kg$
Weight of the fully filled cylinder =$29.0kg$
Weight of the cylinder after the gas has been consumed = $23.2kg$
The amount of gas consumed = Weight of the fully filled cylinder – Weight of the cylinder after gas consumption
So, the amount of gas consumed = $(29.0 - 23.2)kg = 5.8kg = 5800g$
The molecular mass of $n -$butane (${C_4}{H_{10}}$ ) = $58g$
The pressure at standard conditions = $1atm$
The room temperature =$298K$
Applying the ideal gas equation, we have:
$PV = nRT$
Here, number of moles = $n = \dfrac{w}{{{M_w}}} = \dfrac{{5800g}}{{58g}}$
Substituting the values in the ideal gas equation, we have:
$1 \times V = \dfrac{{5800}}{{58}} \times 0.0821 \times 298$
On solving, we get:
$V = 2447L = 2.447{m^3}$
Thus, the correct option is D. $V = 2.447{m^3}$.

Note:
The ideal gas equation is a combination of Boyle’s law, Charle’s law, Gay Lussac’s law and Avogadro law. The various gas laws are explained as:
(i) Boyle’s law: At a given temperature, the pressure of a gas is inversely proportional to the volume of the gas.
(ii) Charle’s law: The volume of a gas at constant temperature is directly proportional to the temperature of the gas.
(iii) Gay-Lussac’s law: The pressure of a given mass of gas is directly proportional to the absolute temperature of the gas, when the volume is kept constant.
(iv) Avogadro law: Equal volumes of all gases, at the same temperature and pressure, have the same number of molecules.