Question

A $1$ liter of an ideal gas at ${27^ \circ }C$ is heated at constant pressure to ${297^ \circ }C$ , Its final volume will be:
(A) $11$ litres
(B) $1.1\,$ litres
(C) $3.4$ litres
(D) $1.9$ litres

Answer 1

Hint:From the constant pressure ideal gas equation, substitute the known values of the volume and the temperature to obtain the final volume of the ideal gas. The volume must be substituted in cubic centimeters and the temperature must be substituted in kelvin.

Useful formula:
The equation of the ideal gas at the constant pressure is given by
$\dfrac{{{V_1}}}{{{V_2}}} = \dfrac{{{T_1}}}{{{T_2}}}$
Where ${V_1}$ is the initial volume of the ideal gas, ${V_2}$ is the final volume of the ideal gas, ${T_1}$ is the initial temperature of the ideal gas and ${T_2}$ is the final temperature of the ideal gas.

Complete step by step solution:
It is given that the
Initial volume of the ideal gas, ${V_1} = 1\,l = 1000\,cc$
Initial temperature of the ideal gas, ${T_1} = {27^ \circ }\,C = 300\,K$
Final temperature of the ideal gas, ${T_2} = {297^ \circ }\,C = 570\,K$
By using the equation of the ideal gas at the constant pressure, we get
$\dfrac{{{V_1}}}{{{V_2}}} = \dfrac{{{T_1}}}{{{T_2}}}$
Substituting the initial and the final temperature and the initial volume of the ideal gas to obtain the final volume.
$\dfrac{{1000}}{{{V_2}}} = \dfrac{{300}}{{570}}$
By grouping the known terms in one side and the unknown term in one side of the equation, we get
${V_2} = \dfrac{{1000 \times 570}}{{300}}$
BY performing various arithmetic operations, we get
${V_2} = 1900\,cc$
By converting the cubic centimeter into the litre, we get
${V_2} = \dfrac{{1900}}{{1000}} = 1.9\,l$
Hence the final volume of the ideal gas is obtained as $1.9$ litre.

Thus the option (D) is correct.

Note:The ideal gas equation is $PV = mRT$ , the constant pressure equation is formed from it. For converting the litre into the cubic centimeter, multiply the volume with the thousand and for converting the Celsius into the Kelvin, add the temperature with the number $273$ .