Question

If a given mass of gas occupies a volume of \[10{\rm{ cc}}\] at \[1\] atmospheric pressure and temperature \[100^\circ {\rm{ C}}\]. What will be its volume at \[4\] atmospheric pressure, the temperature being the same?
A. \[100{\rm{ cc}}\]
B. \[400{\rm{ cc}}\]
C. \[1.04{\rm{ cc}}\]
D. \[2.5{\rm{ cc}}\]

Answer 1

Hint: From the concept of an ideal gas undergoing a thermodynamic process, we can say that the ratio of the product of pressure and volume to temperature is constant. We will rewrite this concept between the initial and final state of the gas.

Complete step by step answer:
Given:
The initial volume of the gas is \[{V_1} = 10{\rm{ cc}}\].
The initial pressure of the gas is \[{P_1} = 1{\rm{ atm}}\].
The final pressure of the gas is \[{P_2} = 4{\rm{ atm}}\].
We have to find the gas's final volume: the volume of the gas when its pressure is increased from \[1\] atmospheric pressure to \[4\] atmospheric pressure.
Write the relation between temperature, pressure, and volume of the gas's initial state and final state.
\[\dfrac{{{P_1}{V_1}}}{{{T_1}}} = \dfrac{{{P_2}{V_2}}}{{{T_2}}}\]……(1)
Here \[{T_1}\] is the initial temperature, \[{T_2}\]is the final temperature, and \[{V_2}\]is the gas's final volume.
It is given that the initial temperature and final temperature of the gas are the same, which is equal to \[100^\circ {\rm{ C}}\] that means the given process is isothermal.

Substitute \[1{\rm{ atm}}\] for \[{P_1}\], \[4{\rm{ atm}}\] for \[{P_2}\], \[10{\rm{ cc}}\] for \[{V_1}\] and \[100^\circ {\rm{ C}}\] for \[{T_1}\]& \[{T_2}\] in equation (1).
\[\begin{array}{l}
\dfrac{{\left( {1{\rm{ atm}}} \right)\left( {10{\rm{ cc}}} \right)}}{{100^\circ {\rm{ C}}}} = \dfrac{{\left( {{\rm{4 atm}}} \right){V_2}}}{{100^\circ {\rm{ C}}}}\\
\Rightarrow {V_2} = 2.5{\rm{ cc}}
\end{array}\]
Therefore, based on the above calculation, if a gas's pressure having an initial volume \[10{\rm{ cc}}\] increases from \[1\] atmospheric pressure to \[4\] atmospheric pressure, its final volume will be equal to \[2.5{\rm{ cc}}\] provided temperature is kept constant throughout the process

So, the correct answer is “Option D”.

Note:
There is no need to convert units of pressure and volume into their base units because when we substitute their values equation (1), they will cancel out each other. We will only leave with the final volume of the gas in a cubic centimeter.