Question

A gaseous system has a pressure of 580 ${\text{c}}{{\text{m}}^{\text{3}}}$ at a certain pressure. If its pressure is increased by ${\text{0}}{\text{.96 atm}}$, its volume becomes 100${\text{c}}{{\text{m}}^{\text{3}}}$. Determine the pressure of the system.

Answer 1

Hint: This problem is based on Boyle's Law of gas, which states that the pressure of a fixed mass of a gas is inversely proportional to the volume of the gas at constant temperature. We can substitute the given values in the equation to calculate the pressure.

Formula Used:
 $\dfrac{{{{\text{P}}_{\text{1}}}}}{{{{\text{P}}_{\text{2}}}}}{\text{ = }}\dfrac{{{{\text{V}}_2}}}{{{{\text{V}}_1}}}$

Complete step by step answer:
As per the question, the initial pressure of the gas, ${{\text{P}}_{\text{1}}}$= P. the initial volume of the gas,${{\text{V}}_{\text{1}}}$ = 580 ${\text{c}}{{\text{m}}^{\text{3}}}$. The pressure increased to${{\text{P}}_2}$= $\left( {{\text{0}}{\text{.96 + P}}} \right)$ and for that the volume changes to ${{\text{V}}_2}$ = 100 ${\text{c}}{{\text{m}}^{\text{3}}}$.
The mathematical expression of the Boyle’s law = \[{{\text{P}}_{\text{1}}} \times {{\text{V}}_{\text{1}}}{\text{ = }}{{\text{P}}_{\text{2}}} \times {{\text{V}}_{\text{2}}}\]
If this equation is rearranged then we get that, $\dfrac{{{{\text{P}}_{\text{1}}}}}{{{{\text{P}}_{\text{2}}}}}{\text{ = }}\dfrac{{{{\text{V}}_2}}}{{{{\text{V}}_1}}}$
Putting the values of all the parameters we get,
$\dfrac{{\text{P}}}{{\left( {{\text{0}}{\text{.96 + P}}} \right)}}{\text{ = }}\dfrac{{100}}{{580}}$
$ \Rightarrow \dfrac{{\text{P}}}{{\left( {{\text{0}}{\text{.96 + P}}} \right)}}{\text{ = }}\dfrac{1}{{5.8}}$
Rearranging:
\[ \Rightarrow {\text{5}}{\text{.8 P = }}\left( {{\text{0}}{\text{.96 + P}}} \right)\]
\[ \Rightarrow {\text{ P = }}\dfrac{{{\text{0}}{\text{.96}}}}{{4.{\text{8}}}} = 0.2\]
Hence, the new pressure of the system initially was equal to $0.2$${\text{atm}}$ respectively and when the volume of the system dropped to 100${\text{c}}{{\text{m}}^{\text{3}}}$, then the pressure of the system becomes ${\text{1}}{\text{.16 atm}}$.

Note:
Most of the gases behave as ideal gases at moderate temperatures and pressures and hence this law is mostly applicable to all types of gases. A beautiful example of the application of Boyle's Law is the breathing system of the Human Body. As we breathe in the air, the volume of the air in the lungs increases and the pressure inside it falls so that that lungs expand to accept the air on the hand as one exhales out the air inside the lungs is released, the volume of air decreases and then the pressure of the lungs is increased.