Question

By what percentage should the pressure of a given mass of gas be increased, so as to decrease its volume by $10\%$ at a constant temperature?
A. $11.1\%$
B. $10.1\%$
C. $9.1\%$
D. $8.1\%$

Answer 1

We know that when a gas expands, there would decrease in pressure and increase in volume. This signifies that volume of a gas and pressure of a gas are indirectly related to each other. So, we can use the equation ${P_1}{V_1} = {P_2}{V_2}$.

Complete answer:
From the questions, we can understand that when there is an increase in pressure, the volume gets reduced at constant temperature and this signifies Boyle’s law.
We can state Boyle’s law as “At constant temperature, the pressure of the gas and volume of the gas are indirectly related to each other”.
We can formulate the equation as,
$P \propto \dfrac{1}{V}$
$PV = K$
Here, constant is represented as K.
We can write the relation between pressure and volume as,
${P_1}{V_1} = {P_2}{V_2}$
Based on the question, ${V_1}$ is V and ${V_2}$ is $10\%$ less than original volume at constant temperature.
So, we can write ${V_2}$ as,
${V_2} = \left( {100 - 10} \right) = 90$
${V_2} = 0.90\,{V_1}$
Let us now calculate the increase in pressure.
${P_1}{V_1} = {P_2}{V_2}$
$\left( {{P_1}} \right)\left( {{V_1}} \right) = \left( {{P_2}} \right)\left( {0.90{V_1}} \right)$
From this, we can relate the pressure as,
${P_1} = {P_2} \times 0.90$
So, ${P_1}$ is equal to $0.90$ times of${P_2}$.
We can also write this as,
${P_2} = \dfrac{{{P_1}}}{{0.90}}$
${P_2} = \dfrac{1}{{0.90}}{P_1}$
${P_2} = 1.11\,{P_1}$
We can say that ${P_2}$ is equal to $1.11$ times of ${P_1}$.
We have to calculate the change in pressure using the expression below,
$P = {P_2} - {P_1}$
Let’s calculate the change in pressure by plugging in the values of ${P_2}$ and ${P_1}$
$P = 1.11{P_1} - {P_1}$
$P = 0.111{P_1}$
We have to calculate this value in percentage as,
$0.111{P_1} \times 100 = 11.1\%$
So, $11.1\%$ of pressure should be increased in order to reduce the volume of gas by $10\%$.

So, the correct answer is “Option A”.

Note:
We have to know that we can use this formula in this equation only if the specified condition is at constant temperature because this formula is according to Boyle’s law. We have to know that the other name of Boyle’s law is Boyle-Mariotte law. We have to know that the breathing system is based on Boyle’s law.