At constant temperature, increasing the pressure of a gas by $5 \%$ its volume will decrease by

At constant temperature, increasing the pressure of a gas by $5 \%$ its volume will decrease by
  1. $5 \%$
  2. $5.26\%$
  3. $4.20 \%$
  4. $4.70 \%$

Solution

According to ideal gas law at constant temperature, $\mathrm{P} \propto \frac{1}{\mathrm{~V}}$ $\begin{array}{ll} \therefore \quad & \mathrm{P}_1 \mathrm{~V}_1=\mathrm{P}_2 \mathrm{~V}_2 \\ & \mathrm{P}_1=\mathrm{P} \\ & \mathrm{P}_2=\mathrm{P}+\frac{5}{100} \mathrm{P}=1.05 \mathrm{P} \end{array}$ Substituting the values $\begin{array}{ll} & \mathrm{PV}_1=1.05 \mathrm{PV}_2 \\ \therefore \quad & \frac{\mathrm{V}_2}{\mathrm{~V}_1}=\frac{1}{1.05} \\ \therefore \quad & \frac{\mathrm{V}_2-\mathrm{V}_1}{\mathrm{~V}_1} \times 100=\frac{1-1.05}{1.05} \times 100 \\ \therefore \quad & \frac{\Delta \mathrm{V}}{\mathrm{V}_1}=-4.76 \% \approx-4.7 \% \end{array}$ The negative sign indicates that the volume is decreasing.

Asked in: MHT CET 2023 (14 May Shift 2)

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