Question

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

• A. $5%$
• B. $5.26%$
• C. $4.20%$
• D. $4.70%$

Hint:

For small percentage changes in pressure, the percentage change in volume is approximately equal in magnitude but opposite in sign. For precise results, always use the formula $P_1 V_1 = P_2 V_2$.

Answer 1

The Correct Option is D

Step 1: For an ideal gas at constant temperature, pressure $P$ and volume $V$ follow Boyle's Law: \[ P_1 V_1 = P_2 V_2 \]
Step 2: Let the initial pressure be $P_1 = P$ and initial volume be $V_1 = V$. If the pressure is increased by $5%$, the new pressure $P_2$ is: \[ P_2 = P + 0.05P = 1.05P \]
Step 3: Substitute these values into Boyle's Law equation to find the new volume $V_2$: \[ PV = (1.05P) V_2 \] \[ V_2 = \frac{V}{1.05} \]
Step 4: The percentage decrease in volume is calculated as: \[ \text{Decrease %} = \frac{V_1 - V_2}{V_1} \times 100 \] \[ \text{Decrease %} = \frac{V - \frac{V}{1.05{V} \times 100 = \left(1 - \frac{1}{1.05}\right) \times 100 \]
Step 5: Simplify the expression: \[ \text{Decrease %} = \frac{0.05}{1.05} \times 100 = \frac{5}{105} \times 100 \approx 4.76% \] The closest matching option is $4.70%$.