The volume of a given gas is V cm$^3$ at a pressure P mm Hg. what is the new volume of the gas if the pressure is reduced to half at constant temperature?

Analysis

The question asks for the relationship between the volume and pressure of a gas when the temperature is held constant. This is a classic application of the gas laws.

Concept: Boyle's Law

Boyle's Law states that for a fixed mass of an ideal gas, the volume ($V$) of the gas is inversely proportional to its pressure ($P$), provided the temperature remains constant.

Mathematically, this is expressed as:

Where:

• $P_1$ = Initial pressure
• $V_1$ = Initial volume
• $P_2$ = Final pressure
• $V_2$ = Final volume

Step-by-Step Solution

1. Identify the initial values: • Initial pressure ($P_1$) = $P$ • Initial volume ($V_1$) = $V$

2. Identify the final values: • The pressure is reduced to half, so the new pressure ($P_2$) = $\frac{P}{2}$ • The new volume ($V_2$) = ?

3. Apply Boyle's Law formula: $$P \times V = \left(\frac{P}{2}\right) \times V_2$$

4. Solve for $V_2$: • Divide both sides by $P$: $$V = \frac{1}{2} \times V_2$$ • Multiply both sides by 2 to isolate $V_2$: $$2V = V_2$$

The new volume is $2V$.

Conclusion

When the pressure of a gas is halved at a constant temperature, the volume must double to maintain the constant product of $PV$.

Exam Tip: Whenever you see "constant temperature" in a gas law question, immediately think of Boyle's Law. Remember: Pressure and Volume are like a see-saw—if one goes down, the other must go up!