Question:
Consider a cycle tyre being filled with air by a pump. Let V be the volume of the tyre and at each stroke of the pump ∆V of air is transferred to the tube adiabatically. What is the work done when the pressure in the tube is increased from P1 to P2?
Solution:
Following is the equation before and after the stroke:
$P_{1} V_{1}^{\gamma}=P_{2} V_{2}^{\gamma}$
$P(V+\Delta V)^{\gamma}=(P+\Delta P) V^{\gamma} \Rightarrow P V^{\gamma}\left(1+\frac{\Delta V}{V}\right)^{\gamma}=P\left(1+\frac{\Delta P}{P}\right) V^{\gamma}$
$P V^{\gamma}\left(1+\gamma \frac{\Delta V}{V}\right) \approx P V^{\gamma}\left(1+\frac{\Delta P}{P}\right)$
$\gamma \frac{\Delta V}{V}=\frac{\Delta P}{P}$
Therefore, work done is given as
$W=\frac{\left(P_{2}-P_{1}\right) V}{\gamma}$