Consider a cycle tyre being filled with air by a pump. Let V be the volume of the tyre (fixed) and at each stroke of the pump `DeltaV (=V)` of air is transferred to the tube adiabatically. What is the work done when the pressure in the tube is increased from `P_(1)` to `P_(2)` ?

Explanation

`P(V +Delta upsilon)^(gamma) = (P+Deltap)V^(gamma)`
`P[1+ gamma (Delta upsilon)/(V)] = P(1+(Delta p)/(P))`
`gamma (Delta upsilon)/(V) =(Deltap)/(P), (d upsilon)/(dp) = (V)/(gammaP)`
`W.D. = int_(P_(1))^(P_(2)) P d upsilon= int_(P_(1))^(P_(2))"P" (V)/(gammaP) dp = ((P_(2)-P_(1)))/(gamma)V`