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(lt lt 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

If `DeltaP` is small increase in pressure of air, when volume of air increases by a small amount `DeltaV`, then as the change in adiabatic, `P(V+DeltaV)^(gamma) = (P+DeltaP)V^(gamma)`
`PV^(gamma)(1+(DeltaV)/V)^(gamma)= P(1+(DeltaP)/P)V^(gamma)` or `1+gamma(DeltaV)/V= 1+(DeltaP)/P`
or `gamma(DeltaV)/V= (DeltaP)/P` or `DeltaV = 1/(gamma) V/P DeltaP` or `dV= V/(gammaP)dp` ....(i)
Work done in increasing the pressure from `P_(1) to P_(2)`
`W = int_(P_(1))^(P_(2)) PdV`
Using (i),
`W= int_(P_(1))^(P_(2)) P(V)/(gammaP)dp=(V)/(gamma)(P_(2)-P_(1))`