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?
Air is transferred into tyre adiabatically let initial volume of air in tyre V and after pumping one stroke it become (V+dV) and pressure increased from P to (P+dP) then P1V1γ=P2V2γ P(d+dv)γ=(P+dP)Vγ PVγ[1+VdV]γ=P[1+PdP]Vγ As the volume of tire V remains constant PVγ[1+γVdV]=PV[1+PdP] [on expanding by binomial theorem neglecting the higher terms of △V as △V<<V] 1+γVdV=1+PdP dV=γPV dP Integrating both side in limits W1 to W2 and P1→P2 ∫pdV=∫P1P2γV dP ∫W1W2dW=γV(P2−P1)(V=constant) W=γ(P2−P1)V.
