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 V(<<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?

Hint: Use the equation of the adiabatic process.
Step 1: Find the change in volume of the tyre.
Let, the volume is increased by V and pressure is increased by p by a stroke. For just before and after a stroke, we can write,
                                  p1V1γ=p2V2γ
                    p(V+V)γ=(p+p)Vγ                   ( volume is fixed)
               pVγ1+VVγ=p1+ppVγ
                  pVγ1+γVV=pVγ1+pρ                  (v<<v)
               γVV=ppV=1γVpp
                dV=1γVpdp
Step 2: Find the work done.
Hence, the work done in increasing the pressure from p1 to p2:
                          W =p1p2pdV=p1p2p×1γVpdp
    =Vγp1p2dp=Vγ(p2-p1)
                       W=(p2-p1)γV