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 \Delta V\leq 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?

Answers (1)

Air is transferred into tyre adiabatically.

Let the initial volume of air be V and after pumping it becomes V+dV and pressure P+dP

P_1 V_1^\gamma =P_2V_{2}^{\gamma }
P(V+dV)^\gamma =(P+dP)V^\gamma

PV^\gamma \left [1+\frac{dV}{V} \right ]^\gamma =P\left [1+\frac{dP}{P} \right ]V^\gamma

PV^\gamma \left [1+\gamma \frac{dV}{V} \right ]=PV^\gamma\left [ 1+\frac{dP}{P} \right ]

On expanding by binomial theorem and neglecting the higher terms

1+\gamma \frac{dV}{V}=1+\frac{dP}{P}

dV=\frac{VdP}{\gamma P}

\int pdV=\int_{P_{1}}^{P_{2}}\frac{VdP}{\gamma }

\int_{W_{1}}^{W_{2}}dW=\frac{V}{\gamma }(P_2-P_1)

W=\frac{V}{\gamma }\left (P_2-P_1 \right )

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