A bicycle tire tube is inflated using a pump. | vedclass

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(A) In an adiabatic process,the work done $W$ by the gas is given by the formula $W = \frac{P_{i}V_{i} - P_{f}V_{f}}{\gamma - 1}$.Since the volume $V$

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$\frac{P_{2}V - P_{1}V}{\gamma - 1}$

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(A) In an adiabatic process,the work done $W$ by the gas is given by the formula $W = \frac{P_{i}V_{i} - P_{f}V_{f}}{\gamma - 1}$.Since the volume $V$ of the tube remains constant throughout the process,we have $V_{i} = V_{f} = V$.The initial pressure is $P_{1}$ and the final pressure is $P_{2}$.Substituting these values into the work formula,we get $W = \frac{P_{1}V - P_{2}V}{\gamma - 1}$.However,the work done $ON$ the gas (by the pump) is the negative of the work done $BY$ the gas.Therefore,the work done to increase the pressure is $W_{ext} = -W = -\left(\frac{P_{1}V - P_{2}V}{\gamma - 1}\right) = \frac{P_{2}V - P_{1}V}{\gamma - 1}$.

$A$ bicycle tire tube is inflated using a pump. Assume the volume of the tube $V$ is constant and in each stroke,an amount of air $\Delta V$ is pushed into the tube under an adiabatic process. What is the work done to increase the pressure in the tube from $P_{1}$ to $P_{2}$?

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