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Thermodynamics — Physics STD 11 Science — Question

Gujarat BoardEnglish MediumSTD 11 SciencePhysicsThermodynamics5 Marks
Question
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 $\Delta\text{V}(<<\text{V})$ 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 $P_2$?
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Answer

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 increase from P to (P + dP) then $\text{P}_{1}\text{V}_{1}^\gamma=\text{P}_{2}\text{V}_{2}^\gamma$
$\text{P}(\text{d}+\text{dv})^\gamma=(\text{P}+\text{dp})\text{V}^\gamma$
$\text{PV}\Big[1+\frac{\text{dV}}{\text{V}}\Big]^\gamma=\text{P}\Big[1+\frac{\text{dP}}{\text{P}}\Big]\text{V}^\gamma$
As volume of tyre V remains constant $\text{PV}^\gamma\Big[1+\gamma\frac{\text{dV}}{\text{V}}\Big]=\text{PV}^\gamma\Big[1+\frac{\text{dP}}{\text{P}}\Big]$
$\big[$on expanding by binomial theorm neglecting the higher terms of $\Delta\text{V}$ as $\Delta\text{V}<<\text{V}\big]$$1+\gamma\frac{\text{dV}}{\text{V}}=1+\frac{\text{dP}}{\text{P}}$
$\text{dV}=\frac{\text{VdP}}{\gamma\text{P}}$
Integrating both side in limits $W_1$ to $W_2$ and $P_1 → P_2$$\int\text{pdV}=\int\limits^{\text{p}_2}_{\text{p}_1}\frac{\text{VdP}}{\gamma}$
$\int\limits^{\text{w}_2}_{\text{w}_1}\text{dw}=\frac{\text{V}}{\gamma}(\text{P}_{2}-\text{P}_{1})(\text{V}=\text{constant})$
$\text{W}=\frac{(\text{P}_{2}-\text{P}_{1})\text{V}}{\gamma}$

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