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A cylinder with a movable piston contains 3 moles of hydrogen at standard temperature and pressure. The walls of the cylinder are made of a heat insulator, and the piston is insulated by having a pile of sand on it. By what factor does the pressure of the gas increase, if the gas is compressed to half its original volume?
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Verified Answer
The process is adiabatic, as no heat in exchanged
$$
\therefore \quad P_2 V_2^\gamma=P_1 V_1^\gamma \quad \therefore \frac{P_2}{P_1}=\left(\frac{V_1}{V_2}\right)^\gamma
$$
Given $V_2=1 / 2 V_1$
$$
\therefore \quad \frac{P_2}{P_1}=\left(\frac{V_1}{V_1 / 2}\right)^{1.4}=2.64
$$
$$
\therefore \quad P_2 V_2^\gamma=P_1 V_1^\gamma \quad \therefore \frac{P_2}{P_1}=\left(\frac{V_1}{V_2}\right)^\gamma
$$
Given $V_2=1 / 2 V_1$
$$
\therefore \quad \frac{P_2}{P_1}=\left(\frac{V_1}{V_1 / 2}\right)^{1.4}=2.64
$$
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