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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

Q12.4 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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As the walls of the cylinder and the piston is insulated the process will be adiabatic. i.e. $PV^{\gamma }$ would be constant.

Hydrogen is a diatomic gas and therefore $\gamma =1.4$

Let the initial and final pressure be P1 and P2 respectively.

Let the initial and final volume be T1 and T2 respectively.

$\\P_{1}V_{1}^{\gamma }=P_{2}V_{2}^{\gamma }\\ \frac{P_{2}}{P_{1}}=\left ( \frac{V_{1}}{V_{2}} \right )^{\gamma }\\ \frac{P_{2}}{P_{1}}=2^{1.4}\\ \frac{P_{2}}{P_{1}}=2.639$

The pressure thus increases by a factor 2.639

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