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 increases if the gas is compressed to half its original volume?
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(a)
The process is adiabatic
$\therefore P _1 V _1^\gamma= P _2 V _2^\gamma$
Given, $\frac{ V _1}{ V _2}=\frac{2}{1}$ and $\gamma=\frac{7}{5}$ (for diatomic gas)
$\therefore \frac{ P _2}{ P _1}=(2)^{7 / 5} \quad \text { i.e., } P _2=(2)^{7 / 5} P _1$
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