$A$ 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?

(2.639) The cylinder is completely insulated from its surroundings. As a result,no heat is exchanged between the system (cylinder) and its surroundings. Thus,the process is adiabatic.
Initial pressure inside the cylinder $= P_1$
Final pressure inside the cylinder $= P_2$
Initial volume inside the cylinder $= V_1$
Final volume inside the cylinder $= V_2 = V_1 / 2$
Ratio of specific heats for hydrogen (diatomic gas),$\gamma = 1.4$
For an adiabatic process,the relation between pressure and volume is given by $P_1 V_1^{\gamma} = P_2 V_2^{\gamma}$.
Substituting the values:
$P_1 V_1^{1.4} = P_2 (V_1 / 2)^{1.4}$
$P_2 / P_1 = (V_1 / (V_1 / 2))^{1.4}$
$P_2 / P_1 = 2^{1.4}$
$P_2 / P_1 \approx 2.639$
Hence,the pressure of the gas increases by a factor of $2.639$.