Question

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?

Answer 1

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 $=V_2$
Ratio of specific heats, $\gamma =\frac{C_P}{C_V}=1.4$
For an adiabatic process, we have:

$P_1{V}_1^{\gamma }=P_2{V}_2^{\gamma }$
The final volume is compressed to half of its initial volume.

$\therefore V_2=V_1/2$

$P_1{V}_1^{\gamma }=P_2(V_1/2{)}^{\gamma }$

$P_2/P_1=V_1^{\gamma }/(V_1/2{)}^{\gamma }$

$=2^{1.4}=2.639$
Hence, the pressure increases by a factor of 2.639.