A cyclinder with a movable piston contains `3 mol s` of hydrogen at standard temperature and pressure. The walls of the cyclinder 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? Given `gamma= 1.4`.

Explanation

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= 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.
`thereforeV_(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)^(gamma)=(2)^(1.4)=2.639`
Hence, the pressure increases by a factor of 2.639.