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?

Now here the cylinder or the container containing the gas and piston mounted are both insulated so no transfer of heat will take place either from system to surrounding and vice versa , so the process will be adiabatic as the heat supplied to or from the system

ΔQ = 0

Now the system consists of 3 Mole Hydrogen gas kept at standard temperature and pressure in an insulated container

As shown in the figure

Now here the gas is compressed to half of its volume, we have to calculate factor of change of pressure of gas when volume is reduced to half

We know for an adiabatic process we have the relation

Where P₁ and P₂ are initial and final pressures of the system respectively, V₁ and V₂ are initial and final volume of the system respectively, γ is the specific heat ratio of the gas since, H₂ is a diatomic gas, for a diatomic gas

γ = 1.4

now let the initial volume of the system be

V₁ = V

Now here the gas is compressed to half of its volume so final volume of the system is

V₂ = V/2

Let initial and final pressures of the system be P₁ and P₂ respectively

Now applying the relation

We have

Or

i.e.

so we can say that the final pressure is 2.639 times initial pressure or the pressure increased by a factor of 2.639