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 = P₁

Final pressure inside the cylinder = P₂

Initial volume inside the cylinder = V₁

Final volume inside the cylinder = V₂

Ratio of specific heats, γ = 1.4

For an adiabatic process, we have:

`P_1V_1^gamma = P_2V_2^gamma`

The final volume is compressed to half of its initial volume.

`:. 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)^gamma = (2)^(1/4) = 2.639`

Hence, the pressure increases by a factor of 2.639.

Answer 2

Here the process is adiabatic compression and `V_2 = V_1/2, P_2 = 1` atm and for hydrogen

(a diatomic gas)`gamma = 1.4`

∵ `P_1V_1^gamma = P_2V_2^gamma`,

Hence `P_2 = P_2(V_1/V_2)^gamma = 1 atm (v_1/(V_1/2))^(1.4)`

`=> P_2  = (2)^(1.4) atm`

= 2.64 atm