Ncert Exercises & Solutions

12.4 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 of its original volume?

No heat is exchanged between the system (cylinder) and its surroundings as the cylinder is completely insulated from its surroundings. Hence, the process is adiabatic.

For a diatomic gas:

The ratio of specific heats, $γ = 1.4$

For an adiabatic process,

$P_{1} V_{1}^{γ} = P_{2} V_{2}^{γ}$

As it is given in the question that:

$V_{2} = \frac{V_{1}}{2} P_{1} V_{1}^{γ} = P_{2} {(\frac{V_{1}}{2})}^{γ} \frac{P_{2}}{P_{1}} = \frac{V_{1}^{γ}}{{(\frac{V_{1}}{2})}^{γ}} = 2^{1.4} = 2.639$

Hence, the pressure increases by a factor of 2.639.

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