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Problem 3.101: 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 ?

Solution:

The cylinder initially contains hydrogen at at standard temperature and pressure.

So, Initial pressure $P_1$ = $P_a$.

Initial volume $V_1$= $V$.

Initial temperature $T_1$= $T$.

Given that the cylinder is made of heat insulator and the piston is insulated by having a pile of sand on it. This prevents any exchange of heat between the hydrogen inside and the surroundings. So the process is adiabatic in nature.

When hydrogen is compressed,

Final volume $V_2$= $\frac{V}{2}$.

As the process is adiabatic, we can modify the ideal gas equation into the form $PV^{\gamma}\:=\:constant$.

The initial and final states can be related as

$P_1V_1^{\gamma}\:=\:P_2V_2^{\gamma}$

From this the final pressure, $P_2\:=\:P_1\times \frac{V_1^{\gamma}}{V_2^{\gamma}}$.

Or   $P_2\:=\:P_1\times\left(\frac{V_1}{V_2}\right)^\gamma$

Here $V_1$= $V$ and $V_2$= $\frac{V}{2}$.

Therefore, $P_2\:=\:P_1\times\left(\frac{V }{V/2}\right)^{\gamma}$

Or, $P_2\:=\:P_1\times2^{\gamma}$

Value of $\gamma$ for hydrogen is $\frac{7}{5}$.

The final pressure, $P_2\:=\:P_1\times2^{\frac{7}{5}}\:=\:2.63P_1$.

Hence the adiabatic compression will increase the pressure by an factor of $2.63$ approximately.