Question

When an ideal gas with pressure $P$ and volume $V$ is compressed isothermally to one fourth of its volume, the pressure is $P_1$. When the same gas is compressed polytropically according to the equation $PV^{1.5} =$ constant to one fourth of its initial volume, the pressure is $P_2$. The ratio $\frac{P_{1}}{P_{2}}$ is

• A. $\frac{1}{2}$
• B. $\frac{1}{2^{1.5}}$
• C. $2$
• D. $2^{1.5}$

Answer 1

Answer: (A)

For isothermal process : $PV =$ constant
or $PV=P_{1} \frac{V}{4} \Rightarrow P_{1}=4P$
For polytropic process: $PV^{1.5} =$ constant
$PV^{1.5}=P_{2}\left(\frac{V}{4}\right)^{1.5}$
$\Rightarrow P_{2}=\left(2^{2}\right)^{3/2}\,P=8\,P$
$\therefore \frac{P_{1}}{P_{2}}=\frac{1}{2}$