A sample of a diatomic gas at pressure $p$, volume $V$ and temperature $T$ is compressed to $V/2$ isothermally. Its pressure is found to be $p_{i}$. If it is compressed to $V/2$ adiabatically, its pressure comes out to be $p_{a}$. What is the relation between $p_{i}$ and $p_{a}$?

Given : A sample of a diatomic gas at pressure pp, volume VV and temperature TT is compressed to V/2V/2 isothermally. Its pressure is found to be pipi. If it is compressed to V/2V/2 adiabatically, its pressure comes out to be papa. What is the relation between pipi and papa

Solution :

Case 1(Isothermal process)

$V_1= V$ , $V_2=\frac{V}{2}$ , $P_1=P$ , $P_2=P_i$

$P_1V_1=P_2V_2$

$PV=P_i \frac{V}{2}$

$P_i=2P$

Case 2(adiabatic process)

$V_1= V$ , $V_2=\frac{V}{2}$ , $P_1=P$ , $P_2=Pa$

$\gamma =1.4$

$PV^{\gamma} = constant$

$PV^{\gamma} =(P_a \frac{V}{2})^{\gamma}$

$P_a =P (2)^{1.4}$

$P_a = 2.6P$

The Correct Opt = D