Question

An ideal gas with pressure P, volume V, and temperature T is expanded isothermally to a volume $2$X and a final pressure $P_i$. If the same gas is expanded adiabatically to a volume $2V$, the final pressure is $P_a$. The ratio of the specific heats for the gas is $1.67$. The ratio $P_a/P_i$ is __________?

• A. $1^{-0.67}$.
• B. $2^{-0.67}$.
• C. $3^{-0.67}$.
• D. $4^{-0.67}$.

Answer 1

Answer: (B)

$2^{-0.67}$.

Given : An ideal gas with pressure P, volume V, and temperature T is expanded isothermally to a volume 22X and a final pressure PiPi. If the same gas is expanded adiabatically to a volume 2V2V, the final pressure is PaPa. The ratio of the specific heats for the gas is 1.671.67. The ratio Pa/PiPa/Pi is __________?

Solution :

Case 1 (Isothermal expainsion)

For Isothermal expainsion $P_1{V}_1=P_2{V}_2$

Given that $V_2=2V$

then $PV=P_i\left(2V\right)$

$P_i=\frac{P}{2}$

Case 2 ( adiabatic expainsion)

For adiabatic expainsion $P{V}^{\gamma }=constant$

Given that $\gamma =1.67$ and $V_2=2V$

$P{V}^{1.67}=P_a(2V{)}^{1.67}$

$P_a=\frac{P}{2^{1.67}}$

Then $\frac{P_a}{P_i}$=$\frac{\frac{P}{2^{1.67}}}{\frac{P}{2}}$

$\frac{P_a}{P_i}$=$2^{-0.67}$

The Correct Opt=B