Question

Neon gas of a given mass expands isothermally to double of its original volume. What should be the further fractional decrease in pressure, so that the gas when adiabatically compressed from that state, reaches the original state?

• A. $1-2^{-2/3}$
• B. $1-2^{2/3}$
• C. $2^{-5/3}$
• D. $2^{2/3}$

Answer 1

The correct option is A $1-2^{-2/3}$

During isothermal process, as the temperature remains constant, we can say that the equation of state of the gas is
$PV=k$
From this,
$P_1{V}_1=P_2{V}_2$
From the data given in the question,
$PV=P_2\left(2V\right)$
$\Rightarrow P_2=\frac{P}{2}......\left(1\right)$

Equation of state of the gas during adiabatic process in terms of $P$ and $V$ is given by,
$P_3{V}_3^{\gamma }=P_1{V}_1^{\gamma }$
[since gas returns to original state $(P_1,V_1)$]
From the data given in the question, we know that, neon gas is monoatomic.
i.e $\gamma =\frac{5}{3}$
$\Rightarrow P_3\times (2V{)}^{5/3}=P_1\times (V{)}^{5/3}$
$\Rightarrow P_3=P_1\times (2{)}^{-5/3}$

$\therefore$ Fractional decrease in pressure $=\frac{P_2-P_3}{P_2}=\frac{P_2-[P_1\times (2{)}^{-5/3}]}{P_2}$
From $\left(1\right)$, $P_1=P,P_2=\frac{P}{2}$
$\Rightarrow \frac{P_2-P_3}{P_2}=1-(2{)}^{-2/3}$
Thus, option (a) is the correct answer.