Question

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

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

Answer 1

Answer: (A)

$1-2^{-2/3}$

$\text{Solution}:$

Let initial Pressure and Volume be $P_1$ and $V_1$

After Isothermal expansion, the final pressure and volume be $P_2$ and $V_2$

For isothermal expansion, Temperature, T = constant.

$⟹P_1{V}_1=P_2{V}_2$

$⟹P_1{V}_1=P_2\left(2V_1\right)$

$⟹P_2=\frac{P_1}{2}$

After adiabatic compression, let the final pressure and volume be $P_3$ and $V_3$

$P_2{V}_2^{\gamma }=P_3{V}_3^{\gamma }$

where, the adiabatic index, $\gamma =\frac{5}{3}$for Neon

$P_3{V_3}^{\frac{5}{3}}=P_1{V}_1^{\frac{5}{3}}{P}_3(2V_1{)}^{\frac{5}{3}}=P_1{V}_1^{\frac{5}{3}}{P}_3=2^{-\frac{5}{3}}$

Fractional decrease in pressure so that gas is adiabatically compressed to same state is:

$\frac{P_2-P_3}{P_2}=\frac{P_1/2-2^{-\frac{5}{3}}{P}_1}{P_1/2}=1-2^{-2/3}$

Hence, there should be $1-2^{-2/3}$ decrease in pressure so that gas when adiabatically compressed from that state, reaches original state

$\text{Hence A is the correct option}$