Question

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

Answer 1

Answer: $1-2^{-\frac{2}{3}}$

Explanation:

Let initial Pressure and Volume be P₁ and V₁

After Isothermal expansion, the final pressure and volume be P₂ and V₂

For isothermal expansion, Temperature, T = constant.

⇒P₁V₁=P₂V₂

⇒P₁V₁=P₂(2V₁)

⇒P₂=P₁/2

After adiabatic compression, let the final pressure and volume be P₃ and V₃

$P_2V_2^\gamma =P_3V_3^\gamma$

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

${P_3}{V_3}^\frac{5}{3}=P_1V_1^\frac{5}{3}$

$P_3(2V_1)^\frac{5}{3}=P_1V_1^\frac{5}{3}$

$P_3=2^{-\frac{5}{3}}$

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.