Question

Consider a given sample of an ideal gas (Cp/Cv = γ) having initial pressure p0 and volume V0. (a) The gas is isothermally taken to a pressure p0/2 and from there, adiabatically to a pressure p0/4. Find the final volume. (b) The gas is brought back to its initial state. It is adiabatically taken to a pressure p0/2 and from there, isothermally to a pressure p0/4. Find the final volume.

Answer 1

Final volume of the gas when,

(a) from Isothermal to adiabatic process with pressure from  $\frac{P_0}{2}$ to  $\frac{P_0}{4}$ is $2^{\frac{\gamma+1}{\gamma}} \mathrm{V}_{0}$

(b) From adiabatic to isothermal process with pressure from  $\frac{P_0}{2}$ to  $\frac{P_0}{4}$ is $2^{\frac{\gamma+1}{\gamma}} \mathrm{V}_{0}$

Explanation:

Given Data

Initial  gas pressure = p0

Initial gas  volume =  V0

(a)  In case of  isothermal process,  PV = constant

$\mathrm{P}_{1} \mathrm{V}_{1}=\mathrm{P}_{2} \mathrm{V}_{2}$

$\mathrm{P}_{2}=\frac{\mathrm{P}_{0} \mathrm{V}_{0}}{\frac{\mathrm{P}_{0}}{2}}=2 \mathrm{V}_{0}$

Compression with adiabatic effect:

$\mathrm{P}_{3}=\frac{\mathrm{P}_{0}}{4}$

$\mathrm{V}_{3}=?$

$\mathrm{P}_{1} \mathrm{V}_{1}^{\gamma}=\mathrm{P}_{2} \mathrm{V}_{2}^{\gamma}$

$\left(\frac{V_{3}}{V_{2}}\right)^{\gamma}=\frac{P_{2}}{P_{3}}$

$\left(\frac{V_{3}}{V_{2}}\right)^{\gamma}=\frac{\frac{P_{0}}{2}}{\frac{P_{0}}{4}}=2$

$\frac{V_{3}}{V_{2}}=2^{\frac{1}{\gamma}}$

$\mathrm{V}_{3}=\mathrm{V}_{2} 2^{\frac{1}{\gamma}}=2 \mathrm{V}_{0} 2^{\frac{1}{\gamma}}$

     $=2^{\frac{\gamma+1}{\gamma}} \mathrm{V}_{0}$

$\text { (b) } \mathrm{P}_{1} \mathrm{V}_{1}^{\gamma}=\mathrm{P}_{2} \mathrm{V}_{2}^{\gamma}$

$\frac{V_{2}}{V_{1}}=\left(\frac{P_{1}}{P_{2}}\right)^{\frac{1}{\gamma}}$

$\mathrm{V}_{2}=\mathrm{V}_{0}(2)^{\frac{1}{\gamma}}$

Compression with Isothermal effect:

$\mathrm{P}_{2} \mathrm{V}_{2}=\mathrm{P}_{3} \mathrm{V}_{3}$

$\mathrm{V}_{3}=\frac{\mathrm{P}_{2} \mathrm{V}_{2}}{\mathrm{P}_{3}}=2 \times \mathrm{V}_{0}(2)^{\frac{1}{\gamma}}$

$\mathrm{V}_{3}=2^{\frac{\gamma+1}{\gamma}} \mathrm{V}_{0}$

Therefore the final volume of the gas which is taken isothermally with pressure $\frac{P_0}{2}$ and adiabatically as $\frac{P_0}{4}$ is $2^{\frac{\gamma+1}{\gamma}} \mathrm{V}_{0}$ and the final volume when the gas is brought back to its initial state from  adiabatically as  $\frac{P_0}{2}$ to isothermally as $\frac{P_0}{4}$ is $2^{\frac{\gamma+1}{\gamma}} \mathrm{V}_{0}$