Question

Consider a given sample of an ideal gas (C_p/C_v = 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

(a) Given,
Initial pressure of the gas = p₀
Initial volume of the gas = $V_0$
For an isothermal process,
PV = constant
So, P₁V₁ = P₂V₂
$P_{\mathit{2}}\mathit{=}\frac{{\mathit{P}}_{\mathit{0}}{\mathit{V}}_{\mathit{0}}}{\mathit{\left(}{\displaystyle \frac{P_0}{2}}\mathit{\right)}}=2{\mathrm{V}}_0$
For an adiabatic process, P₃ = $\frac{P_0}{4}$, V₃ = ?
P₂V₂^γ = P₃V₃^γ
$$\begin{gathered} \Rightarrow {\left(\frac{V_3}{V_2}\right)}^{\gamma }\mathit{=}\left(\frac{P_2}{P_3}\right) \\ \mathit{\Rightarrow }{\left(\frac{V_3}{V_2}\right)}^{\gamma }\mathit{=}\left(\frac{{\displaystyle \frac{P_0}{2}}}{{\displaystyle \frac{P_0}{4}}}\right)=2 \\ \mathit{\Rightarrow }\frac{V_3}{V_2}=2^{\frac{1}{\gamma }} \\ \therefore V_3\mathit{=}{V}_{2\mathit{}}{\mathit{2}}^{\frac{\mathit{1}}{\gamma }}=2V_0{2}^{\frac{\mathit{1}}{\gamma }} \\ \mathit{=}{2}^{\frac{\gamma \mathit{+}1}{\gamma }}{V}_0 \end{gathered}$$

(b) P₁V₁^γ = P₂V₂^γ
$$\begin{gathered} \text{Or}\left(\frac{V_2}{V_1}\right)={\left(\frac{P_1}{P_2}\right)}^{\frac{1}{\gamma }} \\ \Rightarrow V_2=V_0{2}^{\frac{1}{\gamma }} \end{gathered}$$

Again, for an isothermal process,
P₂V₂ = P₃V₃
$$\begin{gathered} \therefore \mathit{}{V}_{\mathit{3}}\mathit{=}\frac{P_{\mathit{2}}{V}_{\mathit{2}}}{P_{\mathit{3}}}={22}^{\frac{1}{\gamma }}{V}_0 \\ =2^{\frac{\mathrm{\gamma }+1}{\mathrm{\gamma }}}{V}_0 \end{gathered}$$