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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.

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Solution

(a) Given, Initial pressure of the gas = p0 Initial volume of the gas = ${V}_{0}$ For an isothermal process, PV = constant So, P1V1 = P2V2 ${P}_{\mathit{2}}\mathit{=}\frac{{\mathit{P}}_{\mathit{0}}{\mathit{V}}_{\mathit{0}}}{\mathit{\left(}\frac{{P}_{0}}{2}\mathit{\right)}}=2{\mathrm{V}}_{0}$ For an adiabatic process, P3 = $\frac{{P}_{0}}{4}$, V3 = ? P2V2γ = P3V3γ $⇒{\left(\frac{{V}_{3}}{{V}_{2}}\right)}^{\gamma }\mathit{=}\left(\frac{{P}_{2}}{{P}_{3}}\right)\phantom{\rule{0ex}{0ex}}\mathit{⇒}{\left(\frac{{V}_{3}}{{V}_{2}}\right)}^{\gamma }\mathit{=}\left(\frac{\frac{{P}_{0}}{2}}{\frac{{P}_{0}}{4}}\right)=2\phantom{\rule{0ex}{0ex}}\mathit{⇒}\frac{{V}_{3}}{{V}_{2}}={2}^{\frac{1}{\gamma }}\phantom{\rule{0ex}{0ex}}\therefore {V}_{3}\mathit{=}{V}_{2\mathit{}}{\mathit{2}}^{\frac{\mathit{1}}{\gamma }}=2{V}_{0}{2}^{\frac{\mathit{1}}{\gamma }}\phantom{\rule{0ex}{0ex}}\mathit{=}{2}^{\frac{\gamma \mathit{+}1}{\gamma }}{V}_{0}$ (b) P1V1γ = P2V2γ $\text{Or}\left(\frac{{V}_{2}}{{V}_{1}}\right)={\left(\frac{{P}_{1}}{{P}_{2}}\right)}^{\frac{1}{\gamma }}\phantom{\rule{0ex}{0ex}}⇒{V}_{2}={V}_{0}{2}^{\frac{1}{\gamma }}$ Again, for an isothermal process, P2V2 = P3V3 $\therefore \mathit{}{V}_{\mathit{3}}\mathit{=}\frac{{P}_{\mathit{2}}{V}_{\mathit{2}}}{{P}_{\mathit{3}}}={22}^{\frac{1}{\gamma }}{V}_{0}\phantom{\rule{0ex}{0ex}}={2}^{\frac{\mathrm{\gamma }+1}{\mathrm{\gamma }}}{V}_{0}$

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