Question
Two samples of gas initially at the same temperature and pressure are compressed from volume V to $\frac{\text{V}}{2}.$ One sample is compressed isothermally and the other adiabatically. In which case will the pressure be higher? Explain.
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Answer
Let Pa and Pi be the final pressure during adiabatic and isothermal compression respectively. In case isothermal compression.
$\text{PV}=\text{P}_\text{i}\Big(\frac{\text{V}}{2}\Big)$ or $\text{P}_\text{i}=2\text{P}\dots\text{(i)}$
In case of adiabatic compression,
$\text{PV}^\gamma=\text{P}_\text{i}\Big(\frac{\text{V}}{2}\Big)^\gamma$ or $\text{P}_\text{a}=2^\gamma\text{P}\dots\text{(ii)}$
$\therefore\frac{\text{P}_\text{a}}{\text{P}_\text{i}}=\frac{2^\gamma}{2}>1,$ because $\gamma>1$
$\therefore\text{P}_\text{a}>\text{P}_\text{i}$
Hence, final pressure during adiabatic compression is greater than the pressure during isothermal compression.
$\text{PV}=\text{P}_\text{i}\Big(\frac{\text{V}}{2}\Big)$ or $\text{P}_\text{i}=2\text{P}\dots\text{(i)}$
In case of adiabatic compression,
$\text{PV}^\gamma=\text{P}_\text{i}\Big(\frac{\text{V}}{2}\Big)^\gamma$ or $\text{P}_\text{a}=2^\gamma\text{P}\dots\text{(ii)}$
$\therefore\frac{\text{P}_\text{a}}{\text{P}_\text{i}}=\frac{2^\gamma}{2}>1,$ because $\gamma>1$
$\therefore\text{P}_\text{a}>\text{P}_\text{i}$
Hence, final pressure during adiabatic compression is greater than the pressure during isothermal compression.
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