With what minimum pressure must a given volume of an ideal gas ($\gamma$ = 1.4), originally at 400 K

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With what minimum pressure must a given volume of an ideal gas ($\gamma$ = 1.4), originally at 400 K and 100 kPa, be adiabatically compressed in order to raise its temperature up to 700 K?

Accepted Answer

For minimum pressure, compression should be irreversible.

$\Delta U = w \Rightarrow n.\frac{R}{\gamma - 1}\left( T_{2} - T_{1} \right) = - P_{ext}\left( V_{2} - V_{1} \right) = - P_{2}\left( \frac{nRT_{2}}{P_{2}} - \frac{nRT_{1}}{P_{1}} \right)$

$\text{Or, }\frac{T_{2} - T_{1}}{\gamma - 1} = - \left( T_{2} - T_{1}.\frac{P_{2}}{P_{1}} \right) \Rightarrow \frac{700 - 400}{1.4 - 1} = - \left( 700 - 400 \times \frac{P_{2}}{100} \right) $$$\therefore P_{2} = 362.5\text{ kPa}$$

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