Question
Two moles of an ideal gas $\left( C_{v,m} = \frac{5}{2}R \right)$ was compressed adiabatically against constant pressure of 2 atm, which was initially at 350 K and 1 atm. The work done on the gas in this process is
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Solution
$w = \Delta U \Rightarrow - P_{ext}\left( V_{2} - V_{1} \right) = n.C_{v,m}\left( T_{2} - T_{1} \right)$
Or, $- P_{2}\left( \frac{nRT}{P_{2}} - \frac{nRT}{P_{1}} \right) = n \times \frac{5}{2}R\left( T_{2} - T_{1} \right)$
Or, $- \left( T_{2} - T_{1}.\frac{P_{2}}{P_{1}} \right) - \frac{5}{2}\left( T_{2} - T_{1} \right)$
Or $- \left( T_{2} - 350 \times \frac{2}{1} \right) = \frac{5}{2}\left( T_{2} - 350 \right) \Rightarrow T_{2} = \text{450 K}$
Now, $w = \Delta U = n.C_{v,m}\left( T_{2} - T_{1} \right) = 2 \times \frac{5}{2}R \times (450 - 350) = 500\text{ R}$
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