Question

An engine takes in $5$ moles of air at $20^\circ C$ and $1atm,$ and compresses it adiabatically to $\left(\frac{1}{10}\right)^{t h}$ of the original volume. Assuming air to be a diatomic ideal gas made up of rigid molecules, the change in its internal energy during this process comes out to be $xkJ$ . The value of $x$ to the nearest integer is

Answer 1

$T_{2}V_{2}^{r - 1}=T_{1}V_{1}^{\gamma - 1}$
$T_{2}=T_{1}\left(\frac{V_{1}}{ V_{2}}\right)^{\gamma - 1}$
$=293\left(\frac{V}{V / 10}\right)^{\frac{7}{5} - 1}$
$\Delta U=nC_{v}\Delta T=5\times \frac{5}{2}R\left(293 \times \left(10\right)^{\frac{2}{5}} - 293\right)$
$=\frac{25}{2}R\times 293\left(\left(10\right)^{\frac{2}{5}} - 1\right)=\frac{25 R}{2}\times 293\left(\right.2.5-1\left.\right)$
$=45675J=46kJ$