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An engine takes in $5$ moles of air at $20\,^{\circ} C$ and $1$ $atm,$ and compresses it adiabaticaly to $1 / 10^{\text {th }}$ 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 $X\, kJ$. The value of $X$ to the nearest integer is
$46.87$
$45.78$
$55.78$
$50.23$
Solution
Diatomic :
$f=5$
$\gamma=7 / 5$
$T _{ i }= T =273+2 0 =293 K$
$V_{i}=V$
$V _{ f }= V / 10$
Adiabatic $TV ^{\gamma-1}=$ constant
$T _{1} V _{1}^{\gamma-1}= T _{2} V _{2}^{\gamma-1}$
$T \cdot V ^{7 / 5-1}= T _{2}\left(\frac{ V }{10}\right)^{7 / 5-1}$
$\Rightarrow T _{2}= T .10^{2 / 5}$
$\Delta U =\frac{ nfR \left( T _{2}- T _{1}\right)}{2}=\frac{5 \times 5 \times \frac{25}{3} \times\left( T \cdot 10^{2 / 5}- T \right)}{2}$
$=\frac{25 \times 25 \times}{6} T \left(10^{2 / 5}-1\right)$
$=\frac{625 \times 293 \times\left(10^{2 / 5}-1\right)}{6}$
$=4.033 \times 10^{3} \approx 4 kJ$
