An engine takes in 5 moles of air at 20^oC and 1 atm, and compresses it adiabatically to 1/10^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________.

For diatomic ideal gas : <br><br>f = 5 <br><br>$\gamma$ = ${7 \over 5}$ <br><br>T_i = T = 273 + 20 = 293 K <br><br>V_i = V <br><br>V_f = ${V \over {10}}$ <br><br>For adiabatic process TV^{$\gamma$ - 1} = constant <br><br>${T_1}V_1^{\gamma - 1} = {T_2}V_2^{\gamma - 1}$ <br><br>$\Rightarrow$ $$\left( {293} \right){V^{{7 \over 5} - 1}} = {T_2}{\left( {{V \over {10}}} \right)^{{7 \over 5} - 1}}$$ <br><br>$\Rightarrow$ ${T_2} = 293 \times {\left( {10} \right)^{{2 \over 5}}}$ <br><br>$\Delta$U = ${{nfR\left( {{T_2} - {T_1}} \right)} \over 2}$ <br><br>= $${{5 \times 5 \times {{25} \over 3} \times \left( {{{293.10}^{{2 \over 5}}} - 293} \right)} \over 2}$$ <br><br>= ${{625 \times 293 \times \left( {{{10}^{{2 \over 5}}} - 1} \right)} \over 6}$ <br><br>= 46.14 $\times$ 10³ J <br><br>$\simeq$ 46 kJ