Question

An engine takes in $5$ moles of air at $20^{\circ }C$ and $1atm,$ and compresses it adiabaticaly 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 $XkJ$. The value of $X$ to the nearest integer is

Answer 1

Answer: 46.00

Diatomic :
$f=5$
$\gamma =7/5$
$T_i=T=273+20=293K$
$V_i=V$
$V_f=V/10$
Adiabatic $T{V}^{\gamma -1}=$ constant
$T_1{V}_1^{\gamma -1}=T_2{V}_2^{\gamma -1}$
$T.V^{7/5-1}=T_2{\left(\frac{V}{10}\right)}^{7/5-1}$
$\Rightarrow T_2=T.10^{2/5}$
$\Delta U=\frac{nfR\left(T_2-T_1\right)}{2}$
$=\frac{5\times 5\times \frac{25}{3}\times \left(T.10^{2/5}-T\right)}{2}$
$=\frac{25\times 25\times }{6}T\left(10^{2/5}-1\right)$
$=\frac{625\times 293\times \left(10^{2/5}-1\right)}{6}$
$=4.033\times 10^3\approx 46.1kJ$