Question

An engine takes in $5\text{moles}$ of air at $20{}^{\circ }\text{C}$ and $1\text{atm}$, and compresses it adiabatically to ${\left(\frac{1}{10}\right)}^{th}$ 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 $X\text{kJ}$. The value of $X$ to the nearest integer is
$(\text{Use}{100}^{\frac{1}{5}}=2.511)$

Answer 1

For adiabatic process, $T{V}^{\gamma -1}=\text{constant}$

$\Rightarrow T_1{V}_1^{\gamma -1}=T_2{V}_2^{\gamma -1}$

Given that,
$T_1=20{}^{\circ }\text{C}+273=293\text{K},V_2=\frac{V_1}{10}\text{and}\gamma =\frac{7}{5}$

$T_1(V_1{)}^{\gamma -1}=T_2{\left(\frac{V_1}{10}\right)}^{\gamma -1}$

$\Rightarrow 293=T_2{\left(\frac{1}{10}\right)}^{\frac{2}{5}}$

$\Rightarrow T_2=293(10{)}^{\frac{2}{5}}=736\text{K}$

$\Delta T=736-293=443\text{K}$

During the process, change in internal energy

$\Delta U=n{C}_V\Delta T$

$\Rightarrow \Delta U=5\times \frac{5}{2}\times 8.3\times 443$

$\Rightarrow \Delta U\approx 46\times {10}^3\text{J}=46\text{kJ}$

$\therefore X=46$.

Hence, $46$ is the correct answer.