Question

An engine takes in $5$ moles of air at ${20}^{0}C$ and 1atm, and compresses it adiabatically to $\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$10$}\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 $XkJ$. The value of $X$ to the nearest integer is ______.

Answer 1

Step1: Given data.

For a diatomic ideal gas, the degree of freedom, $f=5$

The ratio of specific heat of gas,$\gamma =1+\frac{2}{f}=\frac{7}{5}$

Initial volume, $V_i=V$

Final volume, $V_f=\frac{V}{10}$

Initial temperature, $T_i={20}^{0}C=293K$

Step2: Finding the change in internal energy.

Now, for adiabatic change

$T{V}^{\gamma -1}=$Constant

Therefore

$T_i{V}_i^{\gamma -1}=T_f{V}_f^{\gamma -1}$

$\Rightarrow$$293\times V^{\frac{7}{5}-1}=T_f{\left(\frac{V}{10}\right)}^{\frac{7}{5}-1}$

$\Rightarrow$ $T_f=293\times {10}^{\frac{2}{5}}K$

Again,

Change in internal energy

$∆U=n{C}_v\left(T_f-T_i\right)$

Where $n$ is the number of mole of gas and $C_v$ is constant gas volume

$\Rightarrow$$∆U=5\times \frac{5}{2}\times 8.324\times \left(293\times {10}^{\frac{2}{5}}-293\right)$

$\Rightarrow$$∆U=46$

Thus, the change in its internal energy during this process comes out to be $46kJ$

Hence, the value of X to the nearest integer is $46$.