A vessel contains 3 moles of $\mathrm{He}, 1$ mole of Ar, 5 moles of $\mathrm{N}_2$ and 3 moles of…
A vessel contains 3 moles of $\mathrm{He}, 1$ mole of Ar, 5 moles of $\mathrm{N}_2$ and 3 moles of $\mathrm{H}_2$. If the vibrational modes are ignored, the total internal energy of the system of gases is
$20 R T$
$26 R T$
$25 R T$
$30 R T$
Solution
Translational kinetic energy of a gas volume is
$
U=\frac{f}{2} R T
$
Where, $f=$ degrees of freedom of gas molecules. Now, $f=3$ for He or Ar monoatomic and $f=5$ for $\mathrm{H}_2$ or $\mathrm{N}_2$. Diatomic.
Hence, for 3 moles of He, 1 mole Ar, 5 moles of $\mathrm{N}_2$ and 3 moles of $\mathrm{H}_2$.
$
\begin{aligned}
U & =3 \times \frac{3}{2} R T+1 \times \frac{3}{2} R T+5 \times \frac{5}{2} R T+3 \times \frac{5}{2} R T \\
& =\left(\frac{9}{2}+\frac{3}{2}+\frac{25}{2}+\frac{15}{2}\right) R T \\
& =\frac{52}{2} R T=26 R T
\end{aligned}
$