Question

A mixture of two gases consists of $3$ moles of nitrogen and $6$ moles of neon at temperature $27^{\circ} C$. Neglecting all vibrational modes, the total internal energy of the system is _________$\times 10^{4} J$.
$\left( R =8.32\, J\, mol ^{-1} K ^{-1}\right)$

Answer 1

Internal energy of a gas with $f$ degrees of freedom,
$U =\frac{ f }{2} nRT$
Now, $f _{ N _{2}}=\frac{5}{2}, f _{ Ne }=\frac{3}{2}$
$\therefore U _{\text {total }}= U _{ N _{2}}+ U _{ Ne }$
$=\frac{5}{2} \times 3 \times 8.32 \times(27+273)$
$+\frac{3}{2} \times 6 \times 8.32 \times(27+273)$
$=3 \times 8.32 \times 300 \times\left(\frac{5}{2}+3\right)$
$=4.16 \times 900 \times 11$
$=41184\, J$
$\therefore U_{\text {total }} =4.12 \times 10^{4} J$