A gas mixture consists of $2$ moles of oxygen and $4$ moles of neon at temperature $T$. Neglecting all vibrational modes, the total internal energy of the system will be:

The internal energy of a gas is given by $U=n{C}_{V}T$.

A diatomic gas has $3$ translational and $2$ rotational degrees of freedom. A monoatomic gas has $3$ translational degrees of freedom.

Hence, the internal energy of the mixture $=n_1{C}_{V_1}T+n_2{C}_{V_2}T$

$$\begin{gathered} =2\times \left(\frac{5R}{2}\right)T+4\times \frac{3R}{2}T \\ =5RT+6RT \end{gathered}$$

$=11RT$