Solution:

For a gas mixture of oxygen (\(\text{O}_2\)) and argon (\(\text{Ar}\)):

1. **Oxygen (\(\text{O}_2\))** is diatomic, so its internal energy per mole is:
\[
U_{\text{O}_2} = \frac{5}{2} RT
\]
For 2 moles, \( U_{\text{O}_2} = 2 \times \frac{5}{2} RT = 5 RT \).

2. **Argon (\(\text{Ar}\))** is monatomic, so its internal energy per mole is:
\[
U_{\text{Ar}} = \frac{3}{2} RT
\]
For 4 moles, \( U_{\text{Ar}} = 4 \times \frac{3}{2} RT = 6 RT \).

3. **Total internal energy**:
\[
U_{\text{total}} = U_{\text{O}_2} + U_{\text{Ar}} = 5 RT + 6 RT = 11 RT
\]

So, the total internal energy of the system is \( 11 RT \).