1. Internal energy of an ideal gas mixture

For an ideal gas, internal energy is the sum over all components:

$U = \sum n_i \left(\frac{f_i}{2}RT\right)$

where $f_i$ is the number of active degrees of freedom.

2. For oxygen ($O_2$)

Oxygen is a diatomic gas with rigid bond.

• Translational degrees of freedom $=3$
• Rotational degrees of freedom $=2$
• Vibrational modes are neglected because the bond is rigid

So,

$f_{O_2}=5$

For $3$ moles of oxygen,

$U_{O_2}=3\cdot \frac{5}{2}RT=\frac{15}{2}RT$

3. For argon (Ar)

Argon is a monoatomic gas.

• Translational degrees of freedom $=3$

So,

$f_{Ar}=3$

For $5$ moles of argon,

$U_{Ar}=5\cdot \frac{3}{2}RT=\frac{15}{2}RT$

4. Total internal energy

$U=U_{O_2}+U_{Ar}$

$U=\frac{15}{2}RT+\frac{15}{2}RT=15RT$

Thus, in units of $RT$, the total internal energy is:

$\boxed{15}$

5. Checking options

• A: $11$ ❌
• B: $20$ ❌
• C: $15$ ✅
• D: $13$ ❌

So the correct option is C.