A gas mixture coinsists of (2) moles of oxygen and (4) moles of argon at temperature (T). Neglecting all vibrational modes, the total internal energy of the system is (jee 1999)
(a) 4 RT (b) 15 RT ( c) 9 RT (d) 11 RT.

Explanation

To solve the problem of finding the total internal energy of a gas mixture consisting of 2 moles of oxygen and 4 moles of argon at temperature T, we can follow these steps:

### Step 1: Identify the degrees of freedom for each gas.
- **Oxygen (O2)** is a diatomic gas, which has degrees of freedom given by the formula:
\[ F = 2n + 1 \]
where \( n \) is the number of atoms in the molecule. For O2, \( n = 2 \):
\[ F_{O2} = 2 \times 2 + 1 = 5 \]

- **Argon (Ar)** is a monatomic gas, which has degrees of freedom given by:
\[ F = 2n \]
where \( n \) is the number of atoms. For Ar, \( n = 1 \):
\[ F_{Ar} = 2 \times 1 = 2 \]

### Step 2: Calculate the internal energy for each gas.
The internal energy \( U \) of a gas can be calculated using the formula:
\[ U = \frac{1}{2} n F R T \]
where \( n \) is the number of moles, \( F \) is the degrees of freedom, \( R \) is the universal gas constant, and \( T \) is the temperature.

- **Internal Energy of Oxygen (U_O2)**:
\[ U_{O2} = \frac{1}{2} \times 2 \times 5 \times R \times T = 5RT \]

- **Internal Energy of Argon (U_Ar)**:
\[ U_{Ar} = \frac{1}{2} \times 4 \times 3 \times R \times T = 6RT \]

### Step 3: Find the total internal energy of the system.
The total internal energy \( U_{total} \) of the gas mixture is the sum of the internal energies of the individual gases:
\[ U_{total} = U_{O2} + U_{Ar} = 5RT + 6RT = 11RT \]

### Conclusion:
Thus, the total internal energy of the system is:
\[ \boxed{11RT} \]