A gas mixture consists of 2 moles of oxygen and 4 of Argon at temperature T. Neglecting all vibrational modes, the total internal energy of the system is

Options

A. 4 RT
B. 9 RT
C. 11 RT
D. 15 RT

Explanation

To find 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 components of the gas mixture
- We have 2 moles of oxygen (O₂) and 4 moles of argon (Ar).

### Step 2: Understand the degrees of freedom for each gas
- For a diatomic gas like oxygen (O₂), the degrees of freedom (F) can be calculated using the formula:
\[ F = 2n + 1 \]
where \( n \) is the number of atoms in the molecule. For O₂, \( n = 2 \):
\[ F_{O_2} = 2(2) + 1 = 5 \]

- For a monatomic gas like argon (Ar), the degrees of freedom is:
\[ F = 3 \]

### Step 3: Write the expression for internal kinetic energy
The internal kinetic energy (U) for a gas can be expressed as:
\[ U = \frac{1}{2} n F R T \]
where:
- \( n \) = number of moles
- \( F \) = degrees of freedom
- \( R \) = universal gas constant
- \( T \) = temperature

### Step 4: Calculate the internal kinetic energy for oxygen
Using the formula for internal kinetic energy:
\[ U_{O_2} = \frac{1}{2} \times 2 \times 5 \times R \times T = 5RT \]

### Step 5: Calculate the internal kinetic energy for argon
Using the same formula:
\[ U_{Ar} = \frac{1}{2} \times 4 \times 3 \times R \times T = 6RT \]

### Step 6: Calculate the total internal energy of the gas mixture
The total internal energy (U_total) is the sum of the internal kinetic energies of both gases:
\[ U_{total} = U_{O_2} + U_{Ar} = 5RT + 6RT = 11RT \]

### Final Answer
The total internal energy of the gas mixture is:
\[ \boxed{11RT} \]