A gas mixture consists 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

Options

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

Explanation

To find the total internal energy of the gas mixture consisting of 2 moles of oxygen and 4 moles of argon at temperature T, we will use the formula for the internal kinetic energy of ideal gases. Here’s the step-by-step solution:

### Step 1: Identify the number of moles and the type of gases
We have:
- 2 moles of oxygen (O₂)
- 4 moles of argon (Ar)

### Step 2: Determine the degrees of freedom for each gas
- For diatomic gases like oxygen (O₂), the degrees of freedom (F) is given by:
\[ F = 2N + 1 \]
where N is the number of atoms in the molecule. For O₂, N = 2, so:
\[ F_{O_2} = 2 \times 2 + 1 = 5 \]

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

### Step 3: Calculate the internal kinetic energy for each gas
The internal kinetic energy (U) for an ideal gas can be expressed as:
\[ U = \frac{1}{2} N F R T \]
where N is the number of moles, F is the degrees of freedom, R is the gas constant, and T is the temperature.

- For oxygen (2 moles):
\[ U_{O_2} = \frac{1}{2} \times 2 \times 5 \times R \times T = 5RT \]

- For argon (4 moles):
\[ U_{Ar} = \frac{1}{2} \times 4 \times 3 \times R \times T = 6RT \]

### Step 4: Calculate the total internal energy of the gas mixture
Now, we sum the internal energies of both gases:
\[ U_{total} = U_{O_2} + U_{Ar} = 5RT + 6RT = 11RT \]

### Final Answer
The total internal energy of the system is:
\[ U_{total} = 11RT \]