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

Answer

To solve the problem of finding the total internal energy of a gas mixture consisting of 2 moles of oxygen (O₂) and 4 moles of argon (Ar) at temperature T, we can follow these steps:

### Step 1: Identify the Degrees of Freedom
For a diatomic gas like oxygen (O₂), the degrees of freedom are:
- Translational: 3
- Rotational: 2
- Vibrational: Neglected in this case.

Thus, the total degrees of freedom for O₂ = 3 (translational) + 2 (rotational) = 5.

For a monatomic gas like argon (Ar), the degrees of freedom are:
- Translational: 3
- Rotational: 0
- Vibrational: Neglected in this case.

Thus, the total degrees of freedom for Ar = 3 (translational).

### Step 2: Calculate the Internal Energy for Each Gas
The internal energy (U) of an ideal gas can be calculated using the formula:
\[ U = \frac{f}{2} nRT \]
where:
- \( f \) is the degrees of freedom,
- \( n \) is the number of moles,
- \( R \) is the universal gas constant (approximately 8.314 J/(mol·K)),
- \( T \) is the temperature in Kelvin.

#### For Oxygen (O₂):
- Number of moles (n) = 2
- Degrees of freedom (f) = 5

Using the formula:
\[ U_{O_2} = \frac{5}{2} \times 2 \times R \times T = 5RT \]

#### For Argon (Ar):
- Number of moles (n) = 4
- Degrees of freedom (f) = 3

Using the formula:
\[ U_{Ar} = \frac{3}{2} \times 4 \times R \times T = 6RT \]

### Step 3: Calculate the Total Internal Energy of the Mixture
Now, we can find the total internal energy of the gas mixture by adding the internal energies of both gases:
\[ U_{total} = U_{O_2} + U_{Ar} \]
\[ U_{total} = 5RT + 6RT = 11RT \]

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

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