Question

A gas mixture consists of 2 moles of O₂ and 4 moles of Ar at temperature T. Neglecting all vibrational modes, the total internal energy of the system is

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

Answer 1

The Correct Option is D

To determine the total internal energy of the gas mixture, we need to consider the degrees of freedom and energy contributions from each gas component under the given conditions.

Understanding the components:

• The gas mixture consists of 2 moles of O₂ (oxygen) and 4 moles of Ar (argon) both at temperature T.
• Oxygen (O₂) is a diatomic molecule that, at room temperature, contributes translational and rotational energy. Vibrational modes are neglected according to the problem statement.
• Argon (Ar) is a monoatomic gas and contributes only translational energy as it has no rotational or vibrational energy.

Degrees of Freedom:

• Diatomic Gas (O₂): Has 5 degrees of freedom at typical room temperature (3 translational + 2 rotational) without vibrational modes.
• Monoatomic Gas (Ar): Has 3 degrees of freedom (only translational).

Internal Energy Calculations:

• The formula for the internal energy of a gas is given by: U = \frac{f}{2} \cdot nRT, where f is the degrees of freedom, n is the number of moles, R is the ideal gas constant, and T is the temperature.

For the 2 moles of O₂:

• f = 5 (for diatomic oxygen)
• Internal energy: U_{\text{O}_2} = \frac{5}{2} \times 2 \times RT = 5RT

For the 4 moles of Ar:

• f = 3 (for monoatomic argon)
• Internal energy: U_{\text{Ar}} = \frac{3}{2} \times 4 \times RT = 6RT

Total Internal Energy:

By adding the internal energies of both gases, we get:

• U_{\text{total}} = U_{\text{O}_2} + U_{\text{Ar}} = 5RT + 6RT = 11RT

Therefore, the total internal energy of the gas mixture is 11RT.

Conclusion: The correct answer is 11RT.