Gas Mixture Composition:
The gas mixture consists of 3 moles of oxygen and 5 moles of argon.

Internal Energy:
The total internal energy of a system can be calculated by considering the contributions from translational and rotational modes.

Translational Energy:
The translational energy of a gas molecule is given by the equation E_trans = (3/2) * n * R * T, where n is the number of moles, R is the gas constant, and T is the temperature.

For the mixture, the translational energy of oxygen is given by E_trans_oxygen = (3/2) * 3 * R * T, and the translational energy of argon is given by E_trans_argon = (3/2) * 5 * R * T.

Therefore, the total translational energy of the mixture is E_trans_total = E_trans_oxygen + E_trans_argon = (3/2) * 3 * R * T + (3/2) * 5 * R * T = 15 * R * T.

Rotational Energy:
The rotational energy of a gas molecule depends on its moment of inertia. Oxygen molecule (O2) is linear and has 2 rotational degrees of freedom, while argon (Ar) is a monatomic gas and has 3 rotational degrees of freedom.

The rotational energy of oxygen is given by E_rot_oxygen = (2/2) * n * R * T = n * R * T, and the rotational energy of argon is given by E_rot_argon = (3/2) * n * R * T.

Therefore, the total rotational energy of the mixture is E_rot_total = E_rot_oxygen + E_rot_argon = n * R * T + (3/2) * n * R * T = 2n * R * T.

Total Internal Energy:
The total internal energy of the system is the sum of the translational and rotational energies.

E_total = E_trans_total + E_rot_total = 15 * R * T + 2n * R * T = 15 * R * T + 2 * 8.314 * T = 15 * T + 16.628 * T = 31.628 * T.

Given that the correct answer is 15 * R * T, we can equate this with the calculated value and solve for T.

15 * R * T = 31.628 * T
15 * R = 31.628
R ≈ 2.109

Since R is the gas constant, its value is approximately 8.314 J/(mol·K), not 2.109 J/(mol·K).

Therefore, the given answer of '15' is incorrect.