Question

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

• A. 11 RT
• B. 4 RT
• C. 8 RT
• D. 16 RT

Answer 1

The Correct Option is A

To find the total internal energy of the gas mixture, we first need to understand the degrees of freedom each gas contributes and apply the equipartition theorem. We have a mixture of oxygen and neon gases.

1. **Degrees of Freedom and Molar Specific Heat**:

• Oxygen (\(O_2\)) is a diatomic gas, which typically has 5 degrees of freedom (3 translational + 2 rotational) at moderate temperatures. The molar specific heat at constant volume \(C_{v,\text{O}_2}\) for a diatomic gas is: C_{v,\text{O}_2} = \frac{5}{2}R.
• Neon is a monoatomic gas, which has 3 degrees of freedom (all translational). The molar specific heat at constant volume \(C_{v,\text{Ne}}\) for a monoatomic gas is: C_{v,\text{Ne}} = \frac{3}{2}R.

2. **Calculate the Total Internal Energy**:

The internal energy \(U\) of a gas is given by the formula: U = nC_vT where \(n\) is the number of moles, \(C_v\) is the molar specific heat at constant volume, and \(T\) is the temperature.

• For Oxygen: U_{\text{O}_2} = 2 \times \frac{5}{2}RT = 5RT
• For Neon: U_{\text{Ne}} = 4 \times \frac{3}{2}RT = 6RT

3. **Total Internal Energy of the System**:

Combining the energies: U_{\text{total}} = U_{\text{O}_2} + U_{\text{Ne}} = 5RT + 6RT = 11RT.

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