A gas mixture consists of 2 moles of `O_2 ` and 3 moles of Ar at temperature T. Neglecting all vibrational modes, the total internal energy of the system is

Options

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

Answer

To find the total internal energy of the gas mixture consisting of 2 moles of \(O_2\) and 3 moles of \(Ar\) at temperature \(T\), we will use the formula for internal energy based on the degrees of freedom of the gas molecules.

### Step-by-Step Solution:

1. **Identify the Degrees of Freedom:**
- For \(O_2\) (oxygen), which is a diatomic molecule, the degrees of freedom \(F\) are:
- 3 translational (movement in x, y, z directions)
- 2 rotational (rotation about two axes)
- Total \(F_{O_2} = 5\)
- For \(Ar\) (argon), which is a monatomic molecule, the degrees of freedom \(F\) are:
- 3 translational (movement in x, y, z directions)
- Total \(F_{Ar} = 3\)

2. **Calculate the Internal Energy for Each Gas:**
- The internal energy \(U\) for a gas can be calculated using the formula:
\[ U = \frac{F}{2} nRT \]
- For \(O_2\):
\[ U_{O_2} = \frac{5}{2} \times 2 \times R \times T = 5RT \]
- For \(Ar\):
\[ U_{Ar} = \frac{3}{2} \times 3 \times R \times T = \frac{9}{2} RT \]

3. **Calculate the Total Internal Energy:**
- The total internal energy \(U\) of the mixture is the sum of the internal energies of \(O_2\) and \(Ar\):
\[ U = U_{O_2} + U_{Ar} = 5RT + \frac{9}{2}RT \]
- To combine these, convert \(5RT\) to a fraction:
\[ 5RT = \frac{10}{2}RT \]
- Now add:
\[ U = \frac{10}{2}RT + \frac{9}{2}RT = \frac{19}{2}RT \]
- Thus, the total internal energy of the system is:
\[ U = 9.5RT \]

4. **Final Answer:**
- The total internal energy of the gas mixture is \(9.5RT\).