A gas mixture consists of 3 moles of oxygen and 5 moles of argon at temperature T . Assuming the gases to be ideal and the oxygen bond to be rigid, the total internal energy ( in units of RT ) of the mixture is :

Options

A. 11
B. 13
C. 20
D. 15

Answer

To find the total internal energy of the gas mixture consisting of 3 moles of oxygen and 5 moles of argon at temperature T, we will follow these steps:

### Step 1: Determine the degrees of freedom for each gas
- **Oxygen (O₂)** is a diatomic gas. Its degrees of freedom (F) are given by:
\[ F_{\text{O}_2} = 3 \text{ (translational)} + 2 \text{ (rotational)} = 5 \]

- **Argon (Ar)** is a monoatomic gas. Its degrees of freedom are:
\[ F_{\text{Ar}} = 3 \text{ (translational)} = 3 \]

### Step 2: Calculate the internal energy for each gas
The internal energy (U) for an ideal gas can be calculated using the formula:
\[ U = \frac{F}{2} nRT \]
where \( n \) is the number of moles, \( R \) is the gas constant, and \( T \) is the temperature.

#### For Oxygen:
Using the formula for internal energy:
\[ U_{\text{O}_2} = \frac{F_{\text{O}_2}}{2} n_{\text{O}_2} RT = \frac{5}{2} \times 3 \times RT = \frac{15RT}{2} \]

#### For Argon:
Similarly, we calculate the internal energy for argon:
\[ U_{\text{Ar}} = \frac{F_{\text{Ar}}}{2} n_{\text{Ar}} RT = \frac{3}{2} \times 5 \times RT = \frac{15RT}{2} \]

### Step 3: Calculate the total internal energy of the mixture
The total internal energy (U_total) of the mixture is the sum of the internal energies of oxygen and argon:
\[ U_{\text{total}} = U_{\text{O}_2} + U_{\text{Ar}} = \frac{15RT}{2} + \frac{15RT}{2} = 15RT \]

### Conclusion
The total internal energy of the gas mixture is:
\[ U_{\text{total}} = 15RT \]

### Final Answer
Thus, the total internal energy (in units of RT) of the mixture is **15**.

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