A mixture of ideal gases has 2 moles of He, 4 moles of oxygen and 1 mole of ozone at absolute temperature T. The internal energy of mixture is

Options

A. 13 RT
B. 11 RT
C. 16 RT
D. 14 RT

Answer

To find the internal energy of a mixture of ideal gases, we can follow these steps:

### Step 1: Understand the formula for internal energy
The internal energy \( U \) of an ideal gas can be expressed as:
\[ U = n \cdot f \cdot \frac{RT}{2} \]
where:
- \( n \) = number of moles of the gas
- \( f \) = degrees of freedom of the gas
- \( R \) = universal gas constant
- \( T \) = absolute temperature

### Step 2: Identify the gases and their properties
We have the following gases in the mixture:
- Helium (He): 2 moles, monoatomic (f = 3)
- Oxygen (O₂): 4 moles, diatomic (f = 5)
- Ozone (O₃): 1 mole, triatomic (f = 6)

### Step 3: Calculate the internal energy for each gas
1. **For Helium (He)**:
\[ U_{He} = n_{He} \cdot f_{He} \cdot \frac{RT}{2} = 2 \cdot 3 \cdot \frac{RT}{2} = 3RT \]

2. **For Oxygen (O₂)**:
\[ U_{O_2} = n_{O_2} \cdot f_{O_2} \cdot \frac{RT}{2} = 4 \cdot 5 \cdot \frac{RT}{2} = 10RT \]

3. **For Ozone (O₃)**:
\[ U_{O_3} = n_{O_3} \cdot f_{O_3} \cdot \frac{RT}{2} = 1 \cdot 6 \cdot \frac{RT}{2} = 3RT \]

### Step 4: Sum the internal energies of all gases
Now, we can find the total internal energy of the mixture by adding the internal energies of all the gases:
\[ U_{total} = U_{He} + U_{O_2} + U_{O_3} = 3RT + 10RT + 3RT = 16RT \]

### Final Answer
The total internal energy of the mixture is:
\[ U_{total} = 16RT \]