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 use the formula for the internal energy of each gas and then sum them up. The internal energy (U) of an ideal gas can be calculated using the formula:

\[ U = n \cdot \frac{F}{2} R T \]

where:
- \( n \) = number of moles of the gas
- \( F \) = degrees of freedom of the gas
- \( R \) = universal gas constant
- \( T \) = absolute temperature

### Step-by-step Solution:

1. **Identify the gases and their moles:**
- Helium (He): 2 moles
- Oxygen (O₂): 4 moles
- Ozone (O₃): 1 mole

2. **Determine the degrees of freedom (F) for each gas:**
- Helium (He) is a monoatomic gas, so \( F = 3 \).
- Oxygen (O₂) is a diatomic gas, so \( F = 5 \).
- Ozone (O₃) is a triatomic gas, so \( F = 6 \).

3. **Calculate the internal energy for each gas:**
- For Helium (He):
\[ U_{He} = n \cdot \frac{F}{2} R T = 2 \cdot \frac{3}{2} R T = 3 R T \]

- For Oxygen (O₂):
\[ U_{O_2} = n \cdot \frac{F}{2} R T = 4 \cdot \frac{5}{2} R T = 10 R T \]

- For Ozone (O₃):
\[ U_{O_3} = n \cdot \frac{F}{2} R T = 1 \cdot \frac{6}{2} R T = 3 R T \]

4. **Sum the internal energies of all gases:**
\[ U_{total} = U_{He} + U_{O_2} + U_{O_3} = 3 R T + 10 R T + 3 R T = 16 R T \]

5. **Final Result:**
The internal energy of the mixture is:
\[ U_{total} = 16 R T \]