The degrees of freedom of a molecule of a triatomic gas are

Options

A. 2
B. 4
C. 6
D. 8

Answer

To determine the degrees of freedom of a molecule of a triatomic gas, we need to consider two cases: linear and nonlinear triatomic molecules.

### Step-by-Step Solution:

1. **Identify the Type of Triatomic Gas**:
- A triatomic gas can either be linear or nonlinear.
- Linear molecules have atoms arranged in a straight line, while nonlinear molecules have a bent or angular structure.

2. **Degrees of Freedom for Linear Triatomic Gas**:
- For a linear triatomic gas, the degrees of freedom can be calculated as follows:
- **Translational Degrees of Freedom**: 3 (movement in x, y, and z directions)
- **Rotational Degrees of Freedom**: 2 (rotation about two perpendicular axes; rotation about the molecular axis does not count)
- **Total for Linear**:
\[ \text{Total Degrees of Freedom} = 3 \text{ (translational)} + 2 \text{ (rotational)} = 5 \]

3. **Degrees of Freedom for Nonlinear Triatomic Gas**:
- For a nonlinear triatomic gas, the degrees of freedom are calculated as follows:
- **Translational Degrees of Freedom**: 3
- **Rotational Degrees of Freedom**: 3 (rotation about three perpendicular axes)
- **Total for Nonlinear**:
\[ \text{Total Degrees of Freedom} = 3 \text{ (translational)} + 3 \text{ (rotational)} = 6 \]

4. **Consideration of Vibrational Modes**:
- At moderate temperatures, vibrational modes are generally not considered. However, at high temperatures, vibrational modes become significant.
- For both types of triatomic gases, the vibrational modes can be added:
- **Vibrational Degrees of Freedom**: 3 (each bond can vibrate)
- Therefore, at high temperatures:
- **Linear Triatomic Gas**:
\[ \text{Total Degrees of Freedom} = 5 + 3 = 8 \]
- **Nonlinear Triatomic Gas**:
\[ \text{Total Degrees of Freedom} = 6 + 3 = 9 \]

5. **Conclusion**:
- The degrees of freedom for a triatomic gas depend on its structure and temperature:
- **Linear Triatomic Gas**: 5 (at moderate temperature), 8 (at high temperature)
- **Nonlinear Triatomic Gas**: 6 (at moderate temperature), 9 (at high temperature)
- Since the problem does not specify the type of triatomic gas, both cases are valid. However, the general assumption is to consider the nonlinear case at moderate temperatures, which gives us a degree of freedom of 6.

### Final Answer:
- The degrees of freedom of a molecule of a triatomic gas are:
- **5 for linear triatomic gas**
- **6 for nonlinear triatomic gas**
- **8 for linear at high temperature**
- **9 for nonlinear at high temperature**