The degree of freedom of a linear triatomic gas varies depending on the temperature, typically being 5 at room temperature and increasing to 9 at high temperatures due to the excitation of vibrational modes.
Understanding Degrees of Freedom in Gases
The degree of freedom (DOF) refers to the number of independent ways in which a molecule can move or store energy. For a gas molecule, these motions are broadly categorized into:
- Translational Motion: Movement of the entire molecule from one point to another in 3D space (along X, Y, and Z axes). All molecules, regardless of their structure, possess 3 translational degrees of freedom.
- Rotational Motion: Rotation of the molecule about its center of mass.
- Vibrational Motion: Oscillations of atoms within the molecule relative to each other.
The total number of degrees of freedom for a molecule with N atoms is generally 3N. This 3N accounts for all possible independent movements of each atom in three dimensions.
Linear Triatomic Gas Molecules
A linear triatomic gas molecule consists of three atoms arranged in a straight line, such as carbon dioxide (CO₂) or nitrous oxide (N₂O). This linear arrangement impacts its rotational and vibrational degrees of freedom.
For a linear molecule with N atoms:
- Translational Degrees of Freedom: Always 3 (for movement in x, y, and z directions).
- Rotational Degrees of Freedom: 2 (rotations about two axes perpendicular to the molecular axis. Rotation along the molecular axis itself does not store energy as there is no moment of inertia).
- Vibrational Degrees of Freedom: The remaining degrees of freedom, calculated as 3N - (Translational + Rotational) = 3N - 5.
For a linear triatomic gas (N=3):
Vibrational Degrees of Freedom = (3 * 3) - 5 = 9 - 5 = 4.
Temperature Dependency of Degrees of Freedom
The number of active degrees of freedom depends significantly on the temperature, as higher temperatures provide enough energy to excite different modes of motion.
-
At Room Temperature:
At typical room temperatures, molecules primarily exhibit translational and rotational motion. The energy available is usually not sufficient to excite vibrational modes.- Translational: 3
- Rotational: 2
- Vibrational: 0 (negligible)
- Total Degrees of Freedom = 3 + 2 = 5
-
At High Temperatures:
As temperature increases, molecules gain enough thermal energy to excite their vibrational modes. These vibrations then contribute to the total degrees of freedom.- Translational: 3
- Rotational: 2
- Vibrational: 4 (all possible vibrational modes become active)
- Total Degrees of Freedom = 3 + 2 + 4 = 9
The table below summarizes the degrees of freedom for a linear triatomic gas at different temperature ranges:
| Type of Motion | Contribution to Degrees of Freedom (Linear Triatomic) | At Room Temperature | At High Temperature |
|---|---|---|---|
| Translational | 3 | 3 | 3 |
| Rotational | 2 | 2 | 2 |
| Vibrational | 4 (calculated as 3N - 5) | 0 (negligible) | 4 |
| Total | 3N | 5 | 9 |
Practical Implications
Understanding the degrees of freedom is crucial for calculating a gas's specific heat capacity at constant volume ($C_v$) and constant pressure ($C_p$), as well as the adiabatic index ($\gamma$). Each active degree of freedom contributes approximately $(1/2)kT$ to the internal energy of the molecule, where $k$ is Boltzmann's constant and $T$ is the absolute temperature. This explains why the specific heat capacity of gases increases with temperature, particularly for polyatomic gases, as more vibrational modes become active.
