Answer to Question #137017 in Molecular Physics | Thermodynamics for Max

Question

Explain degree of freedom through example.

Answer 1

The degree of freedom of a particle is the the number of directions in which it can move freely or the number of coordinates required to specify completely its position and orientation in space.

• A free particle moving along x-axis moves through only one coordinate, so its degree of freedom is one.
• A free particle moving in space can move through 3 coordinates (x, y, z) and so its degree of freedom is three

There are 3 types of degree of freedom (DOF);

1. Translational degree of freedom: This is a gas molecule's ability to move freely in space. A molecule may move in the x, y, and z directions.
2. Rotational degree of freedom: This is the number of unique ways a molecule may rotate to cause a change in orientation. A monatomic gaseous molecule possesses no rotational degrees of freedom, A diatomic molecule possesses two, while a triatomic molecule may possess two or three depending on its structure.
3. Vibrational degree of freedom: This is the number of unique ways the atoms in a gaseous molecule may move relative to one another, such as in bond stretches or bends. Mathematically, Vibrational DOF = 3N − 6 or 3N - 5(for linear molecules). "where N is the number of atoms in the molecule".

NB: Vibrational DOF is not usually considered active at low temperatures

Examples;

• Helium is a mono-atomic gas.

Its translational DOF is 3 because it can move through space in the x, y, or z axis, its rotational DOF is 0 because it can make no rotation that can cause a unique change in orientation, and it has no vibrational DOF because it's monoatomic, making Helium have a total DOF of 3

• Hydrogen gas is a linear diatomic gas.

At room temperature its translational DOF is 3 because it can move through space in the x, y, or z axis, its rotational DOF is 2, and it has no vibrational DOF making it have a total DOF of 5.

However, at a high temperature, it has an additional vibrational DOF of 1(using the formula 3N - 5), making it have a total DOF of 6 instead.

NB: The vibrational DOF of H₂ can be 2, if we're talking about the energy degrees of motion associated with each of the vibrational modes, making H₂ have a total DOF of 7 instead.

• Water is a non linear triatomic gas.

At room temperature, It has a translational DOF of 3 because it can move through space in the x, y, or z axis, its rotational DOF is 3, and it has no vibrational DOF making it have a total DOF of 6.

However, at a high temperature, it has an additional vibrational DOF of 3 (using the formula 3N - 6) $\therefore$ its total DOF is 9 instead