Degrees of Freedom of Diatomic Molecules and Energy Equipartition

The location of any particle is within three-dimensional space. The direction in which a particle moves is described in terms of three variables-often X, Y, and Z. Particles thus have three translational “degrees of freedom.” As Ken Koehler of the University of Cincinnati informs us, atoms are considered single points not having size, so there are no additional degrees of freedom to be considered for the purposes of this discussion.

Degree of Freedom of Diatomic Molecules

Although it might seem tempting to assume there are only the three translational degrees of freedom for all particles, this would not be true. Some particles are not singular, but are made up of multiple, interrelated components. Take for example, hydrogen molecules. Although hydrogen gas is made up of only one kind of atom, hydrogen molecules consist of two atoms, joined by a chemical bond. This modifies the degrees of freedom of hydrogen particles considerably.

Rotational Degrees of Freedom

The standard model of diatomic molecules resembles a dumbbell with a stiff spring in between the two atoms. The motion of these molecules can still be described by three translational degrees of freedom. There are, however, two rotational degrees of freedom. Holding a pencil horizontally and visualizing two ways the pencil can rotate enables one to see rotation can occur in a clockwise or counterclockwise vertical manner, or in a clockwise or counterclockwise manner horizontally.

Vibrational Degrees of Freedom

There is yet another variety of degree of freedom for diatomic molecules. Since the bond is like a rigid spring, it can be stretched or compressed along the axis of that spring or chemical bond. By visualizing rapid stretching-compressing-stretching-compressing-it can be seen why these degrees of freedom are “vibrational.”

Why is it said “these degrees,” plural, rather than “this degree,” singular? Because if one atom vibrates in one direction, the other atom can either vibrate in the same direction, or in the opposite direction. Thus, the total degrees of freedom describing the motion of a diatomic molecule is not three, for translation only, but seven-two degrees for rotation and two for vibration (some sources, e.g. Charles Kittel, cite only one vibratory degree of freedom kinetically, but include an equivalent amount of energy said to represent a potential energy contribution).

Degrees of Freedom are a Visual and Quantitative Tool

Knowledge of degrees of freedom imparts better understanding of atomic processes. If, for example, the temperature of hydrogen gas is increased, this means there is a corresponding increase of motion of the molecules. Distribution of the energy producing that motion occurs not only to the three translational degrees of freedom, making the molecules move through space more rapidly, but also to the two rotational and two vibrational ones, factors that might not otherwise have been visualized. Since, as Georgia State University points out, equipartition of energy requires each degree of freedom receives an equal portion of the energy, three-sevenths of the energy goes into translation, two-sevenths into rotation, and two-sevenths into vibration. It should added that the degrees of freedom and their visualization are more complicated, the larger the molecular species becomes; however, the principles involved remain the same.

References & Resources

Takada, Kenjiro. 2004 ” Molecular Motions and Heat Capacities ” In Microscopic World – 1 – Mysteries of the Atomic World Kyushu University Accessed June 2010.

UNSW – “Degrees of Freedom and Equipartition” Retrieved June 2010.

Gordon, Prof. Keith. University of Otago – Chem 306, Module 4 – Raman/IR Spectroscopy Accessed June 2010.

Note: Originally published at Suite 101 by author.

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