# How to understand the concept of degree of freedom?

I am not well versed in physics and it is being difficult to understand the degrees of freedom of a system.

Question

I know that degrees of freedom are movements in which a system - like a molecule - can store energy. I can easily count translational degrees but is there any schematic way to see rotational and vibrational degrees of freedom?

Example:

A molecule like water has 3 translational degrees of freedom. I can understand this because we can separate into x, y and z. But how do you see vibrational and rotational degrees of this simple molecule?

In principle the degrees of freedom (dof). of a molecule is the sum of the dof of each atom. That is because we can describe the motion of the molecule as a whole as the sum of the motion of all the atoms - vibration, translation, rotation.

So where a molecule with two atoms might need 6 numbers to describe the velocity of the two atoms (x, y and z for each atom) we can instead think of it as the velocity of the center of mass, plus three numbers needed to describe the relative motion of the atoms: one to describe their relative distance (vibration), and two to describe how they move in the plane perpendicular to the axis (which you can describe as two rotations about axes perpendicular to the axis of the molecule).

The same thing can be done for more complex molecules as well - you add three more degrees of freedom for every atom you add to the molecule.

BUT!!! For some molecules (like$\mathrm{N_2}$) the energy needed to excite vibration along the bond is quite large (compared to $\frac12 kT$). This means that the assumption of equipartition of energy cannot be satisfied - energy cannot be "stored" in that vibration. Consequently when you calculate the heat capacity of the gas you find that you have to assume five degrees of freedom, not six. Because the bond is "so stiff" that it is not truly "free".

This diagram (2D because that is simpler to draw...) may help: • The sixth degree of freedom, vibration of the N2 (or O2) molecule, is not excited at room temperature. In other words the two atoms cannot change the distance between them. With the distance fixed there are only 5 numbers needed to describe the motion of the two atoms. That is the definition of "degrees of freedom", and it means that the heat capacity is calculated from 5, not 6 dof ($\frac52$R instead of 3R per mole) – Floris Oct 8 '17 at 18:13
• But degree of freedom is related to velocity or position or both? – user153036 Oct 8 '17 at 18:15
• Velocity is just the derivative of position. If relative position is fixed, so is relative velocity – Floris Oct 8 '17 at 19:04

Are you just missing that rotating or vibrating objects store energy? Spinning objects have energy

$$E=\frac{I\omega^2}{2}$$

and vibrating objects have energy depending on the vibrational mode.

There is an illustration for ethylene here, and some other images.

• No, I don't know how to see the many vibrational and rotational degrees of a system. – user153036 Oct 8 '17 at 11:40
• What do you mean "see"? You've seen macroscopic objects spin/vibrate, yes? Sorry I'm not (yet) clear what your difficulty is – JMLCarter Oct 8 '17 at 11:43
• Yes, the question is not very good. I have edited. But thanks for your reply.. – user153036 Oct 8 '17 at 11:44
• Each bond is an axis about which the atoms may (if free to do so) rotate. Additionally each bond is a "spring" in that force is exerted to retain the inter-atomic distance. – JMLCarter Oct 8 '17 at 11:50
• added a link for you. – JMLCarter Oct 8 '17 at 12:01

A molecule like water has 3 translational degrees of freedom. I can understand this because we can separate into x, y and z. But how do you see vibrational and rotational degrees of this simple molecule?

You "see" these degrees of freedom as varieties and modifications of simple harmonic motion. Physical pictures of a molecule are misleading, but mathematical models and variations of S.H.M are useful.

You can model the vibrations of the atoms as a mass on a spring, and you can model the rotation of the molecule as a mass on a torsion spring, it winds up and then has a return force.

As far as "seeing" these effects physically, we can do this by measuring the distinctive energy levels associated with each particular molecule.

A good read on this is the book: Vibrations and Waves by Main. He extends the simple physical models to a wide variety of questions like your one.