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?
Best Answer
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: