What will be the degrees of freedom for triatomic linear molecule?

Answer 1

A tri-atomic molecule should have 3 vibrational degrees of freedom (one for each "end" atom vibrating on its bond with the central atom and one for the flexing of the bonds like scissors opening and closing). If it is non-linear, it should also have a three rotational degrees of freedom. All molecules (including a triatomic one) will have 3 degrees of freedom for translational motion. All totaled, it will have 3+3+3 = 9 degrees of freedom. Note that this does not address the question of independence of the degrees of freedom - for example - if the two "end" atoms are identical, not all the rotational degrees of freedom are independent.

Answer 2

There are many triatomic molecules and they do not all have the same number of degrees of freedom. The answer can be 6,7,8 or 9.

Here is how you figure it out.

If you have N individual unconnected atoms, then you have 3N degrees of freedom; three translational degrees of freedom; x, y, z; or the three spatial dimensions, however you want to say it. The keyword here is "freedom," freedom to move. They have three translational degrees of freedom.

If you impose a constrain (or more accurately, if Mother Nature imposes a constrain), then you lose freedom.

Triatomic molecules have three atoms (9 degrees of freedom) and two bonds (constraints subtract degrees of freedom). You can fix one bond or two or you can fix the bond angle and so, lose one, two or three degrees of freedom.

Explanation:

Two atoms:

(a) If you have two atoms and no constraint, then 2 times 3 is 6 and you could have six translational degrees of freedom.

(b) A constraint can remove one degree of freedom if it is fixing a distance, like a bond length. If the two atoms form a dumbell molecule with a fixed bond length, then you have one length constraint and 6 minus 1 leaves 5 degrees of freedom, three for the molecular translational motion and two for molecular rotational motion.

(c) All bonds are not constraints. If the bond is not rigid, the the two atoms can vibrate. Vibration is motion along the line of the coordinate which is the separation distance of the atoms and the length the bond. That is not a constraint then, just a coordinate and it allows motion and motion is freedom, so you don't subtract it. Then, your 6 degrees of freedom are 3 for molecular translation, 2 for molecular rotation and 1 for molecular vibration.

Three atoms:

(a) If you have three atoms and no constraint, then 3 times 3 is 9 and you could have six translational degrees of freedom. If you have three atoms, in one molecule, then you have two bonds and a bond angle. If those bonds are not rigid and if the bond angle is not rigid, then you still allow motion and the bond lenths and angles are coordinates along which motion occurs. You still keep the 9 degrees of freedom, but they get harder to name. They are 3 orientational degrees of freedom (Check out Euler angles), three molecular translational degrees of freedom, one bond angle degree of freedom and two bond lentgh degrees of freedom. (3+3+2+1=9)

(b) If any of those degrees of freedom are then fixed, you have a constraint and you lose that degree of freedom. You can fix one bond or two or you can fix the angle and so, lose one, two or three degrees of freedom.