# How many vibrations are expected for butatriene and cyclobutadiene?

I have butatriene and cyclobutadiene, the formula for finding the number of vibrations expected are

• Linear: $3N-5$
• Nonlinear: $3N-6$

where $N$ is the number of atoms.

Both of these molecules are $\ce{C4H4}$ so they both have 8 atoms (because I think you include the $\ce{H}$ atoms). Butatriene is nonlinear so the vibrations expected would be $$3N-5 = 3(8)-6 = 18,$$ and Cyclobutadiene is nonlinear so $$3N-6 = 3(8)-6 = 18.$$

Is this correct? Or do you only count $C$ atoms when you do this calculation?
Would all nonlinear $\ce{C4H4}$ isomers have the same number of vibrations expected?

• Butatriene is not linear - the hydrogen atoms are not on the axis formed by the four carbon atoms. – snurden Feb 12 '16 at 19:35
• @snurden Oh your right! I didn't think about that, thank you! do you happen to know whether or not H's are included when calculating the vibrations expected? – Charlene Feb 12 '16 at 19:47
• Yes, they should be included (just as you already did) since they are just regular atoms as all the others are. (Or does water have no vibrations? ;)) – snurden Feb 12 '16 at 19:50

Perhaps an explanation of where these mysterious equations come from will help you understand how to apply them properly.

I only need you to make one, single (correct) assumption: all atoms in a molecule are capable of moving in each of the three Cartesian coordinate directions.

That is, each of those little hydrogens can wiggle around in the x, y, and z coordinates as much as it pleases.

That means that we have a total $3N$ ways that the whole molecule can move because each of $N$ atoms can move in three directions.

Now, let's consider how molecules move. If every one of those atoms happened to be moving in the same direction, what would that look like?

Well, that would be translation of the molecule (i.e. moving). Any molecule can do this in three directions.

So, excluding translation we have $3N-3$ ways of moving around.

Now, another thing we know molecules do is rotate. By the same reasoning, we can say that a molecule can rotate in all three coordinate directions.

This leaves us with $3N-6$ ways to move around.

But, what if that molecule was linear? Well, then that molecule can only rotate in two unique directions because spinning around like a cylinder is known to be a zero energy process. Another way of saying that is that it doesn't happen.

So for a linear molecule there $3N-5$ ways of moving around still because we only subtracted out two rotational modes.

Of course, now we see that the only other way our molecules can move is by vibrating.

Thus, for a linear molecule, $$Number\ of\ normal\ modes=3N-5$$and for a nonlinear polyatomic molecule,$$Number\ of\ normal\ modes=3N-6$$

So, using these equations which now hopefully make some more sense, it's quite obvious that we do indeed include the hydrogens in our $N$ and find there are 18 normal modes of vibration for both molecules.

And, if there were an isomer of the formula $\ce{C4H4}$ which was linear then there would be 19 normal modes, but there is no such molecule so, yes all molecules with the molecular formula $\ce{C4H4}$ have 18 vibrational modes.