Question

How many normal modes of longitudinal oscillations are there of a carbon dioxide molecule ?

Answer 1

There are three modes of vibrations in CO2 molecule when it goes to the excited state...
They are, 1. Symmetric 2.Asymmetric and 3.Bent mode of vibrations...

Answer 2

Bent mode is not a longitudinal one.

I think, as there are only 2 spring-like bonds, there are only 2 longitudinal modes of vibration.

Answer 3

@Vincent-Lyon.Fr
But the system contains 3 particles. So shouldn,t the number of modes be 3

Answer 4

True!
3 masses mean 3 longitudinal degrees of freedom BUT, as none of the spring is connected to a fixed point, the velocities of the masses are not independent, since the overall momentum must be constant.

So only 2 of the 3 velocities are truly independent, hence 2 degrees of freedom and 2 modes.

If you had :
wall - spring - mass- spring - mass- spring - mass - spring - wall or
wall - spring - mass- spring - mass- spring - mass

you wuld have 3 modes.

But yours is:
mass- spring - mass- spring - mass
with only 2 modes.