Vibrational frequency of diatomic molecule

[bull0sees]

Homework Statement

Using specific heat data for a nitrogen molecule, estimate then vibrational frequency of the diatomic molecule

Homework Equations

C= 3N_a k = 3R

The Attempt at a Solution

unable to attempt a solution

 

[shamone]

This question interests me. However I am no help with the solution. I would love if someone could answer this question

 

[bull0sees]

thank you for your input shamone. I would also love to find out the answer

 

[shamone]

I am surprised no one has answered. It looks like a straightforward answer but I just cannot come up with a solution, i wish i was smarter

 

[Andrew Mason]

Homework Statement

Using specific heat data for a nitrogen molecule, estimate then vibrational frequency of the diatomic molecule

Homework Equations

C= 3N_a k = 3R

The Attempt at a Solution

unable to attempt a solution

You could start by treating the diatomic molecule as two masses joined by a spring with a certain spring constant. (this may be correct only for small vibrations). You can express the frequency of vibration using the "spring constant", which is a function of the bond strength, and the mass of the N atom. The trick is to find the "spring constant" from the specific heat. I'll have to think about that one.

AM

 

[Andrew Mason]

I think I understand the problem. The N₂ molecule acts as a harmonic oscillator at a frequency determined by the strength of the bond between the two N atoms and the mass of the N atom. But due to quantum effects, it cannot vibrate at any energy. Its modes of vibration are quantized:

$$E_{vib} = (n + 1/2)h\nu$$

At low energies (low temperature < 500K) the energy of vibration is $h\nu/2$ (n=0). The addition of thermal energy is not sufficient to allow many molecules to reach the next energy level (n=1) which is $3h\nu/2$ (ie the number of molecules in the Boltzmann distribution for that temperature with that amount of energy).

However, as T increases the number of molecules able to acquire additional vibrational energy ie. to jump from $h\nu/2 \text{ to } 3h\nu/2$ increases so the specific heat starts increasing. At about 6000 K the specific heat, Cv reaches 3.5R. This means that the addition of any amount of thermal energy adds vibrational energy to the molecules which, I think, means many of the higher n levels are excited.

That should help you figure out the frequency $\nu$. From that you could figure out the force holding the atoms together, too.

AM