# Estimate vibrational frequency of N2 molecule

• Sam J
In summary: For example, if you have a molecule with a rotational energy of 120 degrees, then you can have an effective number of dofs that is between 0 and 119.

## Homework Statement

Experimental data for the heat capacity of N2 as a function of temperature are provided.

Estimate the frequency of vibration of the N2 molecule.

## Homework Equations

Energy of harmonic oscillator = (n+1/2)ħω

C=7/2kB

Average molecular energy = C*T

But this is an expression for the total energy of a molecule.
Presumably the amount of energy within the two vibrational modes is:

E=C*T

with

C=kB

## The Attempt at a Solution

All I can think to do is equate the two expressions for the energy of the oscillator:

kBT=(n+1/2)ħω

But I have no idea where to go now. In order to find frequency I would want to solve for ω, from which frequency can be trivially determined. However, how am I to know which integer value to use for n?

I am also very unsure as to my construction of the problem.

Any help/guidance greatly appreciated.

Sam J said:
Experimental data for the heat capacity of N2 as a function of temperature are provided.
What does this curve look like?

DrClaude said:
What does this curve look like?

T (Kelvin) 170 500 770 1170 1600 2000 2440
C/NkB 2.5 2.57 2.76 3.01 3.22 3.31 3.4

Apologies, not sure how to post a table of data.

Clearly, the heat capacity is not constant, whereas you expect it to be of the form C = f kB T / 2, where f is the number of quadratic degrees of freedom. What could be happening here?

You access rotational energy as well? So you have more degrees of freedom? But this doesn't answer the question "However, how am I to know which integer value to use for n?'...

Astrolover said:
You access rotational energy as well? So you have more degrees of freedom? But this doesn't answer the question "However, how am I to know which integer value to use for n?'...
n is not necessarily an integer. There can be situations where you get an effective number of dofs that is not a whole number.

## 1. What is the equation used to estimate the vibrational frequency of a N2 molecule?

The equation used is v = 1/2π * (k/m)^1/2, where v is the vibrational frequency, k is the force constant, and m is the reduced mass of the molecule.

## 2. How is the force constant of a N2 molecule determined?

The force constant can be determined experimentally through methods such as infrared spectroscopy or theoretical calculations using quantum mechanics.

## 3. What is the reduced mass of a N2 molecule?

The reduced mass is the mass used in the calculation of a molecule's vibrational frequency. For a diatomic molecule like N2, it is equal to half of the sum of the masses of the two atoms.

## 4. Can the estimated vibrational frequency of a N2 molecule be affected by external factors?

Yes, external factors such as temperature and pressure can affect the estimated vibrational frequency of a N2 molecule. This is because they can alter the force constant and the reduced mass of the molecule.

## 5. Is the estimated vibrational frequency of a N2 molecule constant?

No, the vibrational frequency of a N2 molecule can vary depending on the specific conditions it is in, such as temperature and pressure. It can also be affected by any changes in the molecular structure or bonding between the two atoms.