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Chemistry Forums for Students => Physical Chemistry Forum => Topic started by: camariela on April 12, 2006, 11:22:42 PM

Title: Find the moment of inertia and the energy (in kJ/mol) of a nitrogen molecule N2
Post by: camariela on April 12, 2006, 11:22:42 PM

Hello,
I'm confused about the formulas to use and combine to find the moment of inertia and the energy (in kJ/mol) of a nitrogen molecule (N2) in its first excited rotational state.

Thanks
camariela

Title: Re: Moment of Inertia
Post by: Mitch on April 13, 2006, 12:35:48 AM

What's the equation for moment of inertia?

Title: Re: Find the moment of inertia and the energy (in kJ/mol) of a nitrogen molecule
Post by: camariela on April 13, 2006, 12:40:33 AM

I = mr^2

i have the mass, but what do i use for r?

Title: Re: Find the moment of inertia and the energy (in kJ/mol) of a nitrogen molecule N2
Post by: Mitch on April 13, 2006, 12:43:13 AM

The radius of a Nitrogen atom I suppose.

Title: Re: Find the moment of inertia and the energy (in kJ/mol) of a nitrogen molecule N2
Post by: Borek on April 13, 2006, 03:25:57 AM

I would rather go for half a triple bond length (or even better for half a N₂ length). Atomic radius is for free atom and is of no use here.

Title: Re: Find the moment of inertia and the energy (in kJ/mol) of a nitrogen molecule N2
Post by: Mitch on April 13, 2006, 04:16:13 AM

Quote from: Borek on April 13, 2006, 03:25:57 AM

I would rather go for half a triple bond length

I was thinking of rotation down the central axis. Which is the more meaningful?

Title: Re: Find the moment of inertia and the energy (in kJ/mol) of a nitrogen molecule N2
Post by: Borek on April 13, 2006, 04:47:42 AM

Not sure what you mean, but only A variant has any meaning. In B both masses (both nuclei to be precise) lie on the rotation axis and moment of inertia of the particle is zero.

Title: Re: Find the moment of inertia and the energy (in kJ/mol) of a nitrogen molecule
Post by: pantone159 on April 13, 2006, 02:37:49 PM

Yeah, for the B base the only off-axis masses are the electrons and the fact that the nucleus isn't really a point.
Both of these would be some effort to calculcate.

For the A case, the N nuclei masses dominate the calculation, so you can ignore the above complications.

Title: Re: Find the moment of inertia and the energy (in kJ/mol) of a nitrogen molecule
Post by: Borek on April 13, 2006, 02:51:39 PM

Quote from: Mark Kness on April 13, 2006, 02:37:49 PM

Both of these would be some effort to calculcate.

Do you have any idea if anybody ever tried to? AFAIR (but QM and spectroscopy were never my favourite subjects) only A was always taken into account. Assuming electron cloud have some moment of inertia changes in rotational energy could be observable in spectroscopy (although probably for wavelengths comparable with astronomical unit :) ).

Title: Re: Find the moment of inertia and the energy (in kJ/mol) of a nitrogen molecule
Post by: pantone159 on April 13, 2006, 03:54:36 PM

Quote from: Borek on April 13, 2006, 02:51:39 PM

Do you have any idea if anybody ever tried to?

Don't know.

Title: Re: Find the moment of inertia and the energy (in kJ/mol) of a nitrogen molecule
Post by: plu on April 14, 2006, 02:17:23 PM

To calculate the moment of inertia of a diatomic molecule, one must use the formula I = mR², where m is the reduced mass of the molecule (m₁m₂/m₁+m₂) and R is the bond length.

Title: Re: Find the moment of inertia and the energy (in kJ/mol) of a nitrogen molecule N2
Post by: Mitch on April 14, 2006, 04:12:02 PM

You mean half the bond length?

Title: Re: Find the moment of inertia and the energy (in kJ/mol) of a nitrogen molecule
Post by: plu on April 15, 2006, 09:38:17 AM

Actually, no.  R is simply the whole bond length.

Title: Re: Find the moment of inertia and the energy (in kJ/mol) of a nitrogen molecule N2
Post by: Mitch on April 15, 2006, 03:45:20 PM

Good to know. :)

Title: Re: Find the moment of inertia and the energy (in kJ/mol) of a nitrogen molecule
Post by: pantone159 on April 17, 2006, 03:08:35 AM

Quote from: plu on April 15, 2006, 09:38:17 AM

Actually, no.  R is simply the whole bond length.

Using the 'reduced mass' takes care of these halves, as well as counting both nuclei.

Title: Re: Find the moment of inertia and the energy (in kJ/mol) of a nitrogen molecule N2
Post by: Borek on April 17, 2006, 07:22:33 AM

I = ? m_ir_i² = m₁r₁² + m₂r₂²

so we need r₁ and r₂. They can be found as coordinates of mass center, that are described by conditions:

r₁ + r₂ = R
m₁r₁ = m₂r₂

that gives

r₁ = Rm₂/(m₁+m₂)
r₂ = Rm₁/(m₁+m₂)

these must be entered into I formula given above, and when rearranged it gives

I = ?R²

quite interesting exercise if you have not done such things in the last 20 years :)

Mitch, did I told you LaTeX will be handy? ;)