(II) The equilibrium separation of H atoms in the molecule is 0.074 nm (Fig. 29-8). Calculate th

(II) The equilibrium separation of H atoms in the molecule...

Key Concepts

Rotational Energy Levels

In a diatomic molecule, the rotational motion is quantized, meaning that the molecule can only possess certain allowed rotational energy values. These energy levels are typically given by the expression E? = (?²/2I)l(l+1), where l is the rotational quantum number and I is the moment of inertia of the molecule. Understanding these discrete energy levels is fundamental to analyzing rotational spectra and the energy changes associated with rotational transitions.

Moment of Inertia

The moment of inertia for a diatomic molecule is a measure of its resistance to rotational acceleration and depends on the masses of the atoms as well as the bond length. It is calculated using I = ?r², where ? is the reduced mass of the two atoms and r is the bond distance. This parameter is critical for determining the spacing between rotational energy levels and plays a central role in rotational spectroscopy.

Quantum Mechanical Selection Rules

Selection rules in quantum mechanics dictate which transitions between energy levels are allowed. For rotational transitions in diatomic molecules, the primary rule is that the rotational quantum number must change by one unit (?l = ±1). This rule governs the transitions observed in the molecule’s rotational spectrum and helps predict the energies and wavelengths of the photons involved in these transitions.

Photon Energy-Wavelength Relationship

The energy difference between rotational levels is emitted or absorbed as a photon. The relationship between the energy (E) of a photon and its wavelength (?) is given by E = hc/?, where h is Planck's constant and c is the speed of light. This fundamental relation allows one to calculate the wavelength associated with a specific rotational transition from the energy difference between the molecular energy levels.

Transcript

00:01 29 .10, the equilibrium separation of the hydrogens in molecular hydrogen is 0 .074 nanometers.

00:13 So we want to use this information to calculate the energy and wavelengths of the photons that we get out of these rotational transitions of the molecule.

00:27 So what we need is this h squared over 2i, h -bar squared, sorry.

00:40 And since we already know what h -bar is, let's figure out what the moment of inertia is.

00:47 So, you know, our molecule sort of looks something like this.

00:55 This distance is r, and this is the center of mass.

01:01 And since we're considering it, you know, rotating about it center of mass, then, well, first of all, we're going to to have two because we have two point we have a point rotating about some axis and another point rotating about the same axis and they just add and then of course the mass of a hydrogen atom and then the distance squared for one of them because we're counting them both with the two here and so then this just simplifies to one half mh r squared and so then this characteristic rotational energy h bar squared over two i just h bar squared over two times this is mhr squared and so this is 7 .607 times 10 to the negative third electron volts and so further we have that the change in energy between you know some l initial to l final is the initial l times this h far squared over 2 i we just found the value of and then for a photon, its wavelength is hc over delta e.

03:22 And so now using our value for h -bar squared over 2i, we can find, just plug that in to this for each of our l values and also find the wavelength.

03:40 So here our l initial is 1.

03:48 And so just putting that in here, we get the energy...

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