# 9.2 Molecular spectra

 Page 1 / 6
By the end of this section, you will be able to:
• Use the concepts of vibrational and rotational energy to describe energy transitions in a diatomic molecule
• Explain key features of a vibrational-rotational energy spectrum of a diatomic molecule
• Estimate allowed energies of a rotating molecule
• Determine the equilibrium separation distance between atoms in a diatomic molecule from the vibrational-rotational absorption spectrum

Molecular energy levels are more complicated than atomic energy levels because molecules can also vibrate and rotate. The energies associated with such motions lie in different ranges and can therefore be studied separately. Electronic transitions are of order 1 eV, vibrational transitions are of order ${10}^{-2}\phantom{\rule{0.2em}{0ex}}\text{eV},$ and rotational transitions are of order ${10}^{-3}\phantom{\rule{0.2em}{0ex}}\text{eV}\text{.}$ For complex molecules, these energy changes are difficult to characterize, so we begin with the simple case of a diatomic molecule.

According to classical mechanics, the energy of rotation of a diatomic molecule is given by

${E}_{r}=\frac{{L}^{2}}{2I},$

where I is the moment of inertia and L is the angular momentum. According to quantum mechanics, the rotational angular momentum is quantized:

$L=\sqrt{l\left(l+1\right)}\hslash \phantom{\rule{0.2em}{0ex}}\left(l=0,1,2,3\text{,...}\right),$

where l is the orbital angular quantum number. The allowed rotational energy level    of a diatomic molecule is therefore

${E}_{r}=l\left(l+1\right)\frac{{\hslash }^{2}}{2I}=l\left(l+1\right){E}_{0r}\phantom{\rule{0.5em}{0ex}}\left(l=0,1,2,3\text{,...}\right),$

where the characteristic rotational energy of a molecule is defined as

${E}_{0r}=\frac{{\hslash }^{2}}{2I}.$

For a diatomic molecule, the moment of inertia with reduced mass $\mu$ is

$I=\mu {r}_{0}^{2},$

where ${r}_{0}$ is the total distance between the atoms. The energy difference between rotational levels is therefore

$\text{Δ}{E}_{r}={E}_{l+1}-{E}_{l}=2\left(l+1\right)\phantom{\rule{0.2em}{0ex}}{E}_{0r}.$

A detailed study of transitions between rotational energy levels brought about by the absorption or emission of radiation (a so-called electric dipole transition    ) requires that

$\text{Δ}l=±1.$

This rule, known as a selection rule    , limits the possible transitions from one quantum state to another. [link] is the selection rule for rotational energy transitions. It applies only to diatomic molecules that have an electric dipole moment. For this reason, symmetric molecules such as ${\text{H}}_{2}$ and ${\text{N}}_{2}$ do not experience rotational energy transitions due to the absorption or emission of electromagnetic radiation.

## The rotational energy of hcl

Determine the lowest three rotational energy levels of a hydrogen chloride (HCl) molecule.

## Strategy

Hydrogen chloride (HCl) is a diatomic molecule with an equilibrium separation distance of 0.127 nm. Rotational energy levels depend only on the momentum of inertia I and the orbital angular momentum quantum number l (in this case, $l=0$ , 1, and 2). The momentum of inertia depends, in turn, on the equilibrium separation distance (which is given) and the reduced mass, which depends on the masses of the H and Cl atoms.

## Solution

First, we compute the reduced mass. If Particle 1 is hydrogen and Particle 2 is chloride, we have

$\mu =\frac{{m}_{1}{m}_{2}}{{m}_{1}+{m}_{2}}=\frac{\left(1.0\phantom{\rule{0.2em}{0ex}}\text{u}\right)\left(35.4\phantom{\rule{0.2em}{0ex}}\text{u}\right)}{1.0\phantom{\rule{0.2em}{0ex}}\text{u}+35.4\phantom{\rule{0.2em}{0ex}}\text{u}}=0.97\phantom{\rule{0.2em}{0ex}}\text{u}=0.97\phantom{\rule{0.2em}{0ex}}\text{u}\left(\frac{931.5\frac{\text{MeV}}{{c}^{2}}}{1\phantom{\rule{0.2em}{0ex}}\text{u}}\right)=906\frac{\text{MeV}}{{c}^{2}}.$

The corresponding rest mass energy is therefore

$\mu {c}^{2}=9.06\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{8}\phantom{\rule{0.2em}{0ex}}\text{eV}.$

This allows us to calculate the characteristic energy:

${E}_{0r}=\frac{{\hslash }^{2}}{2I}=\frac{{\hslash }^{2}}{2\left(\mu {r}_{0}^{2}\right)}=\frac{{\left(\hslash c\right)}^{2}}{2\left(\mu {c}^{2}\right){r}_{0}^{2}}=\frac{{\left(197.3\phantom{\rule{0.2em}{0ex}}\text{eV}·\text{nm}\right)}^{2}}{2\left(9.06\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{8}\phantom{\rule{0.2em}{0ex}}\text{eV}\right){\left(0.127\phantom{\rule{0.2em}{0ex}}\text{nm}\right)}^{2}}=1.33\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-3}\phantom{\rule{0.2em}{0ex}}\text{eV}\text{.}$

(Notice how this expression is written in terms of the rest mass energy. This technique is common in modern physics calculations.) The rotational energy levels are given by

${E}_{r}=l\left(l+1\right)\frac{{\hslash }^{2}}{2I}=l\left(l+1\right){E}_{0r},$

where l is the orbital quantum number. The three lowest rotational energy levels of an HCl molecule are therefore

$l=0;{E}_{r}=0\phantom{\rule{0.2em}{0ex}}\text{eV}\phantom{\rule{0.2em}{0ex}}\left(\text{no rotation}\right),$
$l=1;{E}_{r}=2\phantom{\rule{0.2em}{0ex}}{E}_{0r}=2.66\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-3}\phantom{\rule{0.2em}{0ex}}\text{eV},$
$l=2;{E}_{r}=6\phantom{\rule{0.2em}{0ex}}{E}_{0r}=7.99\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-3}\phantom{\rule{0.2em}{0ex}}\text{eV}.$

## Significance

The rotational spectrum is associated with weak transitions (1/1000 to 1/100 of an eV). By comparison, the energy of an electron in the ground state of hydrogen is $-13.6\phantom{\rule{0.2em}{0ex}}\text{eV}$ .

A Pb wire wound in a tight solenoid of diameter of 4.0 mm is cooled to a temperature of 5.0 K. The wire is connected in series with a 50-Ωresistor and a variable source of emf. As the emf is increased, what value does it have when the superconductivity of the wire is destroyed?
how does colour appear in thin films
in the wave equation y=Asin(kx-wt+¢) what does k and w stand for.
derivation of lateral shieft
hi
Imran
total binding energy of ionic crystal at equilibrium is
How does, ray of light coming form focus, behaves in concave mirror after refraction?
Sushant
What is motion
Anything which changes itself with respect to time or surrounding
Sushant
good
Chemist
and what's time? is time everywhere same
Chemist
No
Sushant
how can u say that
Chemist
do u know about black hole
Chemist
Not so more
Sushant
DHEERAJ
Sushant
But ask anything changes itself with respect to time or surrounding A Not any harmful radiation
DHEERAJ
explain cavendish experiment to determine the value of gravitational concept.
Cavendish Experiment to Measure Gravitational Constant. ... This experiment used a torsion balance device to attract lead balls together, measuring the torque on a wire and equating it to the gravitational force between the balls. Then by a complex derivation, the value of G was determined.
Triio
For the question about the scuba instructor's head above the pool, how did you arrive at this answer? What is the process?
as a free falling object increases speed what is happening to the acceleration
of course g is constant
Alwielland
acceleration also inc
Usman
which paper will be subjective and which one objective
jay
normal distributiin of errors report
Dennis
normal distribution of errors
Dennis
acceleration also increases
Jay
there are two correct answers depending on whether air resistance is considered. none of those answers have acceleration increasing.
Michael
Acceleration is the change in velocity over time, hence it's the derivative of the velocity with respect to time. So this case would depend on the velocity. More specifically the change in velocity in the system.
Big
photo electrons doesn't emmit when electrons are free to move on surface of metal why?
What would be the minimum work function of a metal have to be for visible light(400-700)nm to ejected photoelectrons?
give any fix value to wave length
Rafi
40 cm into change mm
40cm=40.0×10^-2m =400.0×10^-3m =400mm. that cap(^) I have used above is to the power.
Prema
i.e. 10to the power -2 in the first line and 10 to the power -3 in the the second line.
Prema
there is mistake in my first msg correction is 40cm=40.0×10^-2m =400.0×10^-3m =400mm. sorry for the mistake friends.
Prema
40cm=40.0×10^-2m =400.0×10^-3m =400mm.
Prema
this msg is out of mistake. sorry friends​.
Prema
what is physics?
why we have physics
because is the study of mater and natural world
John
because physics is nature. it explains the laws of nature. some laws already discovered. some laws yet to be discovered.
Yoblaze
physics is the study of non living things if we added it with biology it becomes biophysics and bio is the study of living things tell me please what is this?
tahreem
physics is the study of matter,energy and their interactions
Buvanes
all living things are matter
Buvanes
why rolling friction is less than sliding friction
tahreem
thanks buvanas
tahreem