Rotational (Microwave) Spectroscopy

Question 1

Rotation of molecules?

Answer 1

3 axes of rotation, movement of inertia (I) defines energy of rotation
Ia = Ib, Ic about the molecular axis is very small

Question 2

Energy of rotation depends on?

Answer 2

Mass of atoms
Distance between atoms
Angular velocity

Question 3

Selection rules for rotational spectroscopy?

Answer 3

In order to interact with EM radiation, the molecule must possess an electric dipole which can oscillate at the frequency of the radiation – also called a transition moment

Question 4

Gross selection rule?

Answer 4

Molecules must possess a permanent dipole, only heteronuclear diatomics give a pure rotational spectrum

Question 5

Specific selection rule?

Answer 5

Only transitions between adjacent energy levels can occur DeltaJ ± 1

Question 6

Expression for rotational energies?

Answer 6

EJ = BJ(J+1)

Question 7

Lower reduced mass?

Answer 7

A lower reduced mass will give a larger rotational constant, Thus,H2 ,whichhasthe lowest reduced mass of any molecule, will have a large rotational constant

Question 8

High reduced mass or large r?

Answer 8

Molecules with a high reduced mass, or large r have small B constants, rotational levels are not resolved for very large molecules because they are so close together in energy

Question 9

Absorption spectroscopy?

Answer 9

Spectroscopy looks at transitions, we know the energies and the selection rules, we can predict what the spectrum will look like, absorption occurs when the photon energy matches the difference between energy levels

Question 10

Line spacing in rotational (microwave) spectroscopy?

Answer 10

The levels get further apart as J increases, the spectrum therefore consists of a series of equally spaced lines – separation is 2B, measuring B from the spectrum –> calculate the moment of inertia,
knowing m1 and m2 –> calculate the bond length

Question 11

Why are not all the intensities of the lines the same?

Answer 11

We have to look at the occupancy of the levels, need to look at the population and degeneracy

Question 12

What is degeneracy?

Answer 12

Degeneracy – number of levels with exactly the same energy

Question 13

What does the intensity of the absorption peak depend on?

Answer 13

The intensity of the absorption peak depends on the number of molecules that absorb the radiation i.e. the number in the energy level – the population, usually fewer molecules
in higher energy states, more likely to have higher E states populated if deltaE is small

Question 14

Trends in Boltzmann distribution?

Answer 14

As exponential term tends to 0 where n upper = n lower, change in energy is very small or T is very large
If exponential term is large, the negative sign means n upper &laquo_space;n lower, change in energy is very large or T is very small

Question 15

Boltzmann and populations at higher temperature?

Answer 15

At higher temperatures, there is higher population of higher levels

Question 16

Boltzmann and populations when change in energy is large?

Answer 16

If change in energy is large, only the lowest energy levels will have significant populations

Question 17

Boltzmann and populations when change in energy is small?

Answer 17

If change in energy is small, many energy levels will be populated

Question 18

Population trends?

Answer 18

The population falls off exponentially as change in energy increases

Question 19

Implications of the Boltzmann Distribution when the energy gap is large?

Answer 19

Large energy gap - most population will be in the ground state unless the temperature is very high, e.g. at room temperature, most molecules are in their ground electronic state - energy spacing between electronic states is large compared to kBT

Question 20

Implications of the Boltzmann Distribution when the energy gap is small?

Answer 20

Small energy gap - higher levels are populated at moderate temps e.g. at room temperature, there is a significant population of higher rotational levels - energy spacing between rotational states is comparable with (or smaller than) kBT

Question 21

Implications of the Boltzmann Distribution when two levels have the same degeneracy?

Answer 21

For two levels with same degeneracy as T tends to infinity, then populations of upper and lower levels becomes the same, the upper level can never have a higher population than the lower state in a solely thermal distribution

Question 22

How to account for degeneracy?

Answer 22

Also need to account for the degeneracy – number of states with the same energy,
modify Boltzmann with a simple term reflecting this: gupper/glower

Question 23

Example of degenerate orbital?

Answer 23

2p orbitals, 2 electrons in each orbital – all have the same energy

Question 24

Degeneracy of rotational levels?

Answer 24

Rotational energy levels are also degenerate, degeneracy, for a rigid rotor, there will be a number of levels with identical energy

Question 25

Rotational levels?

Answer 25

Rotational levels: (2J + 1) fold degenerate, in this course, the degeneracy factor is only relevant for populations in Rotational spectroscopy

Question 26

What does intensity of absorption peak depend on?

Answer 26

Intensity of the absorption peak depends on how many molecules are in the J’th state,

Question 27

If change in energy is much less than Boltzmann constant?

Answer 27

If change in energy &laquo_space;kBT so the population ratio for these levels is determined largely by the ratio of the degeneracies

Question 28

What happens to the degeneracy as J increases?

Answer 28

The degeneracy increases as J increases; However, the value of change in energy also increases and eventually becomes greater than kBT - the ratio of the populations becomes < 1 at high values of J, overall, this means that the population of molecules is spread throughout many rotational levels

Question 29

Occupancy of levels?

Answer 29

Occupancy of levels rises and
passes through a maximum
– just as observed in the spectra

Question 30

Diatomic effects of isotopes?

Answer 30

Isotopic substitution does not affect electronic binding, and therefore inter-nuclear distance, r unchanged,
change in mass –> change in I
It is the reduced mass that is affected by isotopic substitution

Question 31

Assumption we have made in rotational spectroscopy?

Answer 31

We have assumed a rigid rotor – chemical bonds are not rigid, this modifies the energy, Bond lengths get slightly longer as J increases, separation between peaks decreases – B isn’t a constant, also not all molecules are diatomic

Question 32

How is rotational energy quantised in microwave rotational spectroscopy?

Answer 32

EJ = BJ(J+1) for a rigid diatomic rotor

Question 33

Selection rules for microwave rotational spectroscopy?

Answer 33

Selection rules: permanent dipole in molecule change in J ±1

Question 34

Rotational spectra correspond to?

Answer 34

Rotational spectra correspond to microwave radiation, recorded in gas phase

Question 35

What does analysis of spectrum allow?

Answer 35

Analysis of the spectrum allows very accurate measurement of bond lengths and/or identification of species

Question 36

Important point of a rotational spectrum?

Answer 36

The absorption’s occur at wavenumber up to approximately 100cm-1, there is a number of sharp absorption’s that occur at regular equally space intervals the line intensities start off small get larger as the energy increases by then pass through a maximum before retuning to very small

Question 37

Populations of rotational energy levels?

Answer 37

The energy difference is much less than KbT so the population ratio for these levels is determined largely by the ratio of degeneracies although the degeneracy of the level increases as J increases the value of the change in energy also increases and eventually becomes greater than KbT. Because of the two opposing factors the ratio of the populations of adjacent levels becomes less than 1 at high values of J

Question 38

Why are the peaks equally space in microwave spectra?

Answer 38

Energy is only absorbed when it exactly matches the difference between energy levels, absolutions occur at values of 2hB, 4hB, 6hB and 8hB and so on the rotational spectrum therefore has a series of regularly spaced lines, the energy difference between successive lines is always 2hB

Question 39

Non rigid rotors?

Answer 39

As the energy increases the molecule rotates fast and the bond lengths get slightly longer this is known as centrifugal disruption to account for this an additional term is added to the equation to described the energy of the rotating diatomic molecule more accurately, the effect of centrifugal distortion in the spectrum is that the spacing between the lines gets smaller as J increases, high accuracy measurements of bond lengths need to incorporate this effect but for most purposes treating molecules as rigid rotors is sufficient

Question 40

Why do measurements normally take place in the gas phase?

Answer 40

Since in a solid molecules are not free to rotate and in a liquid or solution collisions occur too frequently for molecules to rotate freely enough to give rise to spectra

Question 41

What is the importance of J in microwave spectroscopy?

Answer 41

J is the rotational quantum number and can make any value from 0 to infinity, EJ is the energy of the Jth level, the lowest energy has J=0 and molecules in this level have zero rotational energy EJ = 0, the next lowest energy level has J=1 and so on. The wave function for J=0 has the same shape as the s orbitals of the hydrogen atom and so is the same in all directions, there are three wave functions for J=1 so the J=1 level is triply degenerate analogous to the three p orbitals of the hydrogen atom, the J=2 level has a degeneracy of 5, analogous of 5d orbitals, in general the degeneracy of level J is (2J + 1)