Chemistry

Catherine E. Housecroft, Edwin C. Constable

Chapter 12

Vibrational and rotational spectroscopies - all with Video Answers

Educators

Chapter Questions

Problem 1

Give equations for the quantized energy levels of the simple harmonic and anharmonic oscillators and show that the energy levels are equally spaced for the harmonic, but not the anharmonic, oscillator.

Tianyu Li

Tianyu Li

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Problem 2

Calculate the reduced masses of (a) an HD molecule, (b) a $^{12} \mathrm{C}-\mathrm{H}$ unit and (c) a $^{13} \mathrm{C}^{16} \mathrm{O}$ molecule. Exact masses: $^{1} \mathbf{H}, 1.01 ;^{2} \mathrm{H}, 2.01 ;^{12} \mathrm{C}$
$12.00 ;^{13} \mathrm{C}, 13.00 ;^{16} \mathrm{O}, 15.99$.

George Bennett

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Problem 3

The force constants for the bonds in CIF, BrF and BrCl are 448,406 and $282 \mathrm{N} \mathrm{m}^{-1}$ respectively. Predict the trend in their bond enthalpies.

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Problem 4

The stretching mode of NO gives rise to an absorption in the IR spectrum at $1903 \mathrm{cm}^{-1}$. What is the force constant of the bond?

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Problem 5

(a) Determine the reduced masses of $\mathrm{H}^{35} \mathrm{Cl}$ and $\mathrm{H}^{37} \mathrm{Cl}$; exact masses $^{1} \mathrm{H}=1.01$
$^{35} \mathrm{Cl}=34.97 ;^{37} \mathrm{Cl}=36.97 .$ (b) If the force
constants of the bonds are the same, is the ratio of the reduced masses sufficient to cause a shift in the IR absorption assigned to the H-Cl stretch? (c) Would you expect to see any chemical differences between $\mathrm{H}^{35} \mathrm{Cl}$ and $\mathbf{H}^{37} \mathbf{C l} ?$

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Problem 6

Refer to Figure $12.6 .$ In what direction does the molecular dipole moment act as the $\mathrm{CO}_{2}$ molecule
(a) stretches asymmetrically and (b) bends? In either case, is a permanent dipole moment generated?

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Problem 7

What shapes are (a) HCN, (b) $\mathrm{CS}_{2},$ (c) OCS and
(d) $\mathrm{XeF}_{2} ?$ Are the symmetric stretching, asymmetric stretching and bending modes of these molecules IR active?

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Problem 8

Rationalize why the values for the carbonyl stretching mode given in Table 12.1 for carboxylic acids and carboxylate ions are generally lower than those of the corresponding aldehydes:

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Problem 9

For each of the following compounds, state the nature of the functional group present and approximately where in the IR spectrum you would expect to observe a characteristic absorption: (a) acetonitrile; (b) butanol; (c) ethylamine; (d) acetone. Use Tables 1.8 and 1.9 to help you.

George Bennett

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Problem 10

Figure 12.17 shows the IR spectra of high-density polythene, dodecane, hexane and hex-l-ene.
(a) Why are the spectra of high-density polythene, dodecane and hexane similar to one another?
(b) What two features of the spectrum of hex-l-ene allow you to assign this spectrum to an alkene rather than to hexane?

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Problem 11

Figure 12.18 shows four spectra labelled (a) to (d). Assign each spectrum to one of the compounds cyclohexane, benzene, toluene and phenol shown in Figure 12.18.

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Problem 12

How could IR spectroscopy be used to distinguish between each of the following pairs of compounds?

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Problem 13

(a) What is the molecular shape of $\mathrm{SO}_{3} ?$ (b) Draw a diagram to show the symmetric stretching mode of vibration of $\mathrm{SO}_{3}$. (c) Is this mode IR active?

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Problem 14

(a) Why would you not expect to observe an IR spectroscopic absorption for the $\mathrm{C} \equiv \mathrm{C}$ stretch of $\mathrm{C}_{2} \mathrm{H}_{2} ?$ (b) Would you expect to observe such an absorption in the IR spectrum of $\mathrm{CH}_{3} \mathrm{CH}_{2} \mathrm{C} \equiv \mathrm{CH} ?$

George Bennett

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Problem 15

The IR spectrum of $\mathrm{H}_{2} \mathrm{O}$ shows bands at 3756 (asymmetric stretch), 3657 (symmetric stretch) and $1595 \mathrm{cm}^{-1}$ (scissoring). In $\mathrm{D}_{2} \mathrm{O},$ the corresponding vibrational modes give rise to absorptions at 2788 2671 and $1178 \mathrm{cm}^{-1} .$ Account for these observations.

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Problem 16

Methanal, $\mathrm{H}_{2} \mathrm{C}=\mathrm{O},$ exhibits an absorption in its IR spectrum at $2783 \mathrm{cm}^{-1}$ assigned to a $\mathrm{C}-\mathrm{H}$ stretching mode. What will be the shift in vibrational wavenumber for this band on going from $\mathrm{H}_{2} \mathrm{C}=\mathrm{O}$ to $\mathrm{D}_{2} \mathrm{C}=\mathrm{O} ?$ (Assume that the force constants of the $\mathrm{C}-\mathrm{H}$ and $\mathrm{C}-\mathrm{D}$ bonds are the same; for exact masses, see problem $12.2 .$ )

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Problem 17

Calculate $I_{B}$ for $\mathrm{H}^{37} \mathrm{Cl}(r=127 \mathrm{pm})$ and comment on its value in relation to that of $\mathrm{H}^{35} \mathrm{Cl}$ (worked example 12.9 ); accurate masses: $^{1} \mathrm{H}=1.008$ $^{37} \mathrm{Cl}=36.97$.

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Problem 18

Find the moments of inertia $I_{A}, I_{B}$ and $I_{C}$ of $^{79} \mathrm{Br}^{19} \mathrm{F}$ (bond length $=176 \mathrm{pm} ;$ accurate
masses: $\left.^{79} \mathrm{Br}=78.92 ;^{19} \mathrm{F}=19.00\right)$.

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Numerade Educator

Problem 19

The moment of inertia, $I_{B},$ of $^{127} \mathrm{I}^{35} \mathrm{Cl}$ is $2.45 \times 10^{-45} \mathrm{kg} \mathrm{m}^{2} .$ What is the I-Cl bond
distance? (Accurate masses: $^{127} \mathbf{I}=126.90$ $\left.^{35} \mathrm{Cl}=34.97 .\right)$

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Problem 20

Explain why a pure rotational spectrum is observed for HCl but not for $\mathrm{H}_{2}$ or for $\mathrm{Cl}_{2}$.

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Problem 21

Which of the following transitions between rotational energy levels are allowed?
(a) $J=2--J=0 ;$ (b) $J=3--J=2$.
(c) $J=2--J=1 ;$ (d) $J=3--J=1$.

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Problem 22

Explain what is meant by (a) the vibrational ground state and (b) the rotational ground state of a diatomic molecule.

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Problem 23

(a) To what transition does the first line in a rotational absorption spectrum of a diatomic molecule correspond? (b) Is this line at the low- or high-wavenumber end of the spectrum? Is this the low- or high-energy end of the spectrum?

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Problem 24

Predict the positions of the lines in the pure rotational spectrum of $^{35} \mathrm{Cl}^{19} \mathrm{F}(r=163 \mathrm{pm})$
corresponding to the transitions $J=1-\bar{J}=0$ $J=2--J=1$ and $J=3--J=2 .$ What
assumption have you made in your calculations? Accurate masses: $\left.^{35} \mathrm{Cl}=34.97 ;^{19} \mathrm{F}=19.00 .\right)$

Tianyu Li

Numerade Educator

Problem 25

(a) Use the VSEPR model (Section 6.6) to rationalize why $\mathrm{SF}_{6}$ has an octahedral structure.
(b) Draw a diagram that illustrates the symmetric stretch of the $\mathrm{SF}_{6}$ molecule. Why is this not infrared active? (c) The IR spectrum of $\mathrm{SF}_{6}$ exhibits two absorptions at 939 and $614 \mathrm{cm}^{-1}$. Suggest what types of vibrational modes may give rise to these absorptions. (It is not necessary to draw detailed diagrams of the modes of vibration.)

Tianyu Li

Numerade Educator

Problem 26

(a) Use the VSEPR model to predict the shape of an $\mathrm{H}_{2}$ Se molecule. (b) How many degrees of vibrational freedom does an $\mathrm{H}_{2}$ Se molecule possess? (c) The IR spectrum of $\mathrm{H}_{2}$ Se exhibits absorptions at 2358,2345 and $1034 \mathrm{cm}^{-1} .$ Explain how these bands arise. (d) Comment on the connections between the answers to parts (a), (b) and (c).

Tianyu Li

Numerade Educator

Problem 27

(a) The IR spectrum of HCN (12.11) exhibits absorptions at 3311,2097 and $712 \mathrm{cm}^{-1}$. Suggest the origins of the bands at 3311 and $2097 \mathrm{cm}^{-1} .$ After making these assignments, deduce which vibrational mode is responsible for the absorption at $712 \mathrm{cm}^{-1}$
$$
\mathrm{H}-\mathrm{C} \equiv \mathrm{N}
$$
(b) The IR spectra of $\mathrm{NH}_{3}$ and $\mathrm{ND}_{3}$ exhibit bands at 3337 and $2420 \mathrm{cm}^{-1},$ respectively assigned to the symmetric stretch. Explain why the absorption occurs at different wavenumbers.

Tianyu Li

Numerade Educator

Problem 28

Given that the rotational constant of $^{12} \mathrm{C}^{16} \mathrm{O}$ is $1.92 \mathrm{cm}^{-1},$ determine (a) the bond length of $\mathrm{CO}$ and (b) the positions of the first two lines in the pure rotational spectrum of $^{12} \mathrm{C}^{16} \mathrm{O} ;$ accurate masses: $^{12} \mathrm{C}=12.00 ;^{16} \mathrm{O}=15.99 .$ (c) To what transitions do these lines correspond?

Tianyu Li

Numerade Educator

Problem 29

Figure 12.19 shows the structure and IR spectrum of the drug cocaine. (a) Suggest assignments for as many spectroscopic
absorptions as you can. (b) How useful would IR spectroscopy be as a method for customs officers to verify the content of a seizure of cocaine? (c) Could this technique assist in distinguishing between cocaine and a "white powder' that is aspirin, $12.12 ?$

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Numerade Educator

Problem 30

Vegetable oils contain fats. These are triglycerides and contain fatty acid chains (see Section 35.4). In the body, triglycerides are metabolized to fatty acids by enzymes called lipases. During the manufacture of oils in the food industry, the following isomerization may occur in unsaturated
fatty acid chains, i.e. those containing a carboncarbon double bond:
Increased concentrations of the $(E)$ -isomer of an unsaturated fatty acid are associated with heart disease. By using a characteristic IR spectroscopic absorption arising from an $(E)-\mathrm{HC}=\mathrm{CH}$ vibrational mode, it is possible to measure the amount of $(E)$ -isomer in a sample of fat or oil. Figure 12.20 shows the structure of a triglyceride containing $(E)-\mathrm{HC}=\mathrm{CH}$ units, and its IR spectrum. The absorption at $966 \mathrm{cm}^{-1}$ arises from an $(E)-\mathrm{HC}=\mathrm{CH}$ vibrational mode.
(a) Assign the absorptions at $\approx 2900 \mathrm{cm}^{-1}$ and at $1730 \mathrm{cm}^{-1}$
(b) What is the relationship between transmittance and absorbance?
(c) What characteristics of the absorption at $966 \mathrm{cm}^{-1}$ make it suitable for quantifying the amount of triglyceride?
(d) Plot a calibration curve using the following data. What law does the graph illustrate?
(e) The IR spectra of some edible oils and fats were recorded and the areas under the $966 \mathrm{cm}^{-1}$ band are tabulated below. By using the calibration graph, determine the $(E)-$ triglyceride content of each foodstuff, and comment on the relative merits of each in terms of healthy eating.

Anand Jangid

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Problem 31

Forensic analysis of fibres makes use of IR spectroscopy. The polymerization of propionitrile (acrylonitrile) is shown below:
Because fibres of polyacrylonitrile possess poor mechanical properties and are difficult to dye, commercial polymers contain up to $15 \%$ of other monomers (comonomers) to give copolymers which exhibit improved properties. Three common comonomers are:
For this problem, refer to the IR spectrum of polyacrylonitrile in Box 12.2
(a) How does the absorption at $2245 \mathrm{cm}^{-1}$ arise?
(b) The intensity of the absorption at $1730 \mathrm{cm}^{-1}$ depends upon whether any one of comonomers shown above is incorporated into the polymer. Why is this?
(c) The absorption at $1370 \mathrm{cm}^{-1}$ arises from a vibrational mode of the $\mathrm{C}-\mathrm{H}$ bonds in the backbone of the polymer. Assign one other band in the spectrum to a $\mathrm{C}-\mathrm{H}$ vibrational mode.
(d) The ratio of absorbances of pairs of absorptions, $A,$ at, 1730 and $2245 \mathrm{cm}^{-1},$ and 1730 and $1370 \mathrm{cm}^{-1}$ for 20 different, colourless acrylonitrile fibres are tabulated on the next page. Make a scatter plot of the data and assess to what extent this method can be used to discriminate between fibres. How might the data be used to assist in a criminal investigation?

Tianyu Li

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