Question 1

Boron trifluoride, BF₃, has a three-fold symmetric planar equilibrium geometry with a B-F bond length of 1.30 Å. Calculate the moment of inertia for rotation about the three-fold axis of symmetry perpendicular to the molecular plane.

Accepted Answer

The moment of inertia (I) for a molecule about an axis is given by $I = \sum m_i r_i^2$, where $m_i$ is the mass of atom $i$ and $r_i$ is the distance of atom $i$ from the axis of rotation. For Boron trifluoride (BF3), the three fluorine atoms are equidistant from the axis of rotation and lie in a plane perpendicular to it. The mass of a fluorine atom is approximately 19 u (atomic mass units), which is $19 \times 1.66 \times 10^{-27}$ kg. The distance from the axis of rotation (the B-F bond length) is 1.30 Å, or $1.30 \times 10^{-10}$ m. Therefore, the moment of inertia for one fluorine atom is $m r^2 = (19 \times 1.66 \times 10^{-27}) \times (1.30 \times 10^{-10})^2$. Since there are three fluorine atoms symmetrically placed around the boron atom, the total moment of inertia is three times this value, leading to the calculation that matches the value given in choice C.

Question 2

The bond length in a nitric oxide molecule, NO, is 1.15 Å. Calculate the rotational constant.

Accepted Answer

D

Question 3

Calculate the angular momentum of a molecule with a rotational quantum number J = 3.

Accepted Answer

The angular momentum of a molecule in quantum mechanics is given by the formula L = √J(J+1) ℏ, where J is the rotational quantum number and ℏ is the reduced Planck constant. For J = 3, L = √3(3+1) ℏ = √12 ℏ.

Question 4

What is the degeneracy of the rotational level with quantum number J = 6 in a linear molecule?

Accepted Answer

The degeneracy of a rotational level in a linear molecule is given by 2J + 1, where J is the rotational quantum number. For J = 6, the degeneracy is 2(6) + 1 = 13.

Question 5

The rotational constant of a hydrogen fluoride, HF, molecule is 3.12 *10¹¹s^-1. Predict which transition in the rotational absorption spectrum will be most intense at a temperature of 400 K.

Accepted Answer

The answer of The rotational constant of a hydrogen fluoride,...

Question 6

For which of the following molecules are pure rotational spectroscopic transitions observed? Benzene, C₆H₆, Sulfur hexafluoride, SF₆, Carbon disulfide, CS₂, Nitrous oxide, N₂O.

Accepted Answer

Nitrous oxide (N2O) is a polar molecule, which allows it to have a permanent dipole moment necessary for pure rotational transitions to be observed in its rotational spectrum. The other molecules are non-polar and do not exhibit pure rotational spectra.

Question 7

The harmonic vibrational wavenumber of the ³²S¹⁶O isotopomer of sulfur monoxide is 1123.7 cm^-1. Calculate the force constant for the sulfur monoxide bond.

Accepted Answer

The force constant (k) can be calculated using the formula k = (4π²c²μν²), where c is the speed of light, μ is the reduced mass, and ν is the wavenumber. For SO, μ is approximately 1.07 x 10^-26 kg. Plugging in the values, k ≈ 794 N m⁻¹.

Question 8

For chlorine monofluoride, ClF, the harmonic vibrational wavenumber is 793.2 cm^-1 and the anharmonicity is 0.0125. Calculate the wavenumber of the first vibrational overtone in the infrared absorption spectrum of chlorine monofluoride.

Accepted Answer

The wavenumber of the first vibrational overtone can be calculated using the formula for the energy levels of a vibrating diatomic molecule, considering the anharmonicity: $\nu = 2\nu_0 - 2x\nu_0$, where $\nu_0$ is the harmonic vibrational wavenumber and $x$ is the anharmonicity constant. Substituting the given values: $2 \times 793.2 - 2 \times 0.0125 \times 793.2 = 1526.9 \, \text{cm}^{-1}$.

Question 9

Calculate the number of vibrational normal modes in methanol, CH₃OH.

Accepted Answer

Methanol (CH3OH) is a non-linear molecule with 6 atoms. The formula for calculating the number of vibrational normal modes in a non-linear molecule is 3N - 6, where N is the number of atoms. For methanol, this is 3(6) - 6 = 12.

Question 10

Spectroscopic vibration-rotation transitions in the P- and R-branches of the high-resolution infrared spectrum of hydrogen iodide, HI, are observed to be separated by a wavenumber of 6.37 cm^-1. Calculate the length of the bond of a hydrogen iodide molecule.

Accepted Answer

The separation between the P and R branches is approximately twice the rotational constant B. The rotational constant B is related to the bond length (r) by the formula B = h / (8π²cI), where I is the moment of inertia (I = μr², μ is the reduced mass). Solving for r using the given separation (2B = 6.37 cm⁻¹) and the known values for h, c, and μ for HI, we find r ≈ 1.03 Å.

Question 11

A $\pi$- $\pi$* transition in hexa-1,3,5-triene, C₆H₈, corresponds to a separation between energy levels of 4.84 eV. Predict the wavelength of the corresponding maximum absorbance in the UV-visible spectrum of hexatriene.

Accepted Answer

To find the wavelength ($\lambda$) corresponding to the energy difference ($\Delta E$) of 4.84 eV, we use the equation $\Delta E = \frac{hc}{\lambda}$, where $h$ is Planck's constant (4.135667696 x 10^-15 eV·s) and $c$ is the speed of light (3.00 x 10^8 m/s). Rearranging for $\lambda$, we get $\lambda = \frac{hc}{\Delta E}$. Substituting the given values, $\lambda \approx 256$ nm.

Question 12

The absorbance of a solution of path length 10.0 cm that contained two cyanine dyes, A and B, was found to be 24.1 * 10³ at a wavelength of 550 nm and 21.9* 10³ at a wavelength of 650 nm. Use the following data for the molar absorption coefficients of the dyes at these wavelengths to determine the concentration of dye A in the solution. $\quad\quad\quad\quad\quad\quad\varepsilon /\left(10^{5} \mathrm{dm}^{3} \mathrm{~mol}^{-1} \mathrm{~cm}^{-1}\right)$ $\begin{array}{lcc} &\quad\quad 550 \mathrm{~nm} & \quad\quad650 \mathrm{~nm} \ \text { Dye A } & 1.48 & 0.24 \ \text { Dye B } & 0.56 & 2.29 \end{array}$

Accepted Answer

The answer of The absorbance of a solution of path...

Question 13

When monochromatic radiation of energy 21.22 eV from a helium lamp is used to irradiate a sample of nitrogen, N₂, gas in a photoelectron spectrometer, electrons are ejected with a maximum kinetic energy of 1.41 * 10⁶ m s^-1. Calculate the molar ionization energy of N₂.

Accepted Answer

The answer of When monochromatic radiation of energy 21.22 eV...

Question 14

In the photoelectron spectrum of carbon disulfide, CS₂, ionization from the highest occupied, 1 $\pi$_g, molecular orbital results in little vibrational structure. In contrast, ionization from the next lowest occupied 1 $\pi$_u molecular orbital results in a long vibrational progression in the $\pi$₁ symmetric stretching mode of the CS₂⁺ ion. Which of the following descriptions of the bonding in neutral CS₂ best explains these observations?

Accepted Answer

Ionization from a bonding molecular orbital (MO) typically results in a vibrational progression due to the change in bond order and bond length upon ionization. The observation of a long vibrational progression upon ionization from the 1πu MO suggests it is bonding in character, as its removal increases the bond length and decreases the bond order, leading to significant vibrational effects. The lack of vibrational structure upon ionization from the 1πg MO suggests it is non-bonding, as its removal does not significantly affect the bond order or length.

Question 15

When radiation of wavelength 495 nm from a monochromatic flash lamp of power 0.500 kW is used to irradiate a low-pressure sample of iodine vapour, I₂, the rate of production of iodine atoms is 3.24 *10¹⁹ s^-1. Assuming that all of the photons emitted by the lamp are absorbed by the iodine vapour, calculate the quantum yield for dissociation of iodine molecules.

Accepted Answer

The answer of When radiation of wavelength 495 nm from...

Question 16

For a particular substituted stilbene dendrimer the quantum yield for fluorescence following electronic excitation is 0.14, with the observed lifetime of the excited state 4.6 ns. Calculate the rate constant for fluorescence.

Accepted Answer

The rate constant for fluorescence (k_f) can be calculated using the formula: k_f = quantum yield / lifetime. Substituting the given values: k_f = 0.14 / 4.6 ns = 0.14 / 4.6 x 10^-9 s = 30 x 10^6 s^-1.

Question 17

The lifetime of an excited electronic state is 2.9 ns in the absence of a quencher and 1.7 ns if a quencher is present. What is the quantum yield for quenching?

Accepted Answer

The answer of The lifetime of an excited electronic state...

Question 18

The observed lifetime of an excited electronic state of a porphyrin-like metal complex was determined to be 0.35 ns, with rate constants for fluorescence and intersystem crossing of 0.6 * 10⁹ s^-1 and 1.8 * 10⁹ s^-1 respectively. Determine the rate constant for non-radiative internal conversion.

Accepted Answer

The rate constant for non-radiative internal conversion is the difference between the rate constants for all possible deactivation pathways and the observed lifetime of the excited state. In this case, the rate constants for fluorescence and intersystem crossing are 0.6 F1F1F1S1F1F1F10 10⁹ s^-1 and 1.8 F1F1F1S1F1F1F10 10⁹ s^-1 respectively, and the observed lifetime is 0.35 ns. Therefore, the rate constant for non-radiative internal conversion is:k_nr = k_f + k_isc - 1/tau= 0.6 F1F1F1S1F1F1F10 10⁹ s^-1 + 1.8 F1F1F1S1F1F1F10 10⁹ s^-1 - 1/0.35 ns= 0.5 F1F1F1S1F1F1F10 10⁹ s^-1

Question 19

In an experiment to investigate the decay of photoexcited tryptophan, the observed lifetime of the excited state was measured for various concentrations of oxygen, O₂, which acted as a quencher. By constructing a Stern-Volmer plot of the following data, determine the rate constant for fluorescence quenching. $\begin{array}{llllll} {\left[\mathrm{O}_{2}\right] / \mathrm{mol} \mathrm{dm}^{-3}} & 0.02 & 0.04 & 0.06 & 0.08 & 0.10 \ \tau_{\text {pbs }}\left(\mathrm{O}_{2}\right) / \mathrm{ns} & 1.64 & 1.18 & 0.91 & 0.74 & 0.63 \end{array}$

Accepted Answer

The Stern-Volmer equation is:$$\frac{\tau_0}{\tau}=1+k_q\tau_0[Q]$$where $\tau_0$ is the lifetime in the absence of quencher, $\tau$ is the lifetime in the presence of quencher, $k_q$ is the quenching rate constant, and $[Q]$ is the quencher concentration.Plotting $\frac{\tau_0}{\tau}$ against $[Q]$ gives a straight line with slope $k_q\tau_0$.From the data, we can calculate:$$\frac{\tau_0}{\tau}=\frac{1.64}{1.18}=1.39$$$$\frac{\tau_0}{\tau}=\frac{1.64}{0.91}=1.80$$$$\frac{\tau_0}{\tau}=\frac{1.64}{0.74}=2.22$$$$\frac{\tau_0}{\tau}=\frac{1.64}{0.63}=2.60$$Plotting these values against $[Q]$ gives a straight line with slope $k_q\tau_0=12.3\times10^{-9}\text{ mol}^{-1}\text{ dm}^3\text{ s}^{-1}$.

Question 20

In an experiment to investigate the folding of tethered labelled peptides using fluorescence resonant energy transfer, the quantum yield for fluorescence of a rhodamine dye donor was found to decrease, on average, by a factor of 0.21 in the presence of a Texas red dye acceptor. Use the Förster theory of resonant energy transfer to determine the separation of the donor-acceptor pair, given that the Förster distance parameter for the rhodamine-Texas red pair is R₀ = 4.4 nm.

Accepted Answer

The Förster theory of resonant energy transfer states that the efficiency of energy transfer (E) is inversely proportional to the sixth power of the distance (r) between the donor and acceptor, given by $E = 1/(1+(r/R_0)^6)$, where $R_0$ is the Förster distance. Given that the quantum yield decreases by a factor of 0.21, the efficiency of energy transfer is $E = 1 - 0.21 = 0.79$. Rearranging the formula to solve for $r$ and substituting $E = 0.79$ and $R_0 = 4.4$ nm, we find that $r$ is approximately 3.5 nm.

Quiz 11: Molecular Spectroscopy

Boron trifluoride, BF₃, has a three-fold symmetric planar equilibrium geometry with a B-F bond length of 1.30 Å. Calculate the moment of inertia for rotation about the three-fold axis of symmetry perpendicular to the molecular plane.

The bond length in a nitric oxide molecule, NO, is 1.15 Å. Calculate the rotational constant.

Calculate the angular momentum of a molecule with a rotational quantum number J = 3.

What is the degeneracy of the rotational level with quantum number J = 6 in a linear molecule?

The rotational constant of a hydrogen fluoride, HF, molecule is 3.12 *10¹¹s^-1. Predict which transition in the rotational absorption spectrum will be most intense at a temperature of 400 K.

For which of the following molecules are pure rotational spectroscopic transitions observed? Benzene, C₆H₆, Sulfur hexafluoride, SF₆, Carbon disulfide, CS₂, Nitrous oxide, N₂O.

The harmonic vibrational wavenumber of the ³²S¹⁶O isotopomer of sulfur monoxide is 1123.7 cm^-1. Calculate the force constant for the sulfur monoxide bond.

For chlorine monofluoride, ClF, the harmonic vibrational wavenumber is 793.2 cm^-1 and the anharmonicity is 0.0125. Calculate the wavenumber of the first vibrational overtone in the infrared absorption spectrum of chlorine monofluoride.

Calculate the number of vibrational normal modes in methanol, CH₃OH.

Spectroscopic vibration-rotation transitions in the P- and R-branches of the high-resolution infrared spectrum of hydrogen iodide, HI, are observed to be separated by a wavenumber of 6.37 cm^-1. Calculate the length of the bond of a hydrogen iodide molecule.

A $\pi$ - $\pi$ * transition in hexa-1,3,5-triene, C₆H₈, corresponds to a separation between energy levels of 4.84 eV. Predict the wavelength of the corresponding maximum absorbance in the UV-visible spectrum of hexatriene.

When monochromatic radiation of energy 21.22 eV from a helium lamp is used to irradiate a sample of nitrogen, N₂, gas in a photoelectron spectrometer, electrons are ejected with a maximum kinetic energy of 1.41 * 10⁶ m s^-1. Calculate the molar ionization energy of N₂.

For a particular substituted stilbene dendrimer the quantum yield for fluorescence following electronic excitation is 0.14, with the observed lifetime of the excited state 4.6 ns. Calculate the rate constant for fluorescence.

The lifetime of an excited electronic state is 2.9 ns in the absence of a quencher and 1.7 ns if a quencher is present. What is the quantum yield for quenching?

The observed lifetime of an excited electronic state of a porphyrin-like metal complex was determined to be 0.35 ns, with rate constants for fluorescence and intersystem crossing of 0.6 * 10⁹ s^-1 and 1.8 * 10⁹ s^-1 respectively. Determine the rate constant for non-radiative internal conversion.

Quiz 11: Molecular Spectroscopy

Boron trifluoride, BF3, has a three-fold symmetric planar equilibrium geometry with a B-F bond length of 1.30 Å. Calculate the moment of inertia for rotation about the three-fold axis of symmetry perpendicular to the molecular plane.

The bond length in a nitric oxide molecule, NO, is 1.15 Å. Calculate the rotational constant.

Calculate the angular momentum of a molecule with a rotational quantum number J = 3.

What is the degeneracy of the rotational level with quantum number J = 6 in a linear molecule?

The rotational constant of a hydrogen fluoride, HF, molecule is 3.12 *1011 s-1. Predict which transition in the rotational absorption spectrum will be most intense at a temperature of 400 K.

For which of the following molecules are pure rotational spectroscopic transitions observed? Benzene, C6H6, Sulfur hexafluoride, SF6, Carbon disulfide, CS2, Nitrous oxide, N2O.

The harmonic vibrational wavenumber of the 32S16O isotopomer of sulfur monoxide is 1123.7 cm-1. Calculate the force constant for the sulfur monoxide bond.

For chlorine monofluoride, ClF, the harmonic vibrational wavenumber is 793.2 cm-1 and the anharmonicity is 0.0125. Calculate the wavenumber of the first vibrational overtone in the infrared absorption spectrum of chlorine monofluoride.

Calculate the number of vibrational normal modes in methanol, CH3OH.

Spectroscopic vibration-rotation transitions in the P- and R-branches of the high-resolution infrared spectrum of hydrogen iodide, HI, are observed to be separated by a wavenumber of 6.37 cm-1. Calculate the length of the bond of a hydrogen iodide molecule.

A π\piπ - π\piπ * transition in hexa-1,3,5-triene, C6H8, corresponds to a separation between energy levels of 4.84 eV. Predict the wavelength of the corresponding maximum absorbance in the UV-visible spectrum of hexatriene.

When monochromatic radiation of energy 21.22 eV from a helium lamp is used to irradiate a sample of nitrogen, N2, gas in a photoelectron spectrometer, electrons are ejected with a maximum kinetic energy of 1.41 * 106 m s-1. Calculate the molar ionization energy of N2.

For a particular substituted stilbene dendrimer the quantum yield for fluorescence following electronic excitation is 0.14, with the observed lifetime of the excited state 4.6 ns. Calculate the rate constant for fluorescence.

The lifetime of an excited electronic state is 2.9 ns in the absence of a quencher and 1.7 ns if a quencher is present. What is the quantum yield for quenching?

The observed lifetime of an excited electronic state of a porphyrin-like metal complex was determined to be 0.35 ns, with rate constants for fluorescence and intersystem crossing of 0.6 * 109 s-1 and 1.8 * 109 s-1 respectively. Determine the rate constant for non-radiative internal conversion.

Boron trifluoride, BF₃, has a three-fold symmetric planar equilibrium geometry with a B-F bond length of 1.30 Å. Calculate the moment of inertia for rotation about the three-fold axis of symmetry perpendicular to the molecular plane.

The rotational constant of a hydrogen fluoride, HF, molecule is 3.12 *10¹¹s^-1. Predict which transition in the rotational absorption spectrum will be most intense at a temperature of 400 K.

For which of the following molecules are pure rotational spectroscopic transitions observed? Benzene, C₆H₆, Sulfur hexafluoride, SF₆, Carbon disulfide, CS₂, Nitrous oxide, N₂O.

The harmonic vibrational wavenumber of the ³²S¹⁶O isotopomer of sulfur monoxide is 1123.7 cm^-1. Calculate the force constant for the sulfur monoxide bond.

For chlorine monofluoride, ClF, the harmonic vibrational wavenumber is 793.2 cm^-1 and the anharmonicity is 0.0125. Calculate the wavenumber of the first vibrational overtone in the infrared absorption spectrum of chlorine monofluoride.

Calculate the number of vibrational normal modes in methanol, CH₃OH.

Spectroscopic vibration-rotation transitions in the P- and R-branches of the high-resolution infrared spectrum of hydrogen iodide, HI, are observed to be separated by a wavenumber of 6.37 cm^-1. Calculate the length of the bond of a hydrogen iodide molecule.

A $\pi$ - $\pi$ * transition in hexa-1,3,5-triene, C₆H₈, corresponds to a separation between energy levels of 4.84 eV. Predict the wavelength of the corresponding maximum absorbance in the UV-visible spectrum of hexatriene.

When monochromatic radiation of energy 21.22 eV from a helium lamp is used to irradiate a sample of nitrogen, N₂, gas in a photoelectron spectrometer, electrons are ejected with a maximum kinetic energy of 1.41 * 10⁶ m s^-1. Calculate the molar ionization energy of N₂.