Part A: Rotational Energies Within the rigid-rotor approximation, the list of allowed rotational ene — Analytical Chemistry Chemistry Question
Rotational and vibrational energy levels of a diatomic molecule
Part A: Rotational Energies
Within the rigid-rotor approximation, the list of allowed rotational energies of a diatomic molecule, AB(g) in the gas phase, are given by:
Erotation = B J(J+1), J = 0, 1, 2, …
where B = h^2 / (8 * \pi^2 * I) is a characteristic property of the molecule called the “rotational constant” of the molecule. The expression for B is in SI units; h is Planck’s constant, and I is the moment of inertia of the molecule defined by: I = µ R2, where R is the bond length, and µ is called the “reduced mass” of the diatomic molecule. The latter quantity is defined in terms of the masses, mA and mB, of the atoms in the diatomic molecule AB.
µ = (mA * mB) / (mA + mB)
Theory indicates that the bond length, R, does not change when either A or B is replaced by other isotopes of the atoms A or B.
When a sample of gaseous molecules is exposed to microwave radiation, a molecule in the sample that is initially in a rotational energy level with J = Ji may absorb a photon, ending up in a higher energy level with J = Jf. It may be shown that only those rotational transitions in which Jf = Ji + 1 can occur in absorption of light that changes the rotational state.
The rotational constant of the 12C16O molecule has been experimentally determined as B = 23.115 J·mol–1. The isotopic masses of the two atoms in this molecule are known: mass of 12C = 12 amu by definition, and that of 16O = 15.994915 amu. The longest wavelength of electromagnetic radiation that causes a transition between the rotational levels of 12C16O molecule in a sample has been observed to be λ = 0.25876 cm.
Part B: Rotational plus Vibrational Energies
Within the harmonic oscillator approximation, the list of allowed vibrational energies of a diatomic molecule, AB(g) in the gas phase, are given by:
Evibration = \epsilon * (v + 1/2), v = 0, 1, 2, …
where \epsilon is a characteristic vibrational property of the molecule defined by:
\epsilon = (h / (2 * \pi)) * \sqrt{k / µ}
In this expression h is Planck’s constant, k is called the “force constant” of the molecule, and µ is the reduced mass of the diatomic molecule. In SI units, ε is in joules, k in N·m–1, and µ in kg. Theory shows that the force constant k is independent of isotopic substitution in the molecule. When a sample of gas molecules is exposed to infrared (IR) radiation, a molecule in the sample that is initially in a vibrational energy level with v = vi may absorb a photon, ending up in a higher energy vibrational level with v = vf. It may be shown that only those transitions in which vf = vi + 1 can occur in absorption of light that changes the vibrational state.
Absorption of light in the IR region changes not only the vibrational state, but also the rotational state; i.e., a simultaneous change in v and J is involved now. This is because the allowed vibrational plus rotational energies of a molecule are given by:
Erot.+vib.= Erotation + Evibration
What are the values of Ji and Jf for a molecule that absorbs a photon with a wavelength of 0.25876 cm?
Model Answer
The allowed rotational energies are: Erotation= B J(J+1)
Possible ∆E values for the Ji→ Ji+1 transitions are:
∆E = B (Ji+1)( Ji+2) – B Ji(Ji+1) = 2 B (Ji+1), Ji = 0, 1, 2, ...
Only photons with energies hν = 2B, 4B, 6B, ... can be absorbed. The longest wavelength of radiaton absorbed corresponds to a photon with the smallest frequency ν_min = 2B/h.
Hence, radiation with λlongest = 0.25876 cm will be absorbed in the transition Ji = 0 to Jf = 1.
Calculate the moment of inertia and the bond length of the carbon monoxide molecule.
Model Answer
Moment of inertia, I:
B = h^2 / (8 * \pi^2 * I) or I = h^2 / (8 * \pi^2 * B)
B = 23.115 J mol-1 = (23.115 / NA) J = 3.8384 · 10^-23 J
I = (6.6261 · 10^-34)^2 / (8 * \pi^2 * 3.8384 · 10^-23) = 1.4487 · 10^-46 kg m^2
Bond length in 12C16O:
I = µ R2 or R = (I / µ)^(1/2)
µ = (12 * 15.994915) / (12 + 15.994915) = 6.8562087 g mol-1 = 1.1385 · 10^-26 kg
R = (1.4487 · 10^-46 kg m^2 / 1.1385 · 10^-26 kg)^(1/2) = 1.11280 · 10^-10 m = 0.11280 nm
Predict the values of the rotational constants, B, for each of the following three molecules: 12C18O, 13C18O, and 13C16O.
(Additional data: masses of 18O = 17.999159 and 13C=13.003355 amu.)
Model Answer
Let B1 = 23.115 J·mol-1 for 12C16O, B2 be the rotational constant of 12C18O, B3 be that for 13C18O, and B4 be that for 13C16O. Since R = 0.1128 nm has the same value in these 4 molecules and I = µ R2, the B values of these molecules differ only by their reduced masses. One has:
Bi = (µ1 / µi) B1, i = 2, 3, 4.
Reduced masses:
- Molecule 1 (12C16O): µ1 = 6.8562087 amu
- Molecule 2 (12C18O): µ2 = 7.1998654 amu
- Molecule 3 (13C18O): µ3 = 7.5493702 amu
- Molecule 4 (13C16O): µ4 = 7.1724126 amu
With B1 = 23.115 J mol-1 and µ1 = 6.8562087 amu:
B2 = (6.8562087 / 7.1998654) * 23.115 J·mol-1 = 22.012 J·mol-1
B3 = (6.8562087 / 7.5493702) * 23.115 J·mol-1 = 20.993 J mol-1
B4 = (6.8562087 / 7.1724126) * 23.115 J mol-1 = 22.096 J mol-1
Calculate the longest wavelengths of microwave radiation that may be absorbed by each of 12C18O, 13C18O, and 13C16O molecules.
Model Answer
Absorption of the longest λ of radiation leads to an excitation from J = 0 level to J = 1.
With the same numbering of the isotopically substituted molecules as above, one has:
λi = hc / (2Bi) or λi / λ1 = B1 / Bi = µi / µ1 ⇒ λi = (µi / µ1) λ1 with λ1 = 0.25876 cm
Using reduced masses given in part 21.3:
λ2 = (7.1998654 / 6.8562087) * 0.25876 cm = 0.27173 cm
λ3 = (7.5493702 / 6.8562087) * 0.25876 cm = 0.28492 cm
λ4 = (7.1724126 / 6.8562087) * 0.25876 cm = 0.27069 cm
The force constant of the carbon monoxide molecule is 1901.9 N·m–1. Find ε in kJ·mol–1 (to 4 significant figures) for each of the following isotopically related CO molecules:
i. 12C16O
ii. 12C18O
iii. 13C18O
iv. 13C16O
Model Answer
ε_i = (h / (2 * \pi)) * \sqrt{k / µ_i}
where i = 1, 2, 3, 4 and k = 1901.9 N m–1 is the same for all 4 molecules.
µ1 (12C16O) = 6.8562087 g mol–1 = 1.13851 · 10^-26 kg
ε1 = (6.6261 · 10^-34 / (2 * \pi)) * \sqrt{1901.9 / 1.13851 · 10^-26} = 4.3103 · 10^-20 J = 25.96 kJ mol-1
Since k is the same for all, ε_i * µ_i^(1/2) = ε_1 * µ_1^(1/2), giving:
ε_i = ε_1 * (µ1 / µi)^(1/2)
- i. 12C16O: ε1 = 25.96 kJ mol–1
- ii. 12C18O: ε2 = 25.33 kJ mol–1
- iii. 13C18O: ε3 = 24.74 kJ mol–1
- iv. 13C16O: ε4 = 25.38 kJ mol–1
Find the wavelengths (to 4 significant figures) of IR radiation that may be absorbed by a molecule in making a transition from an initial state with (v,J) = (0,0) to a final state with (v,J) = (1,1) for each of the following isotopically related CO molecules:
i. 12C16O
ii. 12C18O
iii. 13C18O
iv. 13C16O
Model Answer
∆E = ∆Erotation + ∆Evibration
∆Erotation(J = 0 to 1) = 2B
∆Evibration(v = 0 to 1) = ε
Hence, ∆E = ε + 2B, and the wavelength of IR radiation that can be absorbed is λ = hc / ∆E.
Due to different reduced masses, ε and B values differ for the 4 molecules, making ∆E different. Values of ε and B (converted to J) and the resulting wavelengths are:
- Molecule 1 (12C16O):
ε = 4.3108 · 10^-20 J, B = 3.8384 · 10^-23 J, ∆E = 4.3185 · 10^-20 J, λ = 4600 nm
- Molecule 2 (12C18O):
ε = 4.2062 · 10^-20 J, B = 3.6552 · 10^-23 J, ∆E = 4.2135 · 10^-20 J, λ = 4714 nm
- Molecule 3 (13C18O):
ε = 4.1082 · 10^-20 J, B = 3.4860 · 10^-23 J, ∆E = 4.1152 · 10^-20 J, λ = 4827 nm
- Molecule 4 (13C16O):
ε = 4.2145 · 10^-20 J, B = 3.6692 · 10^-23 J, ∆E = 4.2218 · 10^-20 J, λ = 4705 nm