Tímadæmi B (Hafdís Inga Ingvarsdóttir): Quantum chemistry of atoms and
molecules
Perform comparison of energy-differences (kJ mol-1) between quantum levels and
absorption wavelengths(nm) relevant to
a) - electron energies of an atom (the H atom)
b) – vibrational energies of a molecule (HCl)
c) – rotational energies of a molecule (HCl)
as:
a) Find energy-differences between the ground and the first excited electronic state
of the H atom (kJ mol-1) and the corresponding absorption wavelength (nm),
b) Find energy-differences between the ground and the first excited vibrational state
of the HCl molecule (kJ mol-1) and the corresponding absorption wavelength
(nm),
c) Find energy-differences between the ground and the first excited rotational state
of the HCl molecule (kJ mol-1) and the corresponding absorption wavelength
(nm).
Perform comparison of the values and specify spectral regions in each case.
Make use of relevant spectroscopic parameters in a database (NIST)
Solution:
a)
 1
1 
E  E 2  E1  13,6   2  2 
n 2 
 n1
E1 = energy for the ground electronic state
n1 = 1 for the ground electronic state
E2 = energy for the first excited electronic state n2 = 2 for the first excited electronic state
1
96,4853kJ / mól
1
E  13,6   2  2   10,2eV 
 984,1501kJ / mól
1eV
2 
1
 1
1 
 RH   2  2 

n 2 
 n1
1

1
 1
1 
RH   2  2 
n2 
 n1
λ = absorption wavelength corresponding to the energy difference (ΔE)
RH = Rydberg constant = 1,0973731∙107 m-1
1

 1,215  10 7 m  121,5nm
1
1
1,0973731  10 7 m 1   2  2 
2 
1
b)
1

  0,1,2,... = vibrational quantum number
E vib      h  v
2

Evib  E1  E0  1,5hv  0,5hv  hv  h  c 
h = Plank´s constant = 6,626∙10-34 Js
c = speed of light = 3,00∙108 m/s
  fundamental frequency of vibration = 2886 cm-1 for HCl
E vib  6,626  10 34 Js  3,00  10 8 m / s  288600m 1  5,737  10 20 J  5,737  10 23 kJ
E vib  5,737  10 23 kJ  6,022  10 23 mól 1  34,55kJ / mól
E vib 
hc


hc
E vib
6,626  10 34 Js  3,00  10 8 m / s

 3,465  10 6 m  3465nm
 20
5,737  10 J
c)
E rot  2 Bh
J = 0→1
B = 10,59341 cm-1 according to NIST database
= 1059,341m-1 ∙ 3,00∙108 m/s = 3,18∙1011 s-1
E rot  2  3,18 1011 s 1  6,626 10 34 Js  4,2110 22 J  4,2110 25 kJ
E rot  4,21  10 25 kJ  6,022  10 23 mól 1  0,254kJ / mól
E rot 

hc


hc
E rot
6,626  10 34 Js  3,00  10 8 m / s
 4,72  10  4 m  472000nm
 22
4,21  10 J
Comparison of the values and spectral regions:
ΔE (kJ/mól)
Electronic state of the
984,1501
H atom
Vibrational state of
34,55
HCl
Rotational state of HCl
0,254
λ (nm)
121,5
Spectral region
UV
3465
Near IR
472000
Far IR