The crystallographic structure of annulene (C18H18) has been determined with x–ray crystallography ( — Physical Chemistry — Kinetics Chemistry Question
Annulene
The crystallographic structure of annulene (C18H18) has been determined with x–ray crystallography (Acta Cryst. 19, 227 (1965)). The following figure shows the electron density in the mean molecular plane; the contours are shown at intervals of 1 e.Å –3. The absorption spectrum of this molecule has been studied too. The location of the absorption maximum can be estimated using the model of "the particle on a ring". The energy of a particle moving on a circular well is given by E = N^2 h^2 / 2 m_e L^2, where h is Planck's constant, me the mass of the electron, L the full path of the electron and N its quantum number. N takes the integer values 0, ±1, ±2, ±3, …, therefore each level has a two–fold degeneracy except for the zero level which is nondegenerate.
Assuming a mean bond length between carbon atoms equal to 1.40 Å, determine the wavelength of the lowest electronic transition.
Model Answer
The number of electrons is 18. Two electrons can occupy each state due to the Pauli exclusion principle. Each state above the lowest is two–fold degenerate. Based on these pieces of information, we can fill this table:
N max number of e– per state total e– up to this state
0 2 2
1 4 6
2 4 10
3 4 14
4 4 18
Sates up to N = 4 are fully occupied with 18 electrons. The lowest possible transition is from state N = 4 to state N = 5. The path forming a circular well is L = 18 × 1.4 Å. Hence ΔE = E5 – E4 = 3.415 × 10^-19 J
The transition frequency is = E / h = 5.1544×10^14 s–1 and the corresponding wavelength is = c / = 581.6 nm.