The straight-chain polyene (····─CH═CH─CH═CH─CH═CH─····) is a chemical moiety present in the molecul — Physical Chemistry — Electrochemistry Chemistry Question
Electronic structure of polyene
The straight-chain polyene (····─CH═CH─CH═CH─CH═CH─····) is a chemical moiety present in the molecules that absorb visible light. Let us consider the behavior of π electrons in the straight-chain polyene.
First, for the sake of simplicity, we use the Hückel approximation to assess the π electrons of 1,3-butadiene, which contains four carbon atoms. We define the normal to the carbon-backbone plane as the z-axis and the atomic orbital (2pz orbital) of each carbon atom as iφ (i = 1, 2, 3, 4). The molecular orbital kψ is expressed as the linear combination of these atomic orbitals according to the following equation:
(1)
The coulomb integral is defined as α and the resonance integral between adjacent atoms is defined as β, where α is the energy of the 2pz orbital of an isolated carbon atom, and β can be determined from the overlap between adjacent 2pz orbitals. The eigenenergy kε and the corresponding molecular orbital are obtained by the variation method, as shown in the following sets:
1 = 1.62ε α – β 1 1 2 3 4= 0.37 0.60 + 0.60 0.37ψ φ – φ φ – φ (2)
2 = 0.62ε α – β = 0.60 0.37 0.37 0.602 1 2 3 4ψ φ – φ – φ + φ (3)
3 = + 0.62ε α β 3 1 2 3 40.60 0.37 0.37 0.60ψ φ φ – φ – φ= + (4)
4 = +1.62ε α β 4 1 2 3 40.37 0.60 0.60 0.37φ φ φ φ= + + +ψ (5)
8.1 Draw the energy levels of the molecular orbitals, and indicate the π electrons in the ground state using arrows while considering the spin direction.
8.2 Using α and β, calculate the photon energy necessary for excitation from the ground state to the first excited state of 1,3-butadiene.
Model Answer
0.62 0.62 1.242 3∆ε = ε – ε = α – β – (α + β) = – β
8.3 Draw the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) of 1,3-butadiene on the basis of the example for ethylene shown in Fig. 2, i.e. draw four 2pz orbitals perpendicular to a horizontal line and indicate the sign for each orbital in black and white. Further, represent nodes by using solid circles. Note that you do not need to consider the differences in the contribution of each 2pz orbital to the molecular orbitals.
Next, consider the behaviour of π electrons in a system where the number of carbon atoms is extremely large, i.e., polyacetylene. Consider a one-dimensional (1D) chain of a certain number (N) of 2 pz orbitals that are aligned perpendicular to the chain with a spacing of α. If we assume a periodic boundary condition of Nα along the chain, the energy state of the π electrons can be described by the following equation:
8.4 Calculate the energy width between the maximum and the minimum π-electron energy levels.
Model Answer
maxkE ( ) = α – 2 β
minkE ( ) = α + 2 β
Therefore, the energy width between Ek(max) and Ek(min) is – 4 β or 4 β
8.5 Since the energy-level spacing of the 1D chain is extremely small, the energy levels form continuum states. Therefore, thermal excitation from HOMO to LUMO is easily induced at room temperature. Although such thermally excited electrons are mobile in the chain and can contribute to electric conductivity, pure polyacetylene is a poor conductor of electricity. This is because the carbon atoms in polyacetylene are not arranged with a periodicity of a; instead, these atoms are arranged with a periodicity of 2a because of the alternating arrangement of single and double bonds. If the HOMO and LUMO of the 1D chain with periodicities of a and 2a are assumed as shown in Fig. 3(a) and (b), respectively, how do the energy levels of the HOMO and LUMO alter when the periodicity changes from a to 2a? Choose the correct answer from the following choices:
a) HOMO is destabilized, while LUMO is stabilized.
b) HOMO is stabilized, while LUMO is destabilized.
c) Both HOMO and LUMO are stabilized.
d) Both HOMO and LUMO are destabilized.
Model Answer
LUMO is destabilized due to proximity of two 2pz orbitals with opposite signs. HOMO is stabilized due to proximity of two 2pz orbitals with same signs.
Answer: (b)
8.6 The 1D chain with a periodicity of 2a shows a gap between continuum states, which results in the formation of a filled valence band and an unfilled conduction band, as illustrated in Fig. 4. Since the valence band is filled with electrons and no unoccupied states are available for conduction, polyacetylene is an insulator. When a chemical substance is added to polyacetylene, the valence electrons become mobile. This chemical is obtained by oxidation of an aqueous solution of an alkali halide, and the number of inner-shell electrons in its constituent atoms is the same as that in argon.
Write the chemical formula of this substance.
Model Answer
Br2