When n = 3, the energy is
E =
h
2 n
2
8ma
2
=
6:626 Â 10
−34
Js
À
Á 2 3
ð Þ
2
8 9:109 Â 10
−31
kg
À
Á
200 Â 10
−9
m
À
Á 2
=
3:951 Â 10
−66
J
2 s
2
2:915 Â 10
−43
kgm
2
= 1:355 Â 10
−23 J
The difference between neighboring energy levels (ΔE) is given by
ΔE = E 2 − E 1 =
h
2
8ma
2
n
2
2 − n
2
1
The subscripts 1 and 2 denote lower and upper energy levels,
respectively. If a increases from 200 nm to 300 nm, ΔE will
decrease by 2
2
/3
2 or about 45%.
The particle-in-a-box model is arguably the simplest quantum mechanical model describing the energy and probability density of an electron.
Despite its simplicity it has been extremely valuable in describing the
absorption properties of simple molecules in which the electron is freely
moving along a line. Such molecules can be linear carbon chains containing alternating single and double bonds (conjugation) in which
electrons are delocalized along the entire chain.
Conjugated molecules have more than one free electron moving along
the chain. For example, hexatriene has six free electrons, as shown in
Figure 5.16. Each carbon atom is sp
2 hybridized, and overlap of these
orbitals forms the s backbone of the carbon chain, including all the C–H
bonds. The electrons due to s-bonding are localized between the two
atoms forming the bond. The particle-in-a-box model does not apply to
these electrons. However, each carbon atom has an unhybridized porbital containing one electron. The p-orbitals on all the carbon atoms are
able to overlap forming the π-framework (shown by the shaded regions in
Figure 5.16a). The six π-electrons are delocalized along the chain and can
be described using the particle-in-a-box model.
In order to deal with the six π-electrons in hexatriene, an energy-level
diagram is constructed like that shown in Figure 5.16b. The energy levels
are filled up with the appropriate number of electrons such that each level
contains a maximum of two electrons. The Pauli exclusion principle tells
us that only two electrons can enter a given energy level, and this pairing occurs with the electrons having opposite spins to each other. In
CHAPTER 5: Intermolecular Interactions and Self-Assembly
166
E =
h
2 n
2
8ma
2
=
6:626 Â 10
−34
Js
À
Á 2 3
ð Þ
2
8 9:109 Â 10
−31
kg
À
Á
200 Â 10
−9
m
À
Á 2
=
3:951 Â 10
−66
J
2 s
2
2:915 Â 10
−43
kgm
2
= 1:355 Â 10
−23 J
The difference between neighboring energy levels (ΔE) is given by
ΔE = E 2 − E 1 =
h
2
8ma
2
n
2
2 − n
2
1
The subscripts 1 and 2 denote lower and upper energy levels,
respectively. If a increases from 200 nm to 300 nm, ΔE will
decrease by 2
2
/3
2 or about 45%.
The particle-in-a-box model is arguably the simplest quantum mechanical model describing the energy and probability density of an electron.
Despite its simplicity it has been extremely valuable in describing the
absorption properties of simple molecules in which the electron is freely
moving along a line. Such molecules can be linear carbon chains containing alternating single and double bonds (conjugation) in which
electrons are delocalized along the entire chain.
Conjugated molecules have more than one free electron moving along
the chain. For example, hexatriene has six free electrons, as shown in
Figure 5.16. Each carbon atom is sp
2 hybridized, and overlap of these
orbitals forms the s backbone of the carbon chain, including all the C–H
bonds. The electrons due to s-bonding are localized between the two
atoms forming the bond. The particle-in-a-box model does not apply to
these electrons. However, each carbon atom has an unhybridized porbital containing one electron. The p-orbitals on all the carbon atoms are
able to overlap forming the π-framework (shown by the shaded regions in
Figure 5.16a). The six π-electrons are delocalized along the chain and can
be described using the particle-in-a-box model.
In order to deal with the six π-electrons in hexatriene, an energy-level
diagram is constructed like that shown in Figure 5.16b. The energy levels
are filled up with the appropriate number of electrons such that each level
contains a maximum of two electrons. The Pauli exclusion principle tells
us that only two electrons can enter a given energy level, and this pairing occurs with the electrons having opposite spins to each other. In
CHAPTER 5: Intermolecular Interactions and Self-Assembly
166