BASICS
1
NATURE AND CHARACTERISTICS OF HYDROCARBONS
2
PHYSICAL AND CHEMICAL EQUILIBRIA
3
SURFACES AND DISPERSE SYSTEMS
4
FLUID DYNAMICS
5
KINETICS AND CATALYSIS
6
PROCESS ENGINEERING ASPECTS
7
COMBUSTION AND DETONATION
8
MATHEMATICAL AND MODELLING ASPECTS
9
MATERIALS
1.1
Theoretical aspects
1.1.1 Overview of the chemical bond
The enormous variety of molecules, with significantly
different structures and properties characterizing the
hydrocarbons, originates in the ability of hydrogen and
carbon atoms to form covalent bonds with one another which
are unusually stable and suitably oriented in space.
Hydrogen, consisting of a proton and an electron, is the
simplest atom in the periodic table of the elements and the
only one for which Schrödinger’s equation, used to calculate
the energy of microscopic systems, can be solved
analytically. In the ground state, the behaviour of the
electron is described by a wavefunction whose values
depend on its coordinates and whose square modulus gives
the probability density of finding it in a particular point in
space; as such, this function determines the value of local
electron density. Carbon 12C, the most widespread isotope,
has a nucleus consisting of six protons and six neutrons,
surrounded by six electrons in the energy levels of the
electronic structure, shown in Fig. 1. Each state has a
corresponding orbital, each of which is characterized by a
specific geometrical configuration which is usually
described by the surface circumscribing 90% of the
electron’s probability density.
Whereas s orbitals have spherical symmetry, there are
three p orbitals (px, py , pz), perpendicular to one another and
with a characteristic two-lobed shape. Electrons occupy the
orbitals in accordance with precise rules that make it
possible to define their electron configuration. The first of
these is Pauli’s exclusion principle, stating that each orbital
can host a maximum of two electrons with antiparallel spin;
the second is the Aufbau principle, stating that orbitals are
occupied following an order of increasing energy; finally,
there is Hund’s rule, stating that electrons occupy degenerate
orbitals (for example px, py , pz) in such a way as to obtain the
maximum number of unpaired electrons.
The formation of a chemical bond of covalent type
between two atoms occurs through the combination of atomic
orbitals, each containing an unpaired electron, into a
molecular orbital whose total energy is lower than that of the
two separate atomic orbitals. In fact, the electronic structure of
a molecule is defined by its wavefunction, whose dependence
on the spatial coordinates is obtained by solving Schrödinger’s
VOLUME V / INSTRUMENTS
equation. In its concise form, the time-independent version of
this equation can be written as follows:
[1]
∧
Hy ⫽Ey
where y⫽y (x1, x2, … xn) is the wavefunction∧dependent on
the xi coordinates associated with n particles, H is the
Hamiltonian differential operator associated with the sum of
the kinetic and potential energy of the electrons in motion
within the electrical field generated by the nuclei and the
electrons, themselves, whilst E is the energy of the system.
This gives an equation with partial derivatives with respect
to function y, that can be solved analytically only for a few
physical situations and which only allows for continuous,
single value and finite solutions for a series of self-values
corresponding to the quantized values of the energy of
system E. As already stated, an analytical solution can only
n⫽2
C
1s22s22p2
n⫽1
E
s
p
Fig. 1. Electronic structure of carbon in the ground state:
two electrons occupy the 1s orbital; two electrons occupy
the 2s orbital (whose calculated spherical structure is shown
top right); whereas the final two are unpaired and localized
in two different 2p orbitals (bottom left and right). The n index
is known as the principal quantum number.
3
NATURE AND CHARACTERISTICS OF HYDROCARBONS
be obtained for the hydrogen atom. The approximate solution
of Schrödinger’s equation, obtainable numerically with
perturbation and variational methods, nonetheless provides
significant information on the energy, structure and
reactivity of molecules. In approximate terms, a molecular
orbital is expressed by a Linear Combination of ci Atomic
Orbitals (LCAO), where i indicates the atom on which the
orbital is centred:
[2]
y ⫽ 冱ci ci
The values of the coefficients of combination ci are
obtained by applying the variation principle according to
which the energy of the ground state must be minimal. Two
different combinations are obtained from two atomic orbitals
cA and cB centred on two different atoms:
[3]
y⫾ ⫽cA cA ⫾cB cB
in which each represents a molecular orbital with bonding
energy EBond and antibonding energy EAntibond , with lower and
higher values, respectively as compared to those of the two
original atomic orbitals. In the ground state, the two electrons
are hosted in orbital y⫹, the most stable orbital since it
corresponds to a combination of the two atomic orbitals
which increases the electron density in the internuclear zone.
As such, a bond is formed because the electrostatic repulsion
between the nuclei is shielded. The first orbital is described
as bonding, whilst the second, having greater energy than the
system would have if the two atoms were infinitely far apart,
is described as antibonding. The energy levels produced by
the combination of two atomic orbitals are usually
represented by diagrams of the following type:
EAntibond E
B
EAntibond
EA
EA
EBond
EA
EBond
To a first and reasonable approximation, the extent of the
overlap between the two lobes of the two orbitals reflects the
strength of the bond.
This description refers to the formation of localized
bonds resulting from a pair of electrons. They are present in
the C⫺C and C⫺H bonds of saturated hydrocarbons, or
alkanes. In fact, molecular orbitals often cover most of the
molecule, or all of it as is the case of benzene, since they
derive from a combination of several atomic orbitals with
mutually compatible energetic and geometric properties.
This situation, as we will see, is of enormous importance in
conjugated and aromatic hydrocarbons. In this case, the
number of molecular orbitals generated is equal to the
number of atomic orbitals involved in the combination and
their occupation in the ground state proceeds in accordance
with the rules already described for atoms. In this situation,
the structure and energy of two specific orbitals take on
special importance: the occupied orbital with the highest
energy (bonding), generally known as HOMO (Highest
Occupied Molecular Orbital), and the unoccupied orbital
with the lowest energy (antibonding), known as LUMO
(Lowest Unoccupied Molecular Orbital). The importance of
HOMO and LUMO orbitals, described as frontier orbitals,
lies in the fact that they are responsible for much of the
molecule’s chemical reactivity and spectroscopic activity.
4
The covalent bonds formed by carbon atoms can be of two
types: s bonds, when the two atomic orbitals participating in
the bond overlap along the ideal axis linking the centres of the
two atoms; p bonds, when the overlap is orthogonal. The latter
involve two p orbitals and overlap with a s bond, thus
contributing to the formation of double and triple bonds.
Carbon atoms have a maximum valency of four, and are
therefore able to form four different bonds. However, as
already seen, the formation of covalent bonds is only
possible through the combination of orbitals containing a
single electron, whilst carbon atoms in the ground state only
have two unpaired electrons in p orbitals. To form four
bonds, therefore, one of the electrons located in the 2s2
orbital must be promoted to a higher energy level to occupy
the third unoccupied 2p orbital; this operation requires an
energy of about 80 kcal/mol (335 kJ/mol). The resulting
tetravalent carbon, therefore, has three electrons in p orbitals
and one in an s orbital. Due to the low energy differential
between 2p and 2s orbitals, the various orbitals can combine
with one another through a process known as hybridization
to generate a corresponding number of new equivalent
orbitals with an identical structure. Depending on the
number of hybridized orbitals, we speak of sp3 hybridization
(3 p orbitals⫹1 s), sp2 hybridization (2 p⫹1 s) or sp
hybridization (1 p⫹1 s). Obviously, the energy expended to
promote electrons to higher energy orbitals is recovered in
the formation of new hybrid orbitals with lower energy.
If a carbon atom forms four s bonds, its orbitals are sp3
hybridized. These sp3 orbitals consist of lobes which are
arranged spatially along the line which joins the atom’s
nucleus to the corners of a tetrahedron. The angles between
the bonding orbitals are identical and measure 109.5°.
If a carbon atom forms three s bonds and one p bond, its
orbitals are sp2 hybridized; this creates a planar trigonal
geometry with 120° angles between the three orbitals. The p
orbital, which does not participate in hybridization, is
perpendicular to the plane described by the three hybrid
orbitals. After the formation of the double bond, the two sp2
orbitals combine in a molecular s bonding orbital and in an
antibonding orbital, both oriented in the direction of the
bond which has formed. The two p orbitals oriented
perpendicular to the plane of the molecule overlap to a far
lesser extent than the sp2 orbitals, giving rise to a molecular
p bonding orbital and an antibonding orbital located above
and below the plane of the molecule; the resulting single
bond is far weaker than the s bond.
Finally, if a carbon atom forms two s bonds and two p
bonds, its orbitals are sp hybridized and positioned at a 180°
angle to one another. The structure of the simplest
hydrocarbons, with sp3, sp2 and sp hybridized carbon atoms
and their corresponding bonding orbitals, is shown in Fig. 2.
The length of C⫺C and C⫺H bonds varies depending
on the type of bond (single, double or triple), the
hybridization of the carbon atom participating in the bond,
and the molecule’s surroundings. The length of C⫺H bonds
gradually decreases as the p characteristic of the carbon to
which the hydrogen is bonded increases; it ranges from
1.091 Å for methane molecules, which have an sp3
hybridized carbon atom, to 1.084 Å for benzene molecules in
which the C⫺C bond is intermediate between single and
double, to 1.07 Å for ethene molecules in which the carbon
atom is sp2 hybridized, to 1.056 Å for ethyne molecules
which have sp hybridization. In molecules with sp3
ENCYCLOPAEDIA OF HYDROCARBONS
THEORETICAL ASPECTS
109.5°
methane
sp3 hybridization - s bond C⫺H
4 sp3 hybridized bonds
sp2 hybridization - s bond C⫺C
sp2 hybridization - 1 p bond
120°
ethylene or ethene
180°
acetylene or ethyne
sp hybridization - s bond C⫺C
sp hybridization - 2 p bonds
Fig. 2. Structure of methane (C⫺H s bonds with bond angles of 109.5°); ethylene (4 C⫺H s bonds, 1 C⫺C s bond with bond
angles of 120°, and 1 C⫺C p bond); acetylene (2 C⫺H s bonds, 1 C⫺C s bond with bond angles of 180°, and 2 C⫺C p bonds)
and their bonding orbitals, calculated by solving Schrödinger’s equation.
hybridized carbon atoms, the length of the C⫺H bond
varies depending on whether the carbon is primary,
secondary or tertiary. Specifically, it decreases in passing
from primary to secondary and tertiary carbons (Table 1).
The length of the single C⫺C bond gradually decreases
depending on whether the two carbon atoms forming the
bond are of type sp3-sp3, sp3-sp2, sp2-sp2, sp3-sp, sp2-sp,
sp-sp, passing from 1.54 Å for paraffins to 1.373 Å for the
single bond in molecules such as 1,3-butadiyne. Double
C⫺C bonds measure 1.337 Å between sp2-sp2 carbon
atoms; 1.395 Å in aromatic molecules (it should be noted
that the bond length is greater than a conventional double
bond, a characteristic in line with the fact that the C⫺C
bond is aromatic); 1.309 Å in cumulated dienes. Triple
C⫺C bonds have a length of 1.204 Å and this value is
VOLUME V / INSTRUMENTS
slightly higher in systems with triple conjugated bonds (see
Table 1).
In a hydrocarbon, the energy of the C⫺H bond has a
mean value of about 100 kcal/mol (Table 2), and varies
depending on the number and type of other bonds formed by
the carbon atom, given the effect which these have on its
electronegativity, in other words its capacity to attract
electrons. Specifically, its energy decreases in inverse
proportion to the degree of substitution from primary carbon
(bonded to four hydrogen atoms or three hydrogen and one
carbon atoms) to secondary carbon (bonded to two hydrogen
and two carbon atoms) and tertiary carbon (bonded to one
hydrogen and three carbon atoms).
It is interesting to observe that the energy of the C⫺H
bond in propylene and propyne is significantly lower than it
5
NATURE AND CHARACTERISTICS OF HYDROCARBONS
Table 1. Length of C⫺C and C⫺H bonds in hydrocarbons (Weast, 1987)
Molecule type
Hybridization of carbon atoms
Length (Å)
C⫺H
methane
sp3
1.091
C⫺H
ethane
sp3-primary
1.101⫾0.003
C⫺H
2-methylpropane
sp3-secondary
1.073⫾0.004
C⫺H
2,2-dimethylpropane
sp3-tertiary
1.070⫾0.007
C⫺H
ethene
sp2
1.07⫾0.01
C⫺H
benzene
sp2
1.084⫾0.006
C⫺H
ethyne
sp
1.056⫾0.003
single C⫺C
paraffin
sp3-sp3
1.540⫾0.003
single C⫺C
propene-toluene
sp3-sp2
1.53⫾0.01
single C⫺C
1,3-butadiene
sp2-sp2
1.47⫾0.01
single C⫺C
propyne
sp3-sp
1.460⫾0.003
single C⫺C
1-buten-3-yne
sp2-sp
1.44⫾0.01
single C⫺C
1,3-butadiyne
sp-sp
1.373⫾0.001
double C⫺C
ethene
sp2-sp2
1.337⫾0.006
double C⫺C
benzene
sp2-sp2
1.395⫾0.003
double C⫺C
allene
sp2-sp2-sp2
1.309⫾0.005
triple C⫺C
ethyne
sp-sp
1.204⫾0.002
triple C⫺C
2,4-hexadiyne
sp-sp-sp-sp
1.206⫾0.004
Bond
is in ethylene and acetylene. This is because the free radicals
formed in the bond separation process are stabilized by an
electron resonance phenomenon (see Section 1.2).
The bond energy between two carbon atoms, as
previously seen for the C⫺H bond, is influenced by the
coordination of the two atoms themselves. Additionally,
carbon, compared to hydrogen, has the ability to form
double bonds (1 s and 1 p) and triple bonds (1 s and 2 p).
The energy of some important C⫺C bonds is reported in
Table 2.
Table 2. C⫺H e C⫺C bond energies
reported for C2 hydrocarbons (Weast, 1987)
The energy of a molecule, that of its molecular bonding
orbitals, its spatial structure and most of its properties are
determined by its wavefunction y, whose form is defined by
Schrödinger’s equation [1]. In order to describe a molecule,
it is legitimate to separate the energy contributions of the
rotational, vibrational and translational motions of the nuclei
from those of the electrons. According to this approximation,
∧
known as the Born-Oppenheimer approximation, H can be
expressed as follows:
1
Z
1
∧
H ⫽冱⫺ 23 ⵜ2i ⫹冱冱⫺ 14a 4 ⫹冱 冱 1
[4]
2
r
r
ai
ij
i
i
a
i⬎j j
Name
Bond energy
(kcal/mol)
H⫺CH2CH3
ethane
100.5
H3C⫺CH3
ethane
89.6
H⫺CHCH2
ethene
110.6
H2C⫽CH2
ethene
172.1
where the first term represents the kinetic energy of the
electrons, the second the Coulomb attraction energy between
electron and nucleus having a positive charge Za and distance
rai , and the last term the electron-electron repulsion energy.
The integration of Schrödinger’s equation is possible
analytically only in the absence of interaction terms between
electrons. Were these to be ignored, the wavefunction y
could be expressed simply as the product of monoelectronic
functions /i , each of which is a solution of the equation:
H⫺CCH
ethyne
111.9
[5]
HC⬅CH
ethyne
229.8
Molecule
6
1.1.2 The Hückel method
1
Z
⫺ 23 ⵜ2i /i ⫺ 冱⫺ 14a4 /i ⫽ei/i
2
r
ai
a
or, in more concise form:
ENCYCLOPAEDIA OF HYDROCARBONS
THEORETICAL ASPECTS
[6]
∧
h/i ⫽ei/i
Among the various approaches proposed for solving
Schrödinger’s equation, the method suggested by Hückel for
the study of conjugated systems deserves special attention.
This method is based on the idea that a generic molecular
orbital y can be expressed as the linear combination of
atomic orbitals ck of equation [2]. The approximation
introduced by∧ Hückel involves the assumption that the
Hamiltonian H can be broken down into the sum of the
monoelectronic Hamiltonians, whose functional form is not
known a priori. Therefore, electron-electron interaction is
apparently ignored; it will be reintroduced later as a
semiempirical parameter. The resolving system of equations
is given by equations [2] and [5] in the unknowns cik . The
energy of each electron can be expressed as the integral of the
product of the complex conjugate of its wavefunction /i by
h/i extended over the whole volume of the system and
normalized:
∧
[7]
∫/*i h/i dt
ei⫽ 1111
∫/*i /i dt
The problem can be resolved by applying the variational
theorem, according to which the correct values of cik are
those with minimal energy, so that:
[8]
⭸e
124
⫽0
⭸ck
The substitution of equations [7] and [5] into equation
[8] results in a system of k equations which take the form:
[9]
冱 cm(Hkm ⫺eSkm)⫽0
m
∧
where Hkm⫽∫ c*k hcm and Skm⫽∫ c*k cm. This equation has
non-trivial solutions if, and only if, the determinant of the
matrix of the coefficients is zero:
[10]
det 兩Hkm ⫺eSkm 兩⫽0
The solution of equation [10] consists of a system of m
equations in m unknowns, where m is the number of atomic
orbitals chosen to describe the molecular orbitals. In the
Hückel method, it is assumed that normalized atomic
orbitals are used, so that the value of the terms Skk is unitary,
and it is essential that the value of the superposition integrals
Skm is zero. Additionally, Hkk is assumed to be equal to a,
whilst the Hkp integrals, dependent on the distance between
atom k and atom p, are assumed to be equal to b when
dealing with two adjacent carbon atoms and to 0 in other
cases. Both a and b are negative. By developing the
determinant in equation [10], a polynomial of order m in the
energy of the system is obtained, which will therefore have
m solutions, corresponding to m possible energy levels of m
different orbitals, of which only those with the lowest energy
will be occupied.
Although molecular energy can currently be calculated
with the direct numerical integration of Schrödinger’s
equation, the Hückel method nonetheless makes it possible to
understand, and in part to quantify, the origin of the
stabilization due to the conjugation of hydrocarbon molecules
in which p bonds are present. To this end, it is worth studying
the case of the butadiene molecule, which has the formula
C4H6 and is characterized by the presence of two double
bonds in positions 1 and 3, separated by a s bond:
VOLUME V / INSTRUMENTS
2
4
1
3
Applying the Hückel method to the four electrons present
in the pz orbitals of carbon atoms 1-4, the determinant defining
the energy of the system is:
a⫺E
b
0
0
b
a⫺E
b
0
0
b
a⫺E
b
0
0
b
a⫺E
[11]
⫽0
which when developed gives the fourth-degree equation:
[12]
(a ⫺E)4 ⫺3b2(a ⫺E)2 ⫹b4 ⫽0
which can be solved analytically. The four permitted energy
levels are therefore:
[13]
E1,2 ⫽a ⫾1.618b
E3,4 ⫽a ⫾0.618b
Remembering that each orbital can contain two electrons
with opposite spin, the two bonding p orbitals of butadiene have
an energy of a⫹1.618b and a⫹0.618b, giving a total energy of
(4a⫹2冪5)b. Since the energy of an electron in a non-conjugated
p orbital is a⫹b, it is possible to calculate the energy deriving
from the conjugation by subtracting the value calculated using
the Hückel method. The energy of stabilization by conjugation
is (2冪5⫺4)b and is described as the energy of molecular
delocalization. As might be expected, the conjugation energy is
proportional to parameter b, which represents the interaction
between pz electrons belonging to adjacent atoms.
Finally, it is important to note that the systematic
application of the Hückel method to aromatic molecules
makes it possible to justify the rule for the stability of
aromatic compounds (see Section 1.2).
Bibliography
Atkins P.W. (1994) Physical chemistry, Oxford, Oxford University
Press.
Atkins P.W., Friedman R. (1997) Molecular quantum mechanics,
Oxford, Oxford University Press.
Graham Solomons T.W. (1993) Chimica organica, Bologna,
Zanichelli.
Morrison R.T., Boyd R.N. (1969) Chimica organica, Milano, Casa
Editrice Ambrosiana.
References
Weast R.C. (editor in chief) (1987) CRC handbook of chemistry and
physics. A ready-reference book of chemical data, Boca Raton
(FL), CRC Press.
Carlo Cavallotti
Davide Moscatelli
Dipartimento di Chimica, Materiali
e Ingegneria chimica ‘Giulio Natta’
Politecnico di Milano
Milano, Italy
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