Kelter P., Mosher M., Scott A. Chemistry. The Practical Science

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2sp

AlH

2sp

c. AlH

geometry: The aluminum atom is sp

hybridized and has a trigonal

planar geometry. The bond angle is 120°.

PRACTICE 9.4

For each of the molecules in Practice 9.3, indicate the geometry about the central

atom.

See Problems 25–28, 35, and 36.

d and sp

Orbitals

Hybrid orbitals can also be constructed to account for expanded octets.A good ex-

ample is phosphorus pentachloride (PCl

), a reactive molecule used to make com-

pounds containing chlorine. The Lewis dot structure model of PCl

indicates that

there should be ﬁve bonds to the phosphorus atom. Because the combination of

all of the s and p orbitals will allow only four bonds, we must include a d orbital

in the hybridization scheme. Five orbitals on the phosphorus can be made by

EXERCISE 9.4 Shapes of the Molecules

For each of the molecules in Exercise 9.3, indicate the geometry around the central

atom.

Solution

a. H

O geometry: sp

hybridized atoms adopt a tetrahedral geometry. Because

two of the sp

orbitals contain lone pairs, the VSEPR model indicates that the

molecule has an overall bent geometry. The bond angle should be less than

109.5° because the lone pairs repel each other more than the bonding pairs.

The angle HOOOH is actually 104.5°.

368 Chapter 9 Advanced Models of Bonding

b. NH

geometry: Again, we should have sp

hybridized orbitals with a tetrahe-

dral geometry. Because one of the sp

orbitals contains a lone pair, ammonia

has a trigonal pyramidal geometry. The bond angle (HONOH) should be

less than 109.5° because of the repulsions between the lone pair and the

bonding pairs of electrons (the angle is actually 107°).

9.2 Hybridization 369

hybridizing the 3s, three 3p, and one of the 3d orbitals. The result is ﬁve degener-

ate 3sp

d orbitals. We can mix the d orbital because it is in the same principal en-

ergy level and is similar in energy to the other orbitals. The resulting valence bond

model of PCl

shows overlap of a 3p orbital from the chlorine atom with one of the

3sp

dorbitals from the phosphorus atom (Figure 9.15). The space-ﬁlling model of

the molecule shows how the orbitals ﬁt together to give the molecule its essential

structure.

Other examples of expanded octets are found in compounds containing

xenon. The ﬁrst example of a compound containing xenon, a noble gas, was made

in 1962. Shortly after the discovery that xenon could make compounds with other

elements, XeF

,XeF

, and XeF

were prepared. The orbital overlap required to

make xenon tetraﬂuoride (XeF

) shows why these compounds are possible. The

Lewis dot structure of XeF

requires six orbitals (four for bonds and two for

lone pairs). We hybridize a 5s, three 5p, and two 5d orbitals to make six degener-

ate 5sp

orbitals. Each bond in XeF

results from overlap of a 2p orbital from the

ﬂuorine atom with a 5sp

orbital from the xenon atom, with the two lone pairs

on the xenon atom occupying two 5sp

orbitals (Figure 9.16).

Sigma and Pi Bonds

Our discussion so far has centered on the use of valence bond theory to address

the formation of single bonds in molecules. However, we know from Chapter 8

that atoms can participate in multiple bonds.

How is this possible if hybridization

provides only orbitals that are oriented directly at the atom with which they are

bonded?

The answer lies in the different ways in which orbitals can overlap in va-

lence bond theory.

Sigma bonds (σ bonds), which make up the framework of a

molecule, result from an end-on overlap of orbitals. Almost every “single bond”

is a sigma bond. In fact, every bond we’ve discussed thus far in this chapter results

from end-on overlaps of orbitals, such as the four σ bonds in CH

, each of which

3sp

5sp

Valence shell electron

configuration of phosphorus

Hybridized valence

shell on phosphorus

3d 3d 3d 3d 3d 3d 3d 3d 3d

3p3p 3p

3sp

d 3sp

FIGURE 9.15

Hybridization of the phosphorus in PCl

. Because

ﬁve bonds are needed to complete the structure,

ﬁve orbitals are hybridized on the phosphorus atom

to give ﬁve new hybrid orbitals.

Valence shell electron

configuration of xenon

Hybridized valence

shell on xenon

5d 5d 5d 5d 5d 5d 5d 5d

5p 5p

5sp

FIGURE 9.16

Hybridization of xenon in XeF

Valence shell electron

configuration of carbon

Hybridization

of carbon

Four bonds can be formed

with these orbitals.

2p 2p

2sp

σ bond

370 Chapter 9 Advanced Models of Bonding

FIGURE 9.17

Methane hybridization

and sigma bonds.

2p 2p 2p 2p

sp sp

Energy

π bondσ bond

FIGURE 9.18

Acetylene pi bonds.

There are two π bonds

in acetylene that result

from the overlap of two

p orbitals from each

carbon atom.

is an end-on overlap of an sp

orbital and an s orbital. The resulting σ bond does

not possess a nodal plane along the axis of the bond (Figure 9.17) and has a shape

reminiscent of a hot dog.

Nodal planes are ﬂat, imaginary planes passing through

the bonded atoms where the orbital does not exist. An example of a nodal plane

is found in many molecules that contain double bonds.

Pi bonds (π bonds), which react with other molecules as if they are electron-

rich bonds, result from the side-to-side overlap of orbitals. This type of bond oc-

curs primarily when p orbitals on adjacent atoms overlap, as is found between

atoms containing a double or triple bond. The resulting π bond has one nodal

plane along the axis of the bond and has the shape of the two halves of a hot dog

bun separated by empty space. That is, there is a plane that exists between the

atoms that is not occupied by π electrons. Acetylene (C

) is a molecule that

contains two π bonds resulting from the overlap of p orbitals (Figure 9.18).

Why do we show the 2sp and 2p orbitals in the diagrams in Figure 9.18 with

one electron each? Shouldn’t we ﬁll the 2sp orbitals before ﬁlling the slightly

higher energy 2p orbitals? Hund’s rule would seem to suggest this, but because the

energy difference between the hybridized and unhybridized orbitals of the same

shell is relatively small, the electrons can be temporarily placed in higher energy

2p orbitals prior to bond formation. The bonds that are created end up with lower

energies than they would have had in the hybridized orbitals. Keep in mind that

because each hybridized orbital will overlap to form a bond, no more than one

electron can be placed in each of the hybridized orbitals. The neighboring atom

will provide the second electron used in the bond. In addition, it is important to

note that although we have discussed the hybridization process as a series of steps,

no hybridization will take place without the formation of bonds. That is, in the

absence of any other atoms, hybridization does not occur in an individual atom.

EXERCISE 9.5 Polymers and Pi Bonds

Ethylene (C

) is used as the starting material for the preparation of polyethylene,

a common polymer used in the production of plastic bags, plastic sheeting, and the

like (see Chapter 13 for more on plastics). Describe the bonding patterns in ethyl-

ene using the valence bond theory.

Video Lesson: Pi Bonds

Solution

After hybridization of the two carbon atoms in ethylene, we observe that the 2sp

orbitals on the carbons can overlap to form a sigma bond. In addition, each carbon

atom still contains a half-ﬁlled 2p orbital. Such orbitals can overlap to create a pi

bond between the atoms. Ethylene, shown in Figure 9.19, has one pi bond and a

total of ﬁve sigma bonds.

PRACTICE 9.5

Describe the valence bond theory for formaldehyde (CH

O), a compound used in

the preservation of biological tissues.

See Problems 11, 12, 37–40, 96, and 99.

Application of Hybridization Theory

The hybridization of orbitals allows us to address the same geometric features

that are explained in the VSEPR approach to modeling. Why do we hybridize the

orbitals in water?

Does it give a better model? Valence bond theory indicates that

if we didn’t hybridize the orbitals of the central oxygen in water, there would still

be two bonds. However, the bonds that would result from the lack of hybridiza-

tion would have 90° angles between each other (overlap of a hydrogen s orbital

with an oxygen p orbital). Hybridization is necessary for the model of water to

agree with the experimentally measured bond angles.

The use of hybridized orbitals also provides information about the relative

lengths of bonds in molecules. We’ve already discussed how s–p orbital overlap

produces a bond that is longer than one containing s–s orbital overlap. This dis-

cussion can be carried further to illustrate the lengths of hybridized orbital

bonds. Table 9.4 lists the relative sizes of these orbitals and some features of the

resulting hybridization. From this table we see that the sp

hybrid orbital makes

bonds that are longer than the sp

. The sp

hybrid orbital is longer than the sp.Ap-

parently, mixing more p orbitals into the hybrid produces longer hybrid orbitals.

And we note that a bond made from sp

–sp

orbital overlap is longer than one

made from s–sp

overlap.The values in the table are average values (bond lengths

and bond angles) based on actual measurements from a collection of similar

molecules.

9.2 Hybridization 371

2p 2p 2p

Energy

Unhybridized p orbitals

π bondσ bond

Hybrid orbitals

FIGURE 9.19

Ethylene and valence bond theory. The overlap of the orbitals between the carbons gives rise to a sigma

bond and a pi bond. The sigma bond results from 2sp

–2sp

overlap. The pi bond results from the side-to-

side overlap of the 2p orbitals.

Visualization: Formation of

CPC Double Bond in Ethylene

EXERCISE 9.6 Bond lengths of C—C Single Bonds

The Lewis dot structures of butane (often used as lighter ﬂuid), 1–butene (a com-

pound used to make plastic wraps), and 1,3-butadiene (the starting material used to

make synthetic rubber for your car’s tires) are shown in Figure 9.20. Which has the

longest COC bond? Which has the shortest COC bond?

372 Chapter 9 Advanced Models of Bonding

CH C

Butane

FIGURE 9.20

Butane, 1-butene, and 1,3-butadiene.

(The bond length in question is indicated

with an arrow.)

First Thoughts

Our ﬁrst goal here is to determine the hybridization of the atoms in each of the mol-

ecules. Then, using this information, we should put together a geometric picture of

each molecule that shows the types of orbital overlap that we expect for each bond.

Solution

The shortest COC bond is found in 1,3-butadiene. It results from the overlap of an

hybridized orbital with another sp

hybridized orbital. The longest bond is

found in butane (sp

–sp

orbital overlap).

Further Insights

The COC bond length in 1,3-butadiene indicates that the electrons in the double

bonds can travel throughout the entire molecule. We can illustrate this by drawing

resonance structures showing a double bond between the two central carbon atoms.

PRACTICE 9.6

Which has a longer NOO bond length, CH

NOH or CH

NHOH?

See Problems 13–15.

Advantages and Disadvantages of Hybridization

We must gain some advantage from spending time developing the hybridized

model of a molecule. In fact, we do. With this model, we are able to explain bond

angles, the lengths of bonds in molecules, and the existence of pi bonds. Unfor-

tunately, valence bond theory and hybridization have some shortcomings. The

relative size of many bond lengths can be accurately predicted, though some are

“mystifyingly” unable to be accurately estimated. For example, why is the CPC

bond longer in 1,3-butadiene than in 1-butene? An even more reﬁned picture of

the orbitals on the molecule needs to be considered.

Organic chemists do put hybridization to good use. Often they use it to

describe the three-dimensional structure of a molecule to others. This approach

enables organic chemists to talk to other organic chemists without drawing

HHHH

CCC C

–

HHHH

CCC C

–

HHHH

CCC C

–

HHHH

CCC

1-Butene

CCC

1,3-Butadiene

pictures on cocktail napkins. It also allows the organic chemist to describe spe-

ciﬁc structural features of a molecule that play an important role in the chemical

and physical properties of that molecule.

EXERCISE 9.7 Using Hybrid Orbital Theory: Will the Molecule Be Solid or Liquid?

Animal fats, such as lard, tend to be solid. Vegetable oils, such as corn oil, tend to be

liquid. Fats and oils are made up of compounds known as triacylglycerols. In turn,

the triacylglycerols are constructed from molecules known as fatty acids. Look at the

two fatty acids, stearic and oleic acids, depicted below. Describe the shapes of these

molecules using valence bond theory, and use this information to determine which

molecule is more likely to exist as a solid at room temperature.

9.2 Hybridization 373

Stearic acid

Oleic acid

CH CHCH

First Thoughts

What makes a compound likely to be solid at room temperature? If the molecules

can pack easily and tightly together, they will be more likely to form a solid. If the

molecules are more rigid and less able to get close together, they will be less likely to

form a solid at room temperature.

Solution

Stearic acid is made up of 17 carbon atoms that are sp

hybridized and 1 carbon

atom that is sp

hybridized. The sp

hybridized carbons involve single bonds to ad-

jacent carbons and can rotate to form a zig-zag pattern of atoms. The sp

hybridized

carbon is more rigid in its trigonal planar conﬁguration. Oleic acid contains mostly

hybridized carbon atoms, but it also has three sp

hybridized carbons. The pres-

ence of two of the sp

hybridized atoms in the middle of the long carbon chain

causes a kink in the zig-zag structure. Steric acid can pack tightly with other steric

acid molecules, so it is a solid at room temperature. Oleic acid, because of its bent,

rigid structure doesn’t stack together well with other molecules. Oleic acid is a

liquid at room temperature.

11-cis-Retinal

Further Insights

Knowing the three-dimensional structure of a molecule can provide information

about the melting temperature of a molecule. But the three-dimensional structure

of a molecule can also yield a wealth of other information. Physical properties, such

as boiling point, melting point, color, taste, and smell, and chemical properties, such

as reactivity with acids or bases, are all dependent on the shape of the molecule.

PRACTICE 9.7

Describe the three-dimensional shape of each of the following molecules. Solely on

the basis of its shape, which do you predict will have the highest boiling point?

a. CH

(pentane)

b. CH

CH(CH

)

(2-methylbutane)

c. C(CH

)

(2,2-dimethylpropane)

See Problems 41, 42, 47, 48, 53, and 54.

HERE’S WHAT WE KNOW SO FAR

■

We can mathematically combine two or more orbitals to provide new orbitals

of equal energy in the process of hybridization.

■

The models of many molecules agree with the experimentally measured bond

angles when hybridization is taken into account.

■

Although the hybridization process itself “runs uphill” energetically, the end

result is still energetically downhill.

■

Hybrid orbitals can be constructed to account for atoms that observe the octet

rule as well as for those with expanded octets.

■

The sigma (σ) bond does not possess a nodal plane along the axis of the bond

and has a shape reminiscent of a hot dog. A pi (π) bond results from the side-

to-side overlap of orbitals and is found between atoms containing a double or

triple bond. It has one nodal plane along the axis of the bond.

9.3 Molecular Orbital Theory

Let’s revisit the molecule of retinal (the light-absorbing portion of the rhodopsin

protein in our eyes) that we introduced in the chapter opener. 11-cis-Retinal is a

molecule containing an extended series of alternating single and double bonds

known as

conjugated π bonds (or conjugation), as shown in Figure 9.21. Electrons

in π bonds (whether conjugated or not) can absorb a photon of light. These elec-

trons become excited (more energetic) when the photon is absorbed and the elec-

trons are promoted to a higher energy state. The models we’ve already discussed

say nothing about these higher energy states. Here are the key ideas we will ex-

plore in this section:

1. Higher energy states can be reached by the absorption of energy.

2. All electrons in a molecule can absorb photons of light.

374 Chapter 9 Advanced Models of Bonding

Application

FIGURE 9.21

11-cis-retinal is a conjugated molecule. Note that there

are alternating single and double bonds through a large

portion of the molecule.

Antibonding

Bonding

Energy

In LCAO–MO theory, atomic orbitals are

added together, both constructively and

destructively, to make molecular orbitals.

The resulting bonding orbital is of lower

energy, and the antibonding orbital is of

higher energy, than the individual atomic

orbitals.

9.3 Molecular Orbital Theory

375

3. To become excited, core electrons require more energy (such as that of

X-rays) than do valence electrons (for which the energy of UV or visible light

is enough).

4. Sigma-bonding electrons require more energy (UV less than 200 nm) than

pi-bonding electrons (visible between 200 and 700 nm) to become excited.

5. Electrons in conjugated π bonds require even less energy than unconjugated

π bonds to promote an electron in a π bond to the higher energy state.

6. As the conjugated π system gets longer, the energy of the photon required to

excite the electrons is reduced, moving from the blue toward the red region

of the visible electromagnetic spectrum.

This ﬁnal property is used by photography chemists in the design of mole-

cules that can absorb a particular wavelength of light. Molecular orbital theory

can also be used to tell us what happens to bonds within a molecule when they

absorb light, why hydrogen exists as a diatomic molecule and helium does not,

and why some molecules (such as O

) are attracted to magnetic ﬁelds and why

some molecules (such as N

) are not.

Molecular Orbital Theory Deﬁned

At about the same time that Linus Pauling worked out valence bond theory,

Robert S. Mulliken (1896–1986) began thinking about how bonds could arise

from delocalized valence orbitals. Erwin Schrödinger (1887–1961) further devel-

oped this approach by devising a mathematical equation that described the hy-

drogen atom. This theory, known as

molecular orbital (MO) theory, is based on the

principles of quantum mechanics that we discussed in Chapter 6. It treats elec-

trons not as particles, but as waves that encompass the entire molecule. As in the

quantum mechanical description of atomic orbitals, molecular orbitals encircle

the atoms in a molecule. The Schrödinger equation deﬁnes the energy of each of

the orbitals of an atom or molecule. Unfortunately, the Schrödinger equation

cannot be solved exactly, except for a few very simple systems, so we usually make

some approximations that allow us to arrive at a usable solution. Probably the

most commonly used approximation is known as the

linear combination of atomic

orbitals–molecular orbitals (LCAO–MO) theory

. In this theory, atomic orbitals are

added together (both constructively and destructively) to make molecular

orbitals.

We must raise several points about the formation of molecular orbitals from

atomic orbitals (see Table 9.5). The most important point is that the combination

of two orbitals gives two new orbitals, just as in the hybridization of atomic or-

bitals. One of the new molecular orbitals, called the

bonding orbital, results from

the addition of two overlapping atomic orbitals; the other, called the

antibonding

orbital

, results from the subtraction of two overlapping atomic orbitals. The

bonding orbital is lower in energy than either of the two atomic orbitals (the for-

mation of a molecular orbital with lower energy than the atomic orbitals drives

the formation of the bond). The bonding orbital indicates that there is some elec-

tron density between adjacent nuclei (a bond exists). The antibonding orbital is

higher in energy than the two atomic orbitals from which it is formed and is typ-

ically represented with an asterisk (

∗

) to distinguish it from the bonding orbital.

The antibonding orbital indicates a lack of electron density between adjacent nu-

clei (no bond exists). Each new molecular orbital (both bonding and antibond-

ing) can contain two electrons.

A similarly important point in mixing atomic orbitals to make molecular or-

bitals is that only atomic orbitals of similar symmetry (shape and orientation)

and energy provide signiﬁcant overlap. This rule means that only 1s orbitals over-

lap to a great degree with 1s orbitals. The 2p

orbital overlaps best with a 2p

or-

bital. Their symmetry and size (i.e., energy) are similar. The 1s orbital doesn’t

Video Lesson: Molecular Orbital

Theory

overlap well with a 3s orbital (the size isn’t a good match). The 2p

orbital does

not overlap well with the 2p

orbital (the symmetry doesn’t allow them to line up

properly).

We can think of this by imagining that overlapping orbitals are a game of

Tetris. In Figure 9.22, we can see that the best game piece to use to ﬁll the 4 × 4

hole at the bottom of the screen is a 4 × 4 rectangle. A 3 × 3 rectangle wouldn’t

do a very good job of ﬁlling the hole, just as a 1s orbital and a 3s orbital don’t

overlap very well. They have the same symmetry (shape), but the size isn’t right.

376 Chapter 9 Advanced Models of Bonding

Key Features of MO Diagram Construction

TABLE 9.5

Good overlap

Poor overlap

Energy

2s 2s

Good overlap

Poor overlap

2s 1s

We restrict ourselves by insisting that only orbitals of

similar energy can mix to make molecular orbitals

(MOs). The 2s orbital, for example, can mix with a 2s

orbital but does not mix well with the 1s orbital.

Mixing atomic orbitals gives the same number of MOs.

• The bonding orbital is lower in energy than the atomic

orbitals from which it was made.

• The antibonding orbital is higher in energy than the

atomic orbitals from which it was made.

Electrons are placed in MOs in order of increasing energy.

Only orbitals that exhibit symmetry (that is, they are

similar in shape and orientation) can mix to make

MOs. For example, the 2p

orbital cannot mix with

the 2p

orbital, since they are oriented differently.

Each MO can hold two electrons with opposite spin.

• Electrons that are placed in bonding MOs promote

bonding.

• Electrons that are placed in antibonding MOs favor

separated atoms.

After mixing of the atomic orbitals, they

cease to exist. Only the molecular orbitals

remain.

In the second image of Figure 9.22, we need a vertical 1 × 3 rectangle to ﬁll the

vertical 1 × 3 hole. A horizontal rectangle of the same size wouldn’t work, just as

a 2p

orbital doesn’t overlap with a 2p

orbital. The symmetry (shape) of the two

orbitals doesn’t match, even though their size does.

Shapes of Orbitals

What about the shapes of the molecular orbitals? Molecular orbitals encompass

entire molecules, and the shapes of the molecular orbitals often look similar to

portions of the entire molecule. We know about their existence by calculation

using some approximations of the Schrödinger equation, but how do we really

know what the shape of a molecule is? X-ray crystallography (see Section 13.1) is

one of the methods used to take a “snapshot” of a molecule. But it relies on de-

termining the average location of the atoms in a crystal. The scanning tunneling

microscope that we discussed in Chapter 6 may actually be the closest we can get

to taking a picture of a single molecule.

The strength of the bond between two atoms can be represented by the

bond

order

, the number of electrons in bonding orbitals minus the number of elec-

trons in antibonding orbitals, divided by 2.

Bond order =

bonding electrons − antibonding electrons

A larger bond order indicates a greater bond strength. The bond order typically

agrees with the Lewis dot structure’s expected number of bonds between adjacent

atoms. However, the bond order between two atoms doesn’t have to be a whole

number.

To summarize, molecular orbital theory holds that when the atoms of a mol-

ecule come together, atomic orbitals on each atom that are similar in symmetry

and energy not only overlap but are transformed into a series of new orbitals that

surround the entire molecule. These molecular orbitals have completely different

shapes, sizes, and energies than the atomic orbitals from which they came.

Although the atomic orbitals cease to exist after the creation of the molecular

orbitals, the new molecular orbitals follow the same rules that guide us in placing

electrons in the atomic orbitals.

MO Diagrams

Just as the civil engineer diagrams the layout of a building with a blueprint, we

represent the placement of electrons in molecular orbitals with our own type of

blueprints called

molecular orbital (MO) diagrams. These diagrams graphically

9.3 Molecular Orbital Theory 377

(a) (b)

FIGURE 9.22

Tetris analogy to overlapping orbitals:

(a) The 4 × 4 square piece has the

same size and shape as the hole. The

3 × 3 square has the same shape but

the wrong size. (b) The 1 × 3 rectangle

has the same size and shape as the

hole. The horizontal 1 × 3 rectangle has

the same size but is the wrong orienta-

tion to ﬁt into the hole.

Visualization: Pi Bonding and

Antibonding Orbitals