5 The Modern View of Electronic Structure: Wave or Quantum Mechanics

6.5  The Modern View of Electronic Structure: Wave or Quantum Mechanics







281



6.5 T

​ he Modern View of Electronic Structure:

Wave or Quantum Mechanics

How does wave–particle duality affect our model of the arrangement of electrons

in atoms? Following World War I, German scientists Erwin Schrödinger (1887–

1961), Max Born (1882–1970), and Werner Heisenberg (1901–1976) provided

the answer.

Erwin Schrödinger received the Nobel Prize in physics in 1933 for a comprehensive theory of the behavior of electrons in atoms. Starting with de Broglie’s hypothesis that an electron could be described as a matter wave, Schrödinger developed a model for electrons in atoms that has come to be called quantum mechanics

or wave mechanics. Unlike Bohr’s model, Schrödinger’s model can be difficult to

visualize, and the mathematical approach is complex. Nonetheless, the consequences of the model are important, and understanding its implications is essential

to understanding the modern view of the atom.

Let us start by thinking of the behavior of an electron in the atom as a standing

wave. If you tie down a string at both ends, as you would the string of a guitar, and

then pluck it, the string vibrates as a standing wave (Figure 6.11). There are only

certain vibrations allowed for the standing wave formed by a plucked guitar string.

That is, the vibrations are quantized. Similarly, as Schrödinger showed, only certain

matter waves are possible for an electron in an atom.

Following the de Broglie concept, we can adopt the quantum-mechanical view

that an electron in an atom behaves as a wave. To describe this wave, physicists introduced the concept of a wavefunction, which is designated by the Greek letter ψ

(psi). Schrödinger built on the idea that the electron in an atom has the characteristics of a standing wave and wrote an equation defining the energy of an electron

in terms of wavefunctions. When this equation was solved for energy—a monumental task in itself—he found the following important outcomes:

•

•



•  Wave Functions and Energy 

In Bohr’s theory, the electron energy for

the H atom is given by E­n = −Rhc/n2.

Schrödinger’s electron wave model

gives the same result.



Only certain wavefunctions are found to be acceptable, and each is associated

with an allowed energy value. That is, the energy of the electron in the atom is

quantized.

The solutions to Schrödinger’s equation for an electron in three-dimensional space

depend on three integers, n, ℓ, and mℓ, which are called quantum numbers. Only

certain combinations of their values are possible, as we shall outline below.



The next step in understanding the quantum mechanical view is to explore the

physical significance of the wavefunction, ψ (psi). Here we owe much to Max Born’s

interpretation. He said that

(a) the value of the wavefunction ψ at a given point in space is the amplitude

(height) of the electron matter wave. This value has both a magnitude and

a sign that can be either positive or negative. (Visualize a vibrating string

Figure 6.11   Standing

waves.  A two-dimensional standing



1/ ␭

2



1␭



Node

3/ ␭

2



Node



kotz_48288_06_0266-0299.indd 281



Node



wave such as a vibrating string must

have two or more points of zero

amplitude (called nodes), and only

certain vibrations are possible. These

allowed vibrations have wavelengths

of n(λ/2), where n is an integer

(n = 1, 2, 3, . . .). In the first vibration

illustrated here, the distance between

the ends of the string is half a wavelength, or λ/2. In the second, the

string’s length equals one complete

wavelength, or 2(λ/2). In the third

vibration, the string’s length is 3(λ/2).



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282



c h a p t er 6   The Structure of Atoms



in a guitar or piano, for example (Figure 6.11). Points of positive amplitude

are above the axis of the wave, and points of negative amplitude are

below it.)

(b)the square of the value of the wavefunction (ψ2) is related to the probability

of finding an electron in a tiny region of space. Scientists refer to ψ2 as a

probability density. Just as we can calculate the mass of an object from the

product of its density and volume, we can calculate the probability of finding

an electron in a tiny volume from the product of ψ2 and the volume. The

Born interpretation of the wavefunction is that, for a given volume, whenever

ψ2 is large, the probability of finding the electron is larger than when ψ2 is

small.

There is one more important concept to touch on as we try to understand the

modern quantum mechanical model. In Bohr’s model of the atom, both the energy and location (the orbit) for the electron in the hydrogen atom can be described accurately. However, Werner Heisenberg postulated that, for a tiny object

such as an electron in an atom, it is impossible to determine accurately both its

position and its energy. That is, any attempt to determine accurately either the

location or the energy will leave the other uncertain. This is now known as Heisenberg’s uncertainty principle: If we choose to know the energy of an electron in an atom

with only a small uncertainty, then we must accept a correspondingly large uncertainty in

its position. The importance of this idea is that we can assess only the likelihood, or

probability, of finding an electron with a given energy within a given region of

space. Because electron energy is the key to understanding the chemistry of an

atom, chemists accept the notion of knowing only the approximate location of the

electron.



•  Orbits and Orbitals 

In Bohr’s model of the H atom, the

electron is confined to a prescribed

path around the nucleus, its orbit, so

we should be able to define its position

and energy at a given moment in time.

In the modern view, the term orbital is

used to describe the fact that we have

a less definite view of the electron’s location. We know its energy but only

the region of space within which it is

probably located, that is, its orbital.



Quantum Numbers and Orbitals

The wavefunction for an electron in an atom describes what we call an atomic orbital. Following the ideas of Born and Heisenberg, we know the energy of this electron, but we only know the region of space within which it is most probably located.

Thus, when an electron has a particular wavefunction, it is said to “occupy” a particular orbital with a given energy, and each orbital is described by three quantum

numbers: n, ℓ, and mℓ. Let us first describe the quantum numbers and the information they provide and then discuss the connection between quantum numbers and

the energies and shapes of atomic orbitals.



n, the Principal Quantum Number (n ​= ​1, 2, 3, . . .)

The principal quantum number n can have any integer value from 1 to infinity. The

value of n is the primary factor in determining the energy of an orbital. It also defines

the size of an orbital: for a given atom, the greater the value of n, the greater the size

of the orbital.

In atoms having more than one electron, two or more electrons may have the

same n value. These electrons are then said to be in the same electron shell.



ℓ, the Orbital Angular Momentum Quantum Number

(ℓ ​= ​0, 1, 2, 3, . . . , n − 1)



•  Electron Energy and Quantum

Numbers  The electron energy in the

H atom depends only on the value of

n. In atoms with more electrons, the

energy depends on both n and ℓ, as

you shall see in Chapter 7.



kotz_48288_06_0266-0299.indd 282



Orbitals of a given shell can be grouped into subshells, where each subshell is characterized by a different value of the quantum number ℓ. The quantum number ℓ, referred to as the “orbital angular momentum” quantum number, can have any integer

value from 0 to a maximum of n ​− ​1. This quantum number defines the characteristic

shape of an orbital; different ℓ values correspond to different orbital shapes.

Because ℓ can be no larger than n ​− ​1, the value of n limits the number of subshells possible for each shell. For the shell with n ​= ​1, ℓ must equal 0; thus, only one

subshell is possible. When n ​= ​2, ℓ can be either 0 or 1. Because two values of ℓ are

now possible, there are two subshells in the n ​= ​2 electron shell.



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6.5  The Modern View of Electronic Structure: Wave or Quantum Mechanics







Subshells are usually identified by letters. For example, an ℓ ​= ​1 subshell is

called a “p subshell,” and an orbital in that subshell is called a “p orbital.”

Value of ℓ

0

1

2

3



Subshell Label

s

p

d

f



283



•  Orbital Symbols  Early studies of

the emission spectra of elements classified lines into four groups on the basis of their appearance. These groups

were labeled sharp, principal, diffuse,

and fundamental. From these names

came the labels we now apply to

orbitals: s, p, d, and f.



mℓ, the Magnetic Quantum Number (mℓ ​= ​0, ±1, ±2, ±3, . . . , ±ℓ)

The magnetic quantum number, mℓ, is related to the orientation in space of the orbitals

within a subshell. Orbitals in a given subshell differ in their orientation in space, not

in their energy.

The value of mℓ can range from +ℓ to −ℓ, with 0 included. For example, when

ℓ ​= ​2, mℓ can have five values: +2, +1, 0, −1, and −2. The number of values of mℓ

for a given subshell (= 2ℓ ​+ ​1) specifies the number of orbitals in the subshell.



Shells and Subshells

Allowed values of the three quantum numbers are summarized in Table 6.1. By analyzing the sets of quantum numbers in this table, you will discover the following:

•

•

•



n ​= ​the number of subshells in a shell

2ℓ ​+ ​1 ​= ​the number of orbitals in a subshell ​= ​the number of values of mℓ

n2 ​= ​the number of orbitals in a shell



The First Electron Shell, n ​= ​1

When n ​= ​1, the value of ℓ can only be 0, so mℓ must also have a value of 0. This

means that, in the shell closest to the nucleus, only one subshell exists, and that

subshell consists of only a single orbital, the 1s orbital.



The Second Electron Shell, n ​= ​2

When n ​= ​2, ℓ can have two values (0 and 1), so there are two subshells in the second shell. One of these is the 2s subshell (n ​= ​2 and ℓ ​= ​0), and the other is the 2p

subshell (n ​= ​2 and ℓ ​= ​1). Because the values of mℓ can be −1, 0, and +1 when



•  Shells, Subshells, and Orbitals—

A Summary  Electrons in atoms are arranged in shells. Within each shell,

there can be one or more electron subshells, each comprised of one or more

orbitals.

Quantum Number

Shell

Subshell

Orbital



n

ℓ

mℓ



Table 6.1  Summary of the Quantum Numbers, Their Interrelationships, and the Orbital Information Conveyed

Principal Quantum

Number



Angular Momentum

Quantum Number



Magnetic Quantum

Number



Number and Type of Orbitals in the Subshell



Symbol = n

Values = 1, 2, 3, . . .



Symbol = ℓ

Values = 0 . . . n − 1



Symbol = mℓ

Values = +ℓ . . . 0 . . . −ℓ



n = number of subshells

Number of orbitals in shell = n2 and number

of orbitals in subshell = 2ℓ + 1



1



0



0



2



0

1



0

+1, 0, −1



3



0

1

2



0

+1, 0, −1

+2, +1, 0, −1, −2



4



0

1

2

3



0

+1, 0, −1

+2, +1, 0, −1, −2

+3, +2, +1, 0, −1, −2, −3



kotz_48288_06_0266-0299.indd 283



one 1s orbital

(one orbital of one type in the n = 1 shell)

one 2s orbital

three 2p orbitals

(four orbitals of two types in the n = 2 shell)

one 3s orbital

three 3p orbitals

five 3d orbitals

(nine orbitals of three types in the n = 3 shell)

one 4s orbital

three 4p orbitals

five 4d orbitals

seven 4f orbitals

(16 orbitals of four types in the n = 4 shell)



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284 



c h a p t er 6 The Structure of Atoms



ℓ  =  1, three 2p orbitals exist. All three orbitals have the same shape. However, because each has a different mℓ value, the three orbitals differ in their orientation in

space.



The Third Electron Shell, n  = 3

When n  =  3, three subshells are possible for an electron because ℓ can have the

values 0, 1, and 2. The first two subshells within the n  =  3 shell are the 3s (ℓ  =  0,

one orbital) and 3p (ℓ  =  1, three orbitals) subshells. The third subshell is labeled

3d (n  =  3, ℓ  =  2). Because mℓ can have five values (−2, −1, 0, +1, and +2) for

ℓ  =  2, there are five d orbitals in this d subshell.



The Fourth Electron Shell, n  = 4

There are four subshells in the n  =  4 shell. In addition to 4s, 4p, and 4d subshells,

there is the 4f subshell for which ℓ  =  3. Seven such orbitals exist because there are

seven values of mℓ when ℓ  =  3 (−3, −2, −1, 0, +1, +2, and +3).

rEvIEW & cHEcK FOr SEctIOn 6.5

1.



What label is given to an orbital with quantum numbers n = 4 and ℓ = 1?

(a)



2.



(b) 4p



(c)



4dd



4f

(d) 4f



(c)



9



(d) 16



How many orbitals are in the n = 4 shell?

(a)



3.



4s



1



(b) 4



Which quantum number, or combination of quantum numbers, is needed to specify a given

subshell in an atom?

(a)



n



(b) ℓ



(c)



n and ℓ



6.6 TheShapesofAtomicOrbitals

We often say the electron is assigned to, or “occupies,” an orbital. But what does this

mean? What is an orbital? What does it look like? To answer these questions, we have

to look at the wavefunctions for the orbitals. (To answer the question of why the

quantum numbers—small, whole numbers—can be related to orbital shape and

energy, see A Closer Look: More About H Atomic Orbital Shapes and Wavefunctions on

page 287.)



s Orbitals

A 1s orbital is associated with the quantum numbers n  =  1 and ℓ  =  0. If we could

photograph a 1s electron at one-second intervals for a few thousand seconds, the

composite picture would look like the drawing in Figure 6.12a. This resembles a

cloud of dots, and chemists often refer to such representations of electron orbitals

as electron cloud pictures. In Figure 6.12a, the density of dots is greater close to the

nucleus, that is, the electron cloud is denser close to the nucleus. This indicates that

the 1s electron is most likely to be found near the nucleus. However, the density of

dots declines on moving away from the nucleus and so, therefore, does the probability of finding the electron.

The thinning of the electron cloud at increasing distance is illustrated in a different way in Figure 6.12b. Here we have plotted the square of the wavefunction for

the electron in a 1s orbital (ψ2), times 4π and the distance squared (r2), as a function of the distance of the electron from the nucleus. This plot represents the probability of finding the electron in a thin spherical shell at a distance r from the nucleus. Chemists refer to the plot of 4πr 2ψ 2 vs. r as a surface density plot or radial

distribution plot. For the 1s orbital, 4πr 2ψ 2 is zero at the nucleus—there is no probability the electron will be exactly at the nucleus (where r = 0)—but the probability



kotz_48288_06_0266-0299.indd 284



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6.6  The Shapes of Atomic Orbitals







rises rapidly on moving away from the nucleus, reaches a maximum a short distance

from the nucleus (at 52.9 pm), and then decreases rapidly as the distance from the

nucleus increases. Notice that the probability of finding the electron approaches

but never quite reaches zero, even at very large distances.

Figure 6.12a shows that, for the 1s orbital, the probability of finding an electron

is the same at a given distance from the nucleus, no matter in which direction you

proceed from the nucleus. Consequently, the 1s orbital is spherical in shape.

Because the probability of finding the electron approaches but never quite

reaches zero, there is no sharp boundary beyond which the electron is never found

(although the probability can be incalculably small at large distances). Nonetheless,

the s orbital (and other types of orbitals as well) is often depicted as having a boundary surface (Figure 6.12c), largely because it is easier to draw such pictures. To create Figure 6.12c, we drew a sphere centered on the nucleus in such a way that the

probability of finding the electron somewhere inside the sphere is 90%. The choice

of 90% is arbitrary—we could have chosen a different value—and if we do, the

shape would be the same, but the size of the sphere would be different.

All s orbitals (1s, 2s, 3s ...) are spherical in shape. However, for any atom, the size

of s orbitals increases as n increases (Figure 6.13). For a given atom, the 1s orbital is

more compact than the 2s orbital, which is in turn more compact than the 3s

orbital.

It is important that you avoid some misconceptions about pictures of orbitals.

•

•



•



•  Surface Density Plot for 1s  The

maximum value of the radial distribution plot for a 1s electron in a hydrogen

atom occurs at 52.9 pm. It is interesting to note that this maximum is at exactly the same distance from the nucleus that Niels Bohr calculated for the

radius of the orbit occupied by the

n = 1 electron.



First, there is not an impenetrable surface within which the electron is

“contained.”

Second, the probability of finding the electron is not the same throughout the

volume enclosed by the surface in Figure 6.12c. (An electron in the 1s orbital

of a H atom has a greater probability of being 52.9 pm from the nucleus than

of being closer or farther away.)

Third, the terms “electron cloud” and “electron distribution” imply that the

electron is a particle, but the basic premise in quantum mechanics is that the

electron is treated as a wave, not a particle.



•  Nodal Surfaces  Nodal surfaces that

cut through the nucleus occur for all

p, d, and f orbitals. These surfaces are

usually flat, so they are often referred

to as nodal planes. In some cases (for

example, dz2), however, the “plane” is

not flat and so is better referred to as a

nodal surface.



p Orbitals



x



r90



y



(a) Dot picture of an electron in a 1s orbital.

Each dot represents the position of the

electron at a different instant in time. Note

that the dots cluster closest to the nucleus. r90

is the radius of a sphere within which the

electron is found 90% of the time.



Probability of finding electron at

given distance from the nucleus



All atomic orbitals for which ℓ ​= ​1 (p orbitals) have the same basic shape. If you enclose 90% of the electron density in a p orbital within a surface, the electron cloud is

often described as having a shape like a weight lifter’s “dumbbell,” and chemists



z



285



z

Most probable distance

of H 1s electron from

the nucleus = 52.9 pm



x



r90



y

0



1



2

3

4

5

Distance from nucleus

(1 unit = 52.9 pm)



(b) A plot of the surface density (4␲r2␺2)

as a function of distance for a hydrogen

atom 1s orbital. This gives the probability

of finding the electron at a given distance

from the nucleus.



6



(c) The surface of the sphere within which the

electron is found 90% of the time for a 1s orbital.

This surface is often called a “boundary surface.” (A

90% surface was chosen arbitrarily. If the choice was

the surface within which the electron is found 50% of

the time, the sphere would be considerably smaller.)



Figure 6.12   Different views of a 1s (n = 1, ℓ = 0) orbital.  In panel  (b)  the horizontal axis is marked in units called “Bohr radii,” where

1 Bohr radius = 52.9 pm. This is common practice when plotting wave functions.



kotz_48288_06_0266-0299.indd 285



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286



c h a p t er 6   The Structure of Atoms



z



x



3px



3py



3pz



2px



2py



2pz



3dxz



3dz2



y



3dyz



3dxy



3dx2– y2



3s

z



x



y

2s

z



x



Figure 6.13   Atomic orbitals.  Boundary surface diagrams for electron densities of 1s, 2s, 2p, 3s,

3p, and 3d orbitals for the hydrogen atom. For the p orbitals, the subscript letter indicates the cartesian

axis along which the orbital lies. (For more about orbitals, see A Closer Look: More About H Atom Orbital

Shapes and Wavefunctions.)



y

1s



•  ℓ and Nodal Surfaces  The number



of nodal surfaces passing through the

nucleus for an orbital = ℓ.



Orbital



ℓ



Number of Nodal

Surfaces through  

the Nucleus



s

p

d

f



0

1

2

3



0

1

2

3



describe p orbitals as having dumbbell shapes (Figures 6.13 and 6.14). A p orbital has a

nodal surface—a surface on which the probability of finding the electron is zero—that

passes through the nucleus. (The nodal surface is a consequence of the wavefunction

for p orbitals, which has a value of zero at the nucleus but which rises rapidly in value

on moving way from the nucleus. See A Closer Look: More About H Atom Orbital Shapes and

Wavefunctions.)

There are three p orbitals in a subshell, and all have the same basic shape with

one nodal plane through the nucleus. Usually, p orbitals are drawn along the x-, y-,

and z-axes and labeled according to the axis along which they lie (px, py, or pz).



d Orbitals

Orbitals with ℓ ​= ​0, s orbitals, have no nodal surfaces through the nucleus, and

p orbitals, for which ℓ ​= ​1, have one nodal surface through the nucleus. The value

of ℓ is equal to the number of nodal surfaces slicing through the nucleus. It follows that

the five d orbitals, for which ℓ ​= ​​2, have two nodal surfaces through the nucleus,

resulting in four regions of electron density. The dxy orbital, for example, lies in



z



yz nodal plane



y



x



px



z



x



xz nodal plane



y



py



z



xy nodal plane



x



y



pz



(a)  The three p orbitals each have one nodal plane (ℓ = 1) that is perpendicular to the axis along

which the orbital lies.



Figure 6.14   Nodal surfaces of p and d orbitals.  A nodal surface is a surface

on which the probability of finding the electron is zero.



kotz_48288_06_0266-0299.indd 286



yz nodal plane



z



xz nodal plane



y



x



dxy



(b)  The dxy orbital. All five d orbitals

have two nodal surfaces (ℓ = 2)

passing through the nucleus. Here,

the nodal surfaces are the xz- and

yz-planes, so the regions of electron

density lie between the x- and y-axes

in the xy-plane.



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287



6.6 The Shapes of Atomic Orbitals







A CLOSER LOOK



More about H Atom Orbital Shapes and Wavefunctions



Value of ψ [ × (52.9 pm)



3/

2]



2.5

2

1.5

1s



1

0.5



2p

0



−0.5



2s

0



2



4



6



8



10



12



14



Distance from nucleus

(1 unit = 52.9 pm)

FiguREA Plot of the wavefunctions for 1s, 

2s, and 2p orbitals versus distance from the 

nucleus. As in other plots of wavefunctions, the

horizontal axis is marked in units called “Bohr

radii,” where 1 Bohr radius = 52.9 pm.



kotz_48288_06_0266-0299.indd 287



16



x



Surface of the

spherical node

Sign of the wave

function is positive

inside this surface.



z



y

2s orbital



Sign of the wave

function is negative



FiguREB Wavefunction for a 2s orbital. A 2s

orbital for the H atom showing the spherical node

(at 2a0 = 105.8 pm) around the nucleus.



For the 2s orbital, you see a node in

Figure A at 105.8 pm (= 2ao) when plotting

the radial portion of the wavefunction.

However, because the angular part of ψ2s

has the same value in all directions, this

means there is a node—a spherical nodal

surface—at the same distance in every direction (as illustrated in Figure B). For any

orbital, the number of spherical nodes is

n − ℓ − 1.

Now let’s look at a p orbital, first the

radial portion and then the angular portion. For a p orbital the radial portion of the

wavefunction is 0 when r = 0. Thus, the

value of ψ2p is zero at the nucleus and a

nodal surface passes through the nucleus

(Figures A and C). This is true for all

p orbitals.

But what happens as you move away

from the nucleus in one direction, say along

the x-axis in the case of the 2px orbital. Now

we see the value of ψ2p rise to a maximum at

105.9 pm (= 2ao) before falling off at greater

distances (Figures A and C).

But look at Figure C for the 2px orbital.

Moving away along the −x direction, the

value of ψ2p is the same but opposite in sign



0.2



3



negative with increasing r before approaching zero at greater distances.

Now to the angular portion of the wavefunction: this reflects changes that occur

when you travel outward from the nucleus

in different directions. It is a function of the

quantum numbers ℓ and mℓ.

As illustrated in Figure 6.12 the value of ψ1s

is the same in every direction. This is a reflection of the fact that, while the radial portion

of the wavefunction changes with r, the angular portion for all s orbitals is a constant. As a

consequence, all s orbital are spherical.



Value of ψ [ × (52.9 pm) /2 ]



There is an important question to answer: How do quantum numbers, which are small, integer numbers, tell us the shape of atomic orbitals? The

answer lies in the orbital wavefunctions,

which are reasonably simple mathematical

equations.

The equations for wavefunctions (ψ) are

the product of two functions: the radial

function and the angular function. You need

to look at each type to get a picture of an

orbital.

Let’s first consider the radial function,

which depends on n and ℓ. This tells us how

the value of ψ depends on the distance from

the nucleus. The radial functions for the

hydrogen atom 1s (n = 1 and ℓ = 0), 2s (n =

2 and ℓ = 0) orbitals, and 2p orbitals (n = 2

and ℓ = 1) are plotted in Figure A. (The horizontal axis has units of ao, where ao is a

constant equal to 52.9 pm).

Waves have crests, troughs, and nodes,

and plots of the wavefunctions show this.

For the 1s orbital of the H atom, the radial

wavefunction ψ1s approaches a maximum at

the nucleus (Figure A), but the wave’s amplitude declines rapidly at points farther

removed from the nucleus. The sign of ψ1s is

positive at all points in space.

For a 2s orbital, there is a different profile: the sign of ψ2s is positive near the

nucleus, drops to zero (there is a node at

r = 2ao = 2 × 52.9 pm), and then becomes



0.15

0.1

0.05

0

−0.05

−0.1

−0.15

−0.2

−15 −10 −5

0

5

10

Distance from nucleus

(1 unit = 52.9 pm)



15



FiguREC Wave functions for a 2p orbital. The

sign of ψ for a 2p orbital is positive on one side of

the nucleus and negative on the other (but it has

a 0 value at the nucleus). A nodal plane separates

the two lobes of this “dumbbell-shaped” orbital.

(The vertical axis is the value of ψ, and the horizontal axis is the distance from the nucleus, where

1 unit = 52.9 pm.)



to the value in the +x direction. The 2p electron is a wave with a node at the nucleus. (In

drawing orbitals, we indicate this with + or

− signs or with two different colors as in

Figure 6.13.)

Now, what about the angular portion of

the wavefunction for 2px? The angular portion for all three p orbitals has the same

general form: c(x/r) for the px orbital, c(y/r)

for the py orbital, and c(z/r) for the pz orbital

(where c is a constant). Consider a 2px

orbital in Figure D. As long as x has a value,

the wavefunction has a value. But when

x = 0 (in the yz plane), then ψ is zero. This is

the nodal plane for the x orbital. Similarly,

the angular portion of the wavefunction for

the 2py orbital means its nodal plane is the

xz plane.



x axis



y axis



x=0



FiguRED The 2px orbital. The wavefunction

has no value when x = 0, that is, the yz plane is a

nodal plane.



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288 



c h a p t er 6 The Structure of Atoms



yz nodal plane



xz nodal plane



z



xy nodal plane



y



the xy-plane, and the two nodal surfaces are the xz- and yz-planes (Figure 6.14).

Two other orbitals, dxz and dyz, lie in planes defined by the xz- and yz-axes, respectively; they also have two, mutually perpendicular nodal surfaces (Figure 6.13).

Of the two remaining d orbitals, the dx2−y2 orbital is easier to visualize. In the

dx2−y2 orbital, the nodal planes bisect the x- and y-axes, so the regions of electron

density lie along the x- and y-axes. The dz2 orbital has two main regions of electron

density along the z-axis, and a “doughnut” of electron density also occurs in the

xy-plane. This orbital has two cone-shaped nodal surfaces.



x

fxyz



FiguRE6.15 One of the seven

possible f orbitals. Notice the

presence of three nodal planes as

required by an orbital with ℓ = 3.



f Orbitals

Seven f orbitals arise with ℓ  =  3. Three nodal surfaces through the nucleus cause

the electron density to lie in up to eight regions of space. One of the f orbitals is illustrated in Figure 6.15.

rEvIEW & cHEcK FOr SEctIOn 6.6

1.



Which of the following is not a correct representation of an orbital?

(a)



2.



3s



(b) 3p



3dd



3f

(d) 3f



Which of the following sets of quantum numbers correctly represents a 4p orbital?

(a)



n = 4, ℓ = 0, mℓ = −1



(c)



(b) n = 4, ℓ = 1, mℓ = 0

3.



(c)



n = 4, ℓ = 2, mℓ = 1



(d) n = 4, ℓ = 1, mℓ = 2



How many nodal planes exist for a 5dd orbital?

(a)



0



(b) 1



(c)



2



(d) 3



6.7 OneMoreElectronProperty:ElectronSpin

There is one more property of the electron that plays an important role in the arrangement of electrons in atoms and gives rise to properties of elements you observe every day: electron spin.



The Electron Spin Quantum Number, ms

In 1921 Otto Stern and Walther Gerlach performed an experiment that probed the

magnetic behavior of atoms by passing a beam of silver atoms in the gas phase

through a magnetic field. Although the results were complex, they were best interpreted by imagining the electron has a spin and behaves as a tiny magnet that can be

attracted or repelled by another magnet. If atoms with a single unpaired electron

are placed in a magnetic field, the Stern-Gerlach experiment showed there are two

orientations for the atoms: with the electron spin aligned with the field or opposed

to the field. That is, the electron spin is quantized, which introduces a fourth quantum

number, the electron spin quantum number, ms. One orientation is associated with

a value of ms of +1⁄2 and the other with ms of −1⁄2.

α

β

ms = +½



ms = −½



When it was recognized that electron spin is quantized, scientists realized that a

complete description of an electron in any atom requires four quantum numbers,

n, ℓ, mℓ, and ms. The important consequences of this fact are explored in Chapter 7.



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6.7 One More Electron Property: Electron Spin







A CLOSER LOOK



289



Paramagnetism and Ferromagnetism



Magnetic materials are relatively common, and many are

important in our economy. For example, a

large magnet is at the heart of the magnetic

resonance imaging (MRI) used in medicine,

and tiny magnets are found in stereo speakers and in telephone handsets. Magnetic

oxides are used in recording tapes and computer disks.



The magnetic materials we use are  ferromagnetic. The magnetic effect of ferromagnetic materials is much larger than that

of paramagnetic ones. Ferromagnetism

occurs when the spins of unpaired electrons

in a cluster of atoms (called a domain) in the

solid align themselves in the same direction.

Only the metals of the iron, cobalt, and

nickel subgroups, as well as a few other metals such as neodymium, exhibit this prop-



erty. They are also unique in that, once the

domains are aligned in a magnetic field, the

metal is permanently magnetized.

Many alloys exhibit greater ferromagnetism than do the pure metals themselves.

One example of such a material is Alnico,

which is composed of aluminum, nickel, and

cobalt as well as copper and iron. The strongest permanent magnet is an alloy of neodymium, iron, and boron (Nd2Fe14B).



© Cengage Learning/Charles D. Winters



(a) Paramagnetism



Magnets. Many common consumer products

such as loudspeakers contain permanent

magnets.



No Magnetic Field



External Magnetic Field



(b) Ferromagnetism



The spins of unpaired electrons

align themselves in the same direction



Magnetism. (a) Paramagnetism: In the absence of an external magnetic field, the unpaired electrons in the

atoms or ions of the substance are randomly oriented. If a magnetic field is imposed, however, these spins will

tend to become aligned with the field. (b) Ferromagnetism: The spins of the unpaired electrons in a cluster of

atoms or ions align themselves in the same direction even in the absence of a magnetic field.



Diamagnetism and Paramagnetism

A hydrogen atom has a single electron. If a hydrogen atom is placed in a magnetic

field, the magnetic field of the single electron will tend to align with the external field

like the needle of a compass aligns with the magnetic lines of force on the Earth.

In contrast, helium atoms, each with two electrons, are not attracted to a magnet. In fact, they are slightly repelled by the magnet. To account for this observation,

we assume the two electrons of helium have opposite spin orientations. We say their

spins are paired, and the result is that the magnetic field of one electron can be canceled out by the magnetic field of the second electron with opposite spin. To account for this, the two electrons are assigned different values of ms.

It is important to understand the relationship between electron spin and magnetism. Elements and compounds that have unpaired electrons are attracted to a magnet.

Such species are said to be paramagnetic. The effect can be quite weak, but, by placing a sample of an element or compound in a magnetic field, it can be observed

(Figure 6.16). For example, the oxygen you breathe is paramagnetic. You can observe this experimentally because liquid oxygen sticks to a magnet of the kind you

may have in the speakers of a music player (Figure 6.16b).

Substances in which all electrons are paired (with the two electrons of each pair

having opposite spins) experience a slight repulsion when subjected to a magnetic

field; they are called diamagnetic. Therefore, by determining the magnetic behavior

of a substance we can gain information about the electronic structure.

In summary, substances in which the constituent ions or atoms contain unpaired electrons

are paramagnetic and are attracted to a magnetic field. Substances in which all electrons

are paired with partners of opposite spin are diamagnetic. This explanation opens

the way to understanding the arrangement of electrons in atoms with more than one

electron as you shall learn in the next chapter.



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290



c h a p t er 6   The Structure of Atoms



Electronic balance

Mass (g)



Mass (g)



Electromagnet

to provide

magnetic field



Electromagnet OFF



Electromagnet ON



© Cengage Learning/Charles D. Winters



Sample sealed

in a glass tube



(a)

balance

(a)  Electromagnetic

A magnetic balance

is used to measure the magnetism of a sample. The sample is first



(b)  Liquid

Liquidoxygen

oxygenclings

(boiling

90.2 K)

(b)

to apoint

magnet.



weighed with the electromagnet turned off. The magnet is then turned on and the sample

reweighed. If the substance is paramagnetic, the sample is drawn into the magnetic field,

and the apparent weight increases.



clings to a strong magnet. Elemental

oxygen is paramagnetic because it has

unpaired electrons. (See Chapter 9.)



Figure 6.16   Observing and measuring paramagnetism.



  and 

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using concepts in this chapter.



chapter goals revisited

Now that you have studied this chapter, you should ask whether you have met the chapter

goals. In particular, you should be able to:

Describe the properties of electromagnetic radiation



a. Use the terms wavelength, frequency, amplitude, and node (Section 6.1).

Study Question: 3.

b. Use Equation 6.1 (c ​= ​λν), relating wavelength (λ) and frequency (ν) of electromagnetic radiation and the speed of light (c). Study Question: 3.

c. Recognize the relative wavelength (or frequency) of the various types of electromagnetic radiation (Figure 6.2).  Study Question: 1.

d. Understand that the energy of a photon, a massless particle of radiation, is

proportional to its frequency (Planck’s equation, Equation 6.2)(Section 6.2).

Study Questions: 5, 7, 8, 9, 12, 14, 56, 57, 58, 63, 64, 66, 72, 73, 83.

Understand the origin of light emitted by excited atoms and its relationship to atomic

structure



a. Describe the Bohr model of the atom, its ability to account for the emission

line spectra of excited hydrogen atoms, and the limitations of the model

(Section 6.3). Study Question: 74.

b. Understand that, in the Bohr model of the H atom, the electron can occupy

only certain energy states, each with an energy proportional to 1/n2

(En ​= ​−Rhc/n2), where n is the principal quantum number (Equation 6.4,

Section 6.3). If an electron moves from one energy state to another, the

amount of energy absorbed or emitted in the process is equal to the difference in energy between the two states (Equation 6.5, Section 6.3). Study

Questions: 16, 18, 19, 21, 22, 60.

Describe the experimental evidence for particle–wave duality



a. Understand that in the modern view of the atom, electrons can be described

either as particles or as waves (Section 6.4). The wavelength of an electron or

any subatomic particle is given by de Broglie’s equation (Equation 6.6).

Study Questions: 23–26, 82.



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291



6.7 One More Electron Property: Electron Spin



Pole of magnet



© Horizon International Images Limited/Alamy



Quantized Spins and MRI



Just as electrons can be

thought of as having a spin,

so do atomic nuclei. In the hydrogen atom,

the single proton can also be thought of having a spin. For most heavier atoms, the

atomic nucleus includes both protons and

neutrons, and the entire entity has a spin.

This property is important, because nuclear

spin allows scientists to detect these atoms

in molecules and to learn something about

their chemical environments.

The technique used to detect the spins

of atomic nuclei is nuclear magnetic resonance (NMR). It is one of the most powerful

methods currently available to determine

molecular structures. About 20 years ago, it

was adapted as a diagnostic technique in

medicine, where it is known as magnetic

resonance imaging (MRI).

Just as electron spin is quantized, so too

is nuclear spin. The H atom nucleus (often

referred to simply as a proton) can spin in

either of two directions. If the H atom is

placed in a strong, external magnetic field,

however, the spinning nuclear magnet can

align itself with or against the external field.

If a sample of ethanol (CH3CH2OH), for

example, is placed in a strong magnetic

field, a slight excess of the H atom nuclei

(and 13C atom nuclei) is aligned with the

lines of force of the field.

The nuclei aligned with the field have a

slightly lower energy than when aligned

against the field. The NMR and MRI technologies depend on the fact that energy in

the radio-frequency region can be absorbed

by the sample and can cause the nuclear

spins to switch alignments—that is, to move



© Paul Burns/Jupiter Images



a closer look



Magnetic resonance imaging. (a) MRI instrument. The patient is placed inside a large magnet, and the tissues to be examined are irradiated with radio-frequency radiation. (b) An MRI image of the human brain.



to a higher energy state. This reemission of

energy is detected by the instrument.

The most important aspect of the magnetic resonance technique is that the difference in energy between two different spin

states depends on the electronic environment of atoms in the molecule. In the case

of ethanol, the three CH3 protons are different from the two CH2 protons, and both sets

are different from the OH proton. These

three different sets of H atoms absorb

radiation of slightly different energies. The

instrument measures the frequencies

absorbed, and a scientist familiar with the

technique can quickly distinguish the three

different environments in the molecule.

The MRI technique closely resembles

the NMR method. Hydrogen is abundant

in the human body as water and in numerous organic molecules. In the MRI device,



Sample tube



the patient is placed in a strong magnetic

field, and the tissues being examined are

irradiated with pulses of radio-frequency

radiation.

The MRI image is produced by detecting

how fast the excited nuclei “relax”; that is,

how fast they return to the lower energy

state from the higher energy state. The

“relaxation time” depends on the type of

tissue. When the tissue is scanned, the H

atoms in different regions of the body show

different relaxation times, and an accurate

“image” is built up.

MRI gives information on soft tissue—

muscle, cartilage, and internal organs—

which is unavailable from x-ray scans. This

technology is also noninvasive, and the

magnetic fields and radio-frequency radiation used are not harmful to the body.



Pole of magnet



CH3CH2OH

CH3

Absorption



OH



Radio-frequency

Detector

transmitter

(a) An NMR spectrometer (see Figure 4.4, page 166.)



Recorder



CH2



6



5



4

3

2

Chemical Shift, ␦ (ppm)



1



0



(b) The NMR spectrum of ethanol



Nuclear magnetic resonance. (a) A schematic diagram of an NMR spectrometer. (b) The NMR spectrum of ethanol showing that the three different types of

protons appear in distinctly different regions of the spectrum. The pattern observed for the CH2 and CH3 protons is characteristic of these groups of atoms and

signals the chemist that they are present in the molecule.



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