CH7 Atomic Structure and Periodicity

Chemistry

Zumdahl, 7th edition

CH7 Atomic Structure and Periodicity

Light refracted through a prism.

7.1 Electromagnetic Radiation 7.2 The Nature of Matter

7.3 The Atomic Spectrum of Hydrogen 7.4 The Bohr Model

7.5 The Quantum Mechanical Model of the Atom 7.6 Quantum Numbers

7.7 Orbital Shapes and Energies

7.8 Electron Spin and the Pauli Principle 7.9 Polyelectronic Atoms

7.10 The History of the Periodic Table

7.11 The Aufbau Principle and the Properties 7.12 Periodic Trends in Atomic Properties

7.13 The Properties of a Group: The Alkali Metals

Contents

Introduction

In the past 200 years, a great deal of experimental evidence has accumulated to support the atomic model.

In fact, for the following 20 centuries, no convincing experimental evidence was available to support the

existence of atoms.

z

The first real scientific data were gathered by Lavoisier

and others from quantitative measurements of chemical

reactions.

z

The results of these stoichiometric experiments led John Dalton to propose the first systematic atomic theory.

z

Dalton’s theory, although crude, has stood the test of time extremely well.

One of the most striking things about the chemistry of

the elements is the periodic repetition of properties. There

are several groups of elements that show great similarities

in chemical behavior.

A radical new theory called quantum mechanics was

developed to account for the behavior of light and atoms.

7.1 Electromagnetic Radiation

One of the ways that energy travels through space is by electromagnetic radiation.

Waves have three primary characteristics: wavelength, frequency, and speed.

Wavelength (symbolized by the lowercase Greek letter lambda, λ) is the distance between two consecutive

peaks or troughs in a wave, as shown in Fig. 7.1.

Figure 7.1

The nature of waves. Note that the radiation with the shortest wavelength has the highest frequency.

The frequency (symbolized by the lowercase Greek letter nu, ν) is defined as the number of waves (cycles) per second that pass a given point in space.

Since all types of electromagnetic radiation travel at the speed of light, short-wavelength radiation must have a

high frequency.

Note that the wave the shortest wavelength (λ

) has

the highest frequency and the wave with the longest

wavelength (λ

) has the lowest frequency.

λν＝ c

•

Where λ is the wavelength in meters, ν is the

frequency in cycles per second, and c is the speed of light (2.9979 × 10

m/s).

•

In the SI system, cycles is understood, and the unit per

second becomes 1/s, or s

, which is called the hertz

(abbreviated Hz).

Electromagnetic radiation is classified as shown in Fig.

7.2. Radiation provides an important means of energy

transfer.

Although the waves associated with light are not obvious to the naked eye, ocean waves provide a familiar source of recreation.

Figure 7.2

Classification of electromagnetic radiation. Spectrum adapted by permission from C. W. Keenan, D. C. Kleinfelter, and J. H.

Wood, General College Chemistry, 6th ed. (New York: Harper

& Row, 1980).

The brilliant red colors seen in fireworks are due to the emission of light with wavelengths around 650 nm when strontium salts such as Sr(NO

)

and SrCO

are heated.

(This can be easily demonstrated in the lab by dissolving one of these salts in methanol that contains a little water and igniting the mixture in an evaporating dish.)

Calculate the frequency of red light of wavelength 6.50 × 10

nm.

Sample Exercise 7.1

Frequency of

Electromagnetic Radiation

Solution

We can convert wavelength to frequency using the equation

where c ＝ 2.9979 × 10

m/s. In this case λ ＝ 6.50 × 10

nm. Changing the wavelength to meters, we have

Sample Exercise 7.1

Sample Exercise 7.1

And

See Exercises 7.31 and 7.32

7.2 The Nature of Matter

Matter was thought to consist of particles, whereas energy in the form of light (electromagnetic radiation) was described as a wave.

Particles were things that had mass and whose position in space could be specified.

Waves were described as massless and delocalized; that

is, their position in space could not be specified.

It also was assumed that there was no intermingling of

matter and light. Everything known before 1900 seemed

to fit neatly into this view.

When alternating current at 110 volts is applied to a dill pickle, a glowing discharge occurs. The current flowing between the electrodes (forks), which is supported by the Na+ and Cl- ions present, apparently cause some sodium atoms to form in an excited state. When these atoms relax to the ground state, they emit visible light at 589 nm, producing the yellow glow reminiscent of sodium vapor lamps.

The first important advance came in 1900 from the German physicist Max Planck (1858-1947).

Planck could account for these observations only by postulating that energy can be gained or lost only in

whole-number multiples of the quantity hν, where h is a constant called Planck’s constant, determined by

experiment to have the value 6.626 × 10

J‧s.

That is, the change in energy for a system ΔE can be represented by the equation

ΔE ＝ hν

where n is an integer (1, 2, 3, …), h is Plack’s constant,

and ν is the frequency of the electromagnetic radiation

absorbed or emitted.

Now it seemed clear that energy is in fact quantized and can occur only in discrete units of size hν.

Each of these small “packets” of energy is called a quantum.

A system can transfer energy only in whole quanta.

The blue color in fireworks is often achieved by heating copper (I) chloride (CuCl) to about 1200℃. Then the compound emits blue light having a wavelength of 450 nm. What is the increment of energy (the quantum) that is emitted at 4.50 × 10

nm by CuCl?

Solution

The quantum of energy can be calculated from the equation

Sample Exercise 7.2

The Energy of a Photon

Sample Exercise 7.2

The frequency ν for this case can be calculated as follows:

So

A sample of CuCl emitting light at 450 nm can lose

energy only in increments of 4.41 × 10

J, the quantum in this case.

See Exercises 7.33 and 7.34

The next important development in the knowledge of atomic structure came when Albert Einstein (see Fig. 7.3) proposed that electromagnetic radiation is itself quantized.

Einstein suggested that electromagnetic radiation can be viewed as a stream of “particles” called photons.

The energy of each photon is given by the expression

ΔE

＝ hν ＝ hc

λ

where h is Plack’s constant, ν is the frequency of the

radiation, and λ is the wavelength of the radiation.

Figure 7.3

Figure 7.3

Albert Einstein (1879-1955) was born in Germany. Nothing in his early development suggested genius; even at the age of 9 he did not speak clearly, and his school principal replied, “it doesn’t

matter; he’ll never make a success of anything.” When he was 10, Einstein entered the Luitpold Gymnaium (high school), which was typical of German schools of that time in being harshly

disciplinarian. There he developed a deep suspicion of authority and a skepticism that encouraged him to question and doubt- valuable qualities in a scientist. In 1905, while a patent clerk in Switzerland, Einstein published a paper explaining the

photoelectric effect via the quantum theory. For this revolutionary thinking he received a Nobel Prize in 1921. Highly regarded by this time, he worked in Germany until 1933, when Hitler’s

Figure 7.3

persecution of the Jews forced him to come to the United States.

He worked at the Instituted for Advanced Studies in Princeton, New Jersey, until his death in 1955.

Einstein was undoubtedly the greatest physicist of our age. Even if someone else had derived the theory of relativity, his other work would have ensured his ranking as the second greatest physicist of his time. Our concepts of space and time were radically

changed by ideas he first proposed when he was 26 years old.

From then until the end of his life, he attempted unsuccessfully to find a single unifying theory that would explain all physical events.

Einstein arrived at this conclusion through his analysis of photoelectric effect (for while he later was awarded the Nobel Prize).

The photoelectric effect refers to the phenomenon in which electrons are emitted from the surface of a metal when light strikes it.

The following observations characterize the photoelectric effect.

The Photoelectric Effect

1. Studies in which the frequency of the light is varied show that no electrons are emitted by a given metal below a specific threshold frequency ν

.

2. For light with frequency lower than the threshold frequency, no electrons are emitted regardless of the intensity of the light.

3. For light with frequency greater than the threshold

frequency, the number of electrons emitted increases

with the intensity of the light.

4. For light with frequency greater than the threshold

frequency, the kinetic energy, of the emitted electrons

increase linearly with the frequency of the light.

Minimum energy required to remove an electron

＝ E

＝ hν

•

Because a photon with energy less than E

(ν＞ν

) cannot remove an electron, light with a frequency less than the threshold frequency produces no electrons.

•

On the other hand, for light where ν＞ν

, the energy

in excess of that required to remove the electron is given

to the electron as kinetic energy (KE):

In a related development, Einstein derived the famous equation

E ＝ mc

In his special theory of relativity published in 1905.

The main significance of this equation is that energy has

mass.

This is more apparent if we rearrange the equation in the following form:

ΔE

＝ hc

λ

In 1922 American physicist Arthur Compton (1892-

1962) performed experiments involving collisions of X

rays and electrons that showed that photons do exhibit the

apparent mass calculated from the preceding equation.

The important conclusions from the work of Planck and Einstein as follows:

Energy is quantized. It can occur only in discrete units called quanta.

Electromagnetic radiation, which was previously

thought to exhibit only wave properties, seems to show certain characteristics of particulate matter as well.

This phenomenon is sometimes referred to as the dual

nature of light and is illustrated in Fig. 7.4.

Figure 7.4

Electromagnetic radiation exhibits wave properties and particulate properties. The energy of each photon of the radiation is related to the wavelength and frequency by the equation E_photon = hν = hc/λ.

This question was raised in 1923 by a young French physicist named Louis de Broglie (1892-1987).

λ＝

This equation, called de Broglie’s equation, allows us to calculate the wavelength for a particle, as shown in Sample Exercise 7.3.

h

mυ

Compare the wavelength for an electron (mass ＝ 9.11 × 10

kg) traveling at a speed of 1.0 × 10

m/s with that for a ball (mass ＝ 0.10 kg) traveling at 35 m/s.

Solution

We use the equation λ＝ h/mυ, where

since

Sample Exercise 7.3

Calculations of

Wavelength

For the electron,

For the ball,

See Exercises 7.41 though 7.44 Sample Exercise 7.3

Diffraction results when light is scattered from a regular array of points or lines.

When X rays are directed onto a crystal of sodium

chloride, with its regular array of Na

and Cl

ions, the

scattered radiation produces a diffraction pattern of

bright spots and dark areas on a photographic plate, as

shown in Fig. 7.5(a).

This occurs because the scattered light can interfere constructively (the peaks and troughs of the beams are in phase) to produce a bright spot [Fig. 7.5(b)] or

destructively (the peaks and troughs are out of phase) to

produce a dark area [Fig. 7.5(c)].

A diffraction pattern can only be explained in terms of waves.

Thus this phenomenon provides a test for the postulate that particles such as electrons have wavelengths.

An experiment to test this idea was carried out in 1927

by C.J. Davisson and L.H. Germer at Bell Laboratories.

(top) The pattern produced by electron diffraction of a titanium/nickel alloy. (bottom) Pattern produced by X-ray diffraction of a beryl crystal.

Figure 7.5

Figure 7.5

(a) Diffraction occurs when electromagnetic radiation is

scattered from a regular array of objects, such as the ions in a crystal of sodium chloride. The large spot in the center is from the main incident beam of X-rays. (b) Bright spots in the

diffraction pattern result from constructive interference of

waves. The waves are in phase; that is, their peaks match. (c) Dark areas result from destructive interference of waves. The waves are out of phase; the peaks of one wave coincide with the troughs of another wave.

7.3 The Atomic Spectrum of Hydrogen

The resulting hydrogen atoms are excited; that is , they contain excess energy, which they release by emitting

light of various wavelengths to produce what is called the emission spectrum of the hydrogen atom.

To understand the significance of the hydrogen

emission spectrum, we must first describe the continuous

spectrum that results when white light is passed through

a prism, as shown in Fig. 7.6(a).

This spectrum, like contains all the wavelengths of visible light.

In contrast, when the hydrogen emission spectrum in the visible region is passed through a prism, as shown in Fig. 7.6(b), we see only a few lines, each of which

corresponds to a discrete wavelength.

The hydrogen emission spectrum is called a line

spectrum.

Figure 7.6

(a) A continuous spectrum containing all wavelengths of

visible light (indicated by the initial letters of the colors of the rainbow)

Figure 7.6

(b) The hydrogen line spectrum contains only a few discrete wavelengths.

What is the significance of the line spectrum of hydrogen?

It indicates that only certain energies are allowed for the electron in the hydrogen atom.

In other words, the energy of the electron in the hydrogen atom is quantized.

Changes in energy between discrete energy levels in hydrogen will produce only certain wavelengths of

emitted light, as shown in Fig. 7.7.

Figure 7.7

A change between two discrete energy levels emits a photon of light.

7.4 The Bohr Model

In 1913, a Danish physicist named Niels Bohr (1885- 1962), aware of the experimental results we have just

discussed, developed a quantum model for the hydrogen atom. Bohr proposed that the electron in a hydrogen atom moves around the nucleus only in certain allowed

circular orbits.

Bohr’s model gave hydrogen atom energy levels

consistent with the hydrogen emission spectrum. The

model is represented pictorially in Fig. 7.8.

Figure 7.8

Figure 7.8

Electronic transitions in the Bohr model for the hydrogen atom.

(a) An energy-level diagram for electronic transitions. (b) An orbit-transition diagram, which accounts for the experimental spectrum. (Note that the orbits shown are schematic. They are not draw to scale.) (c) The resulting line spectrum on a

photographic plate. Note that the lines in the visible region of the spectrum correspond to transitions from higher levels to the n=2 level.

Although we will not show the derivation here, the most important equation to come from Bohr’s model is the expression for the energy levels available to the

electron in the hydrogen atom:

in which n is an integer (the larger the value of n, the

larger is the orbit radius) and Z is the nuclear charge.

Equation (7.1) can be used to calculate the change in energy of an electron when the electron changes orbits.

For example, suppose an electron in level n=6 of an excited hydrogen atom falls back to level n=1 as the

hydrogen atom returns to its lowest possible energy state,

its ground state.

Note that for n=1 electron has a more negative energy than it does for n=6, which means that the electron is more tightly bound in the smallest allowed orbit.

The change in energy ΔE when the electron falls from

n ＝ 6 to n＝1 is

The negative sign for the change in energy indicates that the atom has lost energy and is now in a more stable state.

The wavelength of the emitted photon can be calculated from the equation

Where ΔE represents the change in energy of the atom,

which equals the energy of the emitted photon.

Note that for this calculation the absolute value of ΔE

is used.

Niels Hendrik David Bohr (1885-1962) as a boy lived in the

shadow of his younger brother Harald, who played on the 1908 Danish Olympic Soccer Team and Later became a

distinguished mathematician. In school, Bohr received his

poorest marks in composition and struggles with writing during his entire life. In fact, he wrote so poorly that he was forced to dictate his Ph.D. thesis to his mother. Nevertheless, Bohr was a brilliant physicist. After receiving his Ph. D. in Denmark, he

constructed a quantum model for the hydrogen atom by the time he was 27. Even though his model later proved to be incorrect, Bohr remained a central figure in the drive to understand the atom. He was awarded the Nobel Prize in physics in 1922.

Calculate the energy required to excite the hydrogen

electron from level n ＝ 1 to level n ＝ 2. Also calculate the wavelength of light that must be absorbed by a

hydrogen atom in its ground state to reach this excited state.

Solution

Using Equation (7.1) with Z＝1, we have

Sample Exercise 7.4

Energy Quantization in

Hydrogen

The positive value for ΔE indicates that the system has gained energy. The wavelength of light that must be

absorbed to produce this change is

See Exercises 7.45 and 7.46 Sample Exercise 7.4

At this time we must emphasize two important points about the Bohr model:

1. The model correctly fits the quantized energy levels of the hydrogen atom and postulates only certain allowed circular orbits for the electron.

2. As the electron becomes more tightly bound, its energy becomes more negative relative to the zero-energy

reference state (corresponding to the electron being at

infinite distance from the nucleus). As the electron is

brought closer to the nucleus, energy is released from

Calculate the energy required to remove the electron from a hydrogen atom in its ground state.

Solution

Removing the electron from a hydrogen atom in its

ground state corresponds to taking the electron from n

＝1 to n

＝ ∞. Thus

Sample Exercise 7.5

Electron Energies

The energy required to remove the electron from a hydrogen atom in its ground state is 2.178 × 10

J.

See Exercises 7.51 and 7.52 Sample Exercise 7.5

7.5 The Quantum Mechanical Model of the Atom

By the mid-1920s it had become apparent that the Bohr model could not be made to work.

Three physicists were at the forefront of this effort:

Werner Heisenberg (1901-1976), Louis de Broglie (1892- 1987), and Erwin Schrödinger (1887-1961).

The approach they developed became known as wave

mechanics or, more commonly, quantum mechanics.

The most familiar example of standing waves occurs in association with musical instruments such as guitars or violins, where a string attached at both ends vibrates to produce a musical tone.

The waves are described as “standing” because they are stationary; the waves do not travel along the length of the string.

The motions of the string can be explained as a

combination of simple waves of the type shown in Fig. 7.9.

Figure 7.9

The standing waves caused by the vibration of a guitar string fastened at both ends. Each dot represents a node (a point of zero displacement).

Wave-generating apparatus.

A similar situation results when the electron in the hydrogen atom is imagined to be a standing wave. As shown in Fig. 7.10, only certain circular orbits have a

circumference into which a whole number of wavelengths of the standing electron wave will “fit.”

It is important to recognize that Schrödinger could not be sure that this idea would work.

The physical principles for describing standing waves

were well known in 1925 when Schrödinger decided to

treat the electron in this way.

Figure 7.10

The hydrogen electron visualized as a standing wave around the nucleus. The circumference of a particular circular orbit would have to correspond to a whole

number of wavelengths, as shown in (a) and (b), or else destructive interference occurs, as shown in (c). This is consistent with the fact that only certain electron

energies are allowed; the atom is

quantized. (Although this idea encouraged scientists to use a wave theory, it does not mean that the electron really travels in

Schrödinger’s equation is

where ψ, called the wave function, is a function of the coordinates (x, y, and z) of the electron’s position in three- dimensional space and H represents a set of mathematical instructions called an operator.

When this equation is analyzed, many solutions are

found. Each solution consists of a wave function ψ that is characterized by a particular value of E.

^

A specific wave function is often called an orbital.

To illustrate the most important ideas of the quantum (wave) mechanical model of the atom, we will first

concentrate on the wave function corresponding to the lowest energy for the hydrogen atom.

This wave function is called the 1s orbital.

An orbital is not a Bohr orbit. The electron in the

hydrogen 1s orbital is not moving around the nucleus in a

circular orbit.

T

o help us understand the nature of an orbital, we need to consider a principle discovered by Werner Heisenberg, one of the primary developers of quantum mechanics.

Heisenberg’s mathematical analysis led him to a

surprising conclusion: There is a fundamental limitation to just how precisely we can know both the position and

momentum of a particle at a given time.

This is a statement of the Heisenberg uncertainty principle.

Stated mathematically, the uncertainty principle is

where Δx is the uncertainty in a particle’s position,

Δ(mυ) is the uncertainty in a particle’s momentum, and

h is Planck’s constant. Thus the minimum uncertainty in

the product Δ‧Δ(mυ) is h/4π.

The square of the function indicates the probability of finding an electron near a particular point in space.

Suppose we have two positions in space, one defined by the coordinates x

, y

, and z

and the other by the

coordinates x

, y

, and z

.

The relative probability of finding the electron at

positions 1 and 2 is given by substituting the values of x, y, and z for the two positions into the wave function,

squaring the function value, and computing the following ratio:

The Physical Meaning of a Wave

Function

The quotient N

/N

is the ratio of the probabilities of finding the electron at positions 1 and 2.

The square of the wave function is most conveniently

represented as a probability distribution, in which the

intensity of color is used to indicate the probability value

near a given point in space. The probability distribution

for the hydrogen 1s wave function (orbital) is shown in

Fig. 7.11(a).

This diagram is also known as an electron density map;

electron density and electron probability mean the same thing.

Another way of representing the electron probability distribution for the 1s wave function is to calculate the probability at points along a line drawn outward in any direction from the nucleus. The result is shown in Fig.

7.11(b).

Figure 7.11

(a) The probability distribution for the hydrogen 1s orbital in three-dimensional space. (b) The probability of finding the electron at points along a line drawn from the nucleus

outward in any direction for the hydrogen 1s orbital.

Imagine that the space around the hydrogen nucleus is made up of a series of thin spherical shells (rather like layers in an onion), as shown in Fig. 7.12(a).

When the total probability of finding the electron in each spherical shell is plotted versus the distance from the

nucleus, the plot in Fig. 7.12(b) is obtained.

This graph is called the radial probability distribution.

Figure 7.12

(a) Cross section of the hydrogen 1s orbital probability

distribution divided into successive thin spherical shells. (b) The radial probability distribution. A plot of the total probability of finding the electron in each thin spherical shell as a function of distance from the nucleus.

One more characteristic of the hydrogen 1s orbital that we must consider is its size. As we can see from Fig. 7.11, the size of this orbital cannot be defined precisely, since the probability never becomes zero.

So, in fact, the hydrogen 1s orbital has no distinct size.

The definition most often used by chemists to describe the size of the hydrogen 1s orbital is the radius of the

sphere that encloses 90% of the total electron probability.

7.6 Quantum Numbers

Each of these orbitals is characterized by a series of numbers called quantum numbers,

z

The principal quantum number (n) has integral values: 1, 2, 3,…The principal quantum numbers is related to the size and energy of the orbital. As n

increases, the orbital becomes larger and the electron

spends more time farther from the nucleus. An increase in n also means higher energy, because the electron is less tightly bound to the nucleus, and the energy is less

negative.

z

The angular momentum quantum number (l) has integral values from 0 to n-1 for each value of n. This

quantum number is related to the shape of atomic orbitals.

The value of l for a particular orbital is commonly

assigned a letter: l = 0 is called s; l = 1 is called p; l = 2 is

called d; l = 3 is called f. This system arises from early

spectral studies and is summarized in Table 7.1.

z

The magnetic quantum number (m

) has integral

values between l and –l, including zero. The value to m

is related to the orientation of the orbital in space relative

to the other orbitals in the atom.

TABLE 7.1 The Angular Momentum Quantum

Numbers and Corresponding Letters Used to

Designate Atomic Orbitals

The first four levels of orbitals in the hydrogen atom are listed with their quantum numbers in Table 7.2. Note that each set of orbitals with a given value of l

(sometimes called a subshell) is designated by giving the value of n and the letter for l.

Thus an orbital where n = 2 and l = 1 is symbolized as 2p. There are three 2p orbitals, which have different

orientations in space. We will describe these orbitals in

the next section.

TABLE 7.2 Quantum Numbers for the First Four Levels of Orbitals in the Hydrogen

Atom

For principal quantum level n = 5, determine the number of allowed subshells (different values of l), and give the designation of each.

Solution

For n = 5, the allowed values of l run from 0 to 4 (n－1 = 5 － 1). Thus the subshells and their designations are

Sample Exercise 7.6

Electron Subshells

7.7 Orbital Shapes and Energies

These two types of representations for the hydrogen 1s, 2s, and 3s orbitals are shown in Fig. 7.13.

Note the characteristic spherical shape of each of the s orbitals.

Note also that the 2s and 3s orbitals contain areas of high probability separated by areas of zero probability.

These latter areas are called nodal surfaces, or simply

nodes.

Figure 7.13

Two representations of the hydrogen 1s, 2s, and 3s orbitals. (a) The electron

probability distribution. (b) The surface that contains 90% of the total electron probability (the size of the orbital, by definition).

The two types of representations for the 2p orbitals (there are no 1p orbitals) are shown in Fig. 7.14.

Note that the p orbitals are not spherical like s orbitals but have two lobes separated by a node at the nucleus.

At this point it is useful to remember that mathematical functions have signs.

This behavior is indicated in Fig. 7.14(b) by the

positive and negative signs inside their boundary surfaces.

It is important to understand that these are mathematical signs, not charges.

Just as a sine wave has alternating positive and negative phases, so too p orbitals have positive and negative phases.

The phases of the p

, p

, and p

orbitals are indicated in

Fig. 7.14(b).

Figure 7.14

Representation of the 2p orbitals. (a) The electron probability distribution for a 2p orbital. (Generated from a program by Robert Allendoerfer on Project SERAPHIM disk PC 2402;

reprinted with permission.) (b) The boundary surface

representations all three 2p orbitals. Note that the signs inside the surface indicate the phases (signs) of the orbital in that region of space.

The 3p orbitals have a more complex probability distribution than that of the 2p orbitals (see Fig. 7.15),

The d orbitals ( l = 2) first occur in level n = 3. The five 3d orbitals have the shapes shown in Fig. 7.16.

The d orbitals have two different fundamental shapes.

The f orbitals first occur in level n = 4.

Figure 7.17 shows representations of the 4f orbitals ( l =

3) along with their designations.

Figure 7.15

A cross section of the electron probability distribution for a 3p orbital.

Figure 7.16

Figure 7.16

Representation of the 3d orbital. (a) Electron density plots of selected 3d orbitals. (Generated from a program by Robert Allendoerfer on Project SERAPHIM disk PC 2402; reprinted with permission.) (b) The boundary surfaces of all five 3d

orbitals, with the signs (phases) indicated.

Figure 7.17

Representation of the 4f orbitals in terms of their boundary surfaces.

Thus all orbitals with the same value of n have the same energy-they are said to be degenerate.

This is shown in Fig. 7.18, where the energies for the orbitals in the first three quantum levels for hydrogen are shown.

In the lowest energy state, the ground state, the

electron resides in the 1s orbital. If energy is put into the

atom, the electron can be transferred to a higher-energy

orbital, producing an excited state.

Figure 7.18

Orbital energy levels for the hydrogen atom.

A Summary of the Hydrogen Atom

z In the quantum (wave) mechanical model, the electron is viewed as a standing wave. This representation leads to a series of wave functions (orbitals) that describe the possible energies and spatial distributions available to the electron.

z In agreement with the Heisenberg uncertainty principle, the model cannot specify the detailed electron motions. Instead, the square of the wave function represents the probability distribution of the

electron in that orbital. This allows us to picture orbitals in terms of probability distributions, or electron density maps.

z The size of an orbital is arbitrarily defined as the surface that contains 90% of the total electron probability.

z The hydrogen atom has many types of orbitals. In the ground state, the single electron resides in the 1s orbital. The electron can be excited to higher-energy orbitals if energy is put into the atom.

7.8 Electron Spin and the Pauli Principle

The concept of electron spin was developed by Samuel Goudsmit and George Uhlenbeck.

A spinning charge produces a magnetic moment, the electron could have two spin states, thus producing the two oppositely directed magnetic moments (see Fig.

7.19).

The new quantum number adopted to describe this

phenomenon, called the electron spin quantum number

(m

), can have only one of two values, +1/2 and –1/2.

Figure 7.19

A picture of the spinning electron. Spinning in one direction, the electron

produces the magnetic field oriented as shown in (a).

Spinning in the opposite

direction, it gives a magnetic field of the opposite orientation, as shown in (b).

Austrian physicist Wolfgang Pauli (1900-1958): In a given atom on two electrons can have the same set of four quantum numbers (n, l , m

, and m

).

This is called the Pauli exclusion principle.

Since electrons in the same orbital have the same

values of n, l and m

, this postulate says that they must

have different values of m

.

Then, since only two values of m

are allowed, an

orbital can hold only two electrons, and they must have opposite spins.

This principle will have important consequences as we use the atomic model to account for the electron

arrangements of the atoms in the periodic table.

7.9 Polyelectronic Atoms

polyelectronic atoms : atoms with more than one

electron, let’s consider helium, which has two protons in its nucleus and two electrons:

Three energy contributions must be considered in the description of the helium atom:

(1) the kinetic energy of the electrons as they move

around the nucleus, (2) the potential energy of attraction between the nucleus and the electrons, and (3) the

potential energy of repulsion between the two electrons.

Electron correlation problem: Since the electron

pathways are unknown, the electron repulsions cannot be calculated exactly.

To treat these systems using the quantum mechanical model, we must make approximations.

Most commonly, the approximation used is to treat

each electron as if it were moving in a field of charge that

is the net result of the nuclear attraction and the average

repulsions of all the other electrons.

The sodium atom has 11 electrons:

The electron clearly is attracted to the highly charges nucleus. The electron also feels the repulsions caused by the other 10 electrons.

The net effect is that the electron is not bound nearly as

tightly to the nucleus as it would be if the other electrons

were not present.

The electron is screened or shielded from the nuclear charge by the repulsions of the other electrons.

This picture of polyelectronic atoms leads to hydrogenlike orbitals for these atoms.

One especially important difference between

polyelectronic atoms and the hydrogen atom is that for hydrogen all the orbitals in a given principal quantum level have the same energy (they are said to be

degenerate).

In other words, when electrons are placed in a

particular quantum level, they “prefer” the orbitals in the other s, p, d, and then f.

2s orbital has a lower energy than the 2p orbital in a

polyelectronic atom by looking at the probability profiles of these orbitals (see Fig. 7.20).

Notice that the 2p orbital has its maximum probability

closer to the nucleus than for the 2s.

Figure 7.20

A comparison of the radial probability distributions of the 2s and 2p orbitals.

This penetration effect causes an electron in a 2s orbital to be attracted to the nucleus more strongly than an

electron in a 2p orbital.

That is, the 2s orbital is lower in energy than the 2p orbitals in a polyelectronic atom.

The same thing happens in the other principal quantum levels as well.

Figure 7.21 shows the radial probability profiles for the 3s, 3p, and 3d orbitals. Note again the hump in the 3s

profile very near the nucleus.

Figure 7.21

Figure 7.21

(a) The radial probability distribution for an electron in a 3s orbital. Although a 3s electron is mostly found far from the nucleus, there is a small but significant probability (shown by the arrows) of its being found close to the nucleus. The 3s

electron penetrates the shield of inner electrons. (b) The radial probability distribution for the 3s, 3p, and 3d orbitals. The

arrows indicate that the s orbital (red arrow) allows greater electron penetration than the p orbital (yellow arrow) does; the d orbital allows minimal electron penetration.

The relative energies of the orbitals for n = 3 are

A summary diagram of the orders of the orbital energies

for polyelectronic atoms is represented in Fig. 7.22.

Figure 7.22

The orders of the energies of the orbitals in the first three levels of polyelectronic atoms.

7.10 The History of the Periodic Table

The modern periodic table contains a tremendous amount of useful information.

The periodic table was originally constructed to

represent the patterns observed in the chemical properties of the elements.

The first chemist to recognize patterns was Johann

Dobereiner (1780-1849), who found several groups of

three elements that have similar properties, for example,

chlorine, bromine, and iodine.

The next notable attempt was made by the English chemist John Newlands, who in 1864 suggested that

elements should be arranged in octaves, based on the idea that certain properties seemed to repeat for every eighth element in a way similar to the musical scale, which

repeats for every eighth tone.

The present form of the periodic table was conceived independently by two chemists: the German Julius Lothar Meyer (1830-1895) and Dmitri Ivanovich Mendeleev

(1834-1907), a Russian (Fig. 7.23).

Figure 7.23

Figure 7.23

Dmitri Ivanovich Mendeleev (1834-1907), born in Siberia as the youngest of 17 children, taught chemistry at the University of St.

Petersburg. In 1860 Mendeleev heard the Italian chemist Cannizzaro lecture on a reliable method for determining the correct atomic masses of the elements. This important

development paved the way for Mendeleev’s own brilliant

contribution to chemistry--the periodic table. In 1861 Mendeleev returned to St. Petersburg, where he wrote a book on organic chemistry. Later Mendeleev also wrote a book on inorganic chemistry, and he was struck by the fact that the systematic approach characterizing organic chemistry was lacking in inorganic chemistry. In attempting to systematize inorganic

Figure 7.23

chemistry, he eventually arranged the elements in the form of the periodic table.

Mendeleev was a versatile genius who was interested in many fields of science. He worked on many problems associated with Russia’s natural resources, such as coal, salt, and various

metals. Being particularly interested in the petroleum industry, he visited the United States in 1876 to study the Pennsylvania oil fields. His interests also included meteorology and hot-air

balloons. In 1887 he made an ascent in a balloon to study a total eclipse of the sun.

In 1872 when Mendeleev first published his table (see

Fig. 7.24), the elements gallium, scandium, and germanium were unknown.

The data for germanium (which Mendeleev called

“ekasilicon”) are shown in Table 7.3.

Using his table, Mendeleev also was able to correct several values for atomic masses.

The original atomic mass of 76 for indium was bases on

the assumption that indium oxide had the formula InO.

Figure 7.24

Mendeleev’s early periodic table, published in 1872. Note the spaces left for missing elements with atomic masses 44, 68, 72,

TABLE 7.3 Comparison of the Properties of

Germanium as Predicted by Mendeleev and

as Actually Observed

This atomic mass placed indium, which has metallic properties, among the nonmetals.

Mendeleev assumed the atomic mass was probably incorrect formula, indium has an atomic mass of

approximately 113, placing the element among the metals.

It is still used to predict the properties of elements

recently discovered, as shown in Table 7.4.

TABLE 7.4 Predicted Properties of

Elements 113 and 114

7.11 The Aufbau Principle and the Periodic Table

Our main assumption here is that all atoms have the same type of orbitals as have been described for the hydrogen atom.

As protons are added one by one to the nucleus to build up the elements, electrons are similarly added to these

hydrogenlike orbitals.

This is called the aufbau principle.

Hydrogen has one electron, which occupies the 1s orbital in its ground state. The configuration for hydrogen is

written as 1s1, which can be represented by the following orbital diagram:

The arrow represents an electron spinning in a particular

direction.

The next element, helium, has two electrons.

Lithium has three electrons, two of which can go into the

1s orbital before the orbital is filled.

The next element, beryllium, has four electrons, which occupy the 1s and 2s orbitals:

Boron has five electrons, four of which occupy the 1s

and 2s orbitals. The fifth electron goes into the second type

of orbital with n = 2, the 2p orbitals:

Carbon is the next element and has six electrons. Two electrons occupy the 1s orbital, two occupy the 2s orbital, and two occupy 2p orbitals.

This behavior is summarized by Hund’s rule (named for the German physicist F. H. Hund), which states that the

lowest energy configuration for an atom is the one having

the maximum number of unpaired electrons allowed by the

Pauli principle in a particular set of degenerate orbitals.

The configuration for carbon could be written

1s

2s

2p

2p

to indicate that the electrons occupy separate 2p orbitals.

However, the configuration is usually given as 1s

2s

2p

, and it is understood that the electrons are in different 2p

orbitals.

The orbital diagram for carbon is

Note that the unpaired electrons in the 2p orbitals are

shown with parallel spins.

Figure 7.25 summarizes the electron configurations of the first 18 elements by giving the number of electrons in the type of orbital occupied last.

At this point it is useful to introduce the concept of

valence electrons, the electrons in the outermost principal quantum level of an atom.

The valence electrons of the nitrogen atom, for example,

are the 2s and 2p electrons.

Valence electrons are the most important electrons to chemists because they are involved in bonding, as we will see in the next two chapters. The inner electrons are known as core electrons.

Note in Fig. 7.25 that a very important pattern is

developing: The elements in the same group (vertical

column of the periodic table) have the same valence

electron configuration.

Sodium metal is so reactive that it is stored under kerosene to protect it from the oxygen in the air.

A vial containing potassium metal. The sealed vial contains an inert gas to protect the potassium from reacting with oxygen.

Figure 7.25

The electron configurations in the type of orbital occupied last for the first 18 elements.

The electron configuration of potassium is

The next element, scandium, beings a series of 10

elements (scandium through zinc) called the transition

metals, whose configurations are obtained by adding

electrons to the five 3d orbitals. The configuration of

scandium is

Calcium metal.

Chromium is often used to plate bumpers and hood

ornaments, such as this statue of Mercury found on a 1929 Buick.

The configurations of the transition metals are shown in Fig. 7.26.

The entire periodic table is represented in Fig. 2.27 in terms of which orbitals are being filled. The valence

electron configurations are given in Fig. 7.28. From these two figures, note the following additional points:

1. The (n＋1)s orbitals always fill before the nd orbitals.

For example, the 5s orbitals fill in rubidium and

strontium before the 4d orbitals fill in the second row of

transition metals (yttrium through cadmium). This eartly

Figure 7.26

Electron configurations for potassium through krypton. The transition metals (scandium through zinc) have the general configuration [Ar]4s²3dⁿ, except for chromium and copper.

Figure 7.27

Figure 7.27

The orbitals being filled for elements in various parts of the periodic table. Note that in going along a horizontal row (a

period), the (n+1)s orbital fills before the nd orbital. The group labels indicate the number of valence electrons (ns plus np electrons) for the elements in each group.

filling of the s orbitals can be explained by the

penetration effect. For example, the 4s orbital allows for so muh more penetration to the vicinity of the

nucleus that it becomes lower in energy than the 3d orbital. Thus the 4s fill before the 3d. The same thing can be said about the 5s and 4d, the 6s and 5d, and the 7s and 6d orbitals.

2. After lanthanum, which has the configuration

[Xe]6s

5d

, a group of 14 elements called the

lanthanide series, or the lanthanides, occurs. This series of elements corresponds to the filling of the seven 4f orbitals. Note that sometimes an electron

occupies a 5d orbital instead of a 4f orbital. This occurs because the energies of the 4f and 5d orbitals are very similar.

3. After actinium, which has the configuration [Rn]

7s

6d

, a group of 14 elements called the actinide

series, or the actinides, occurs. This series corresponds

to the filling of the seven 5f orbitals. Note that

sometimes one or two electrons occupy the 6d orbitals instead of the 5f orbitals, because these orbitals have very similar energies.

4. The group labels for Groups 1A, 2A, 3A, 4A, 5A, 6A, 7A, and 8A indicate the total number of valence

electrons for the atoms in these groups. For example,

all the elements in Group 5A have the configuration

ns

np

. (The d electrons fill one period late and are

usually nor counted as valence electrons.) The meaning of the group labels for the transition metals is not as

clear as for the Group A elements, and these will not be used in this text.

5. The groups labeled 1A, 2A, 3A, 4A, 5A, 6A, 7A, and

8A are often called the main-group, or representative,

elements. Every member of these groups has the same

valence electron configuration.

The International Union of Pure and Applied Chemistry (IUPAC), a body of scientists organized to standardize

scientific conventions, has recommended a new form for

the periodic table, which the American Chemical Society

has adopted (see the blue numbers in Fig. 7.28).

Figure 7.28

Figure 7.28

The periodic table with atomic symbols, and partial electron configurations.

Give the electron configurations for sulfur (S), cadmium (Cd), hafnium (Hf), and radium (Ra) using the periodic table inside the front cover of this book.

Solution

Sulfur is element 16 and resides in Period 3, where the 3p orbitals are being filled (see Fig. 7.29). Since sulfur is the fourth among the “3p elements,” it must have four 3p

electrons. Its configuration is

Sample Exercise 7.7

Electron configurations

Cadmium is element 48 and is located in Period 5 at the end of the 4d transition metals, as shown in Fig. 7.29. It is the tenth element in the series and thus has 10 electrons in the 4d orbitals, in addition to the 2 electrons in the 5s orbital. The configuration is

Hafnium is element 72 and is found in Period 6, as shown in Fig.7.29. Note that it occurs just after the lanthanide series. Thus the 4f orbitals are already filled. Hafnium is

Sample Exercise 7.7

the second member of the 5d transition series and has two 5d electrons. The configuration is

Radium is element 88 and is in Period 7 (and Group 2A), as shown in Fig. 7.29. Thus radium has two electrons in the 7s orbital, and the configuration is

See Exercises 7.69 through 7.72 Sample Exercise 7.7

Figure 7.29

The positions of the elements considered in Sample Exercise 7.7.

7.12 Periodic Trends in Atomic Properties

Ionization energy is the energy required to remove an electron from a gaseous atom ot ion:

X(g) X(g)

＋ e

Where the atom or ion is assumed to be in its ground state.

Ionization Energy

The ionization energies are

The first ionization energy I

is the energy required to

remove the highest-energy electron of an atom.

The second electron comes from the 3s orbital (since Al

has the configuration [Ne]3s

). Note that the value of I

is considerably smaller than the value of I

, the second ionization energy.

Table 7.5 gives the values of ionization energies for all

the Period 3 elements. Note the large jump in energy in

each case in going from removal of valence electrons to

removal of core electrons.

TABLE 7.5 Successive Ionization

Energies in Kilojoules per Mole for the

Elements in Period 3

The values of the first ionization energies for the

elements in the first six periods of the periodic table are graphed in Fig. 7.30.

Note that in general as we go across a period from left to right, the first ionization energy increase.

On the other hand, first ionization energy decreases in going down a group.

This can be seen most clearly by focusing on the Group

1A elements (the alkali metals) and the Group 8A elements

(the noble gases), as shown in Table 7.6.

TABLE 7.6 First Ionization Energies for

the Alkali Metals and Noble Gases

Figure 7.30