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

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Balmer’s Analysis and Atomic Structure

Balmer found that all of the wavelengths emitted from the hydrogen atom in the

visible portion of the spectrum could be ﬁtted to an equation that relates the

wavelength (in nanometers) to an integer (n):

= 1.0968 ×10

−2



−



−1

The most signiﬁcant thing about this equation is that it introduces to us a

quantum number, n, that can vary only in whole, positive increments. For example,

the equation predicts an inﬁnite number of wavelengths in the spectrum gener-

ated by n = 3,4,5 ...all the way to inﬁnity. If you don’t quite understand where

the quantum number comes from, you are in good company. Balmer did not un-

derstand it either, but he used the equation, complete with quantum numbers,

because it ﬁts the data from the hydrogen atom. He had no theory or model that

predicted this ﬁt and had no good reason to propose the quantum numbers—

apart from the fact that they worked.

Although in concept an inﬁnite number of wavelengths can be emitted in the

Balmer spectrum, many wavelengths of light cannot be emitted. For example, the

longest wavelength of visible light emitted from the hydrogen atom (656.5 nm) is

found when n =3. The next longest wavelength in the Balmer emission spectrum

(486.1 nm) is at n = 4. No light is emitted with a wavelength between 486.1 nm

and 656.5 nm.

Although Balmer’s series is predominantly in the visible region of the electro-

magnetic spectrum, other hydrogen emission series exist that can all be ﬁtted to

a similar form of an equation. This generalized equation was developed by the

Swedish physicist Johannes Rydberg in 1888 and is known as the Rydberg for-

mula. The main feature of the Balmer series occurs when we set n

=2. The other

quantum number is variable, and we’ll call it n

The Rydberg Formula:

= 1.0968 ×10

−2



−



−1

The Balmer Equation:

= 1.0968 ×10

−2



−



−1

Remember that Balmer didn’t attach a physical meaning to the numbers. He was

just looking at the data from the standpoint of a mathematician.

The physicist Theodore Lyman reported observing a series of wavelengths in

the ultraviolet and X-ray region. Each of those wavelengths could be calculated if

were held constant at 1 and if n

were varied from 2 to 3 to 4 to 5 to ∞. In addi-

tion, there is a series with n

=3, with quantum numbers n

=4,5,6,...∞, which

begins in the infrared region; another with n

= 4 and quantum numbers n

= 5,

6,7,...∞; another with n

= 5 and n

= 6,7,8,...∞; and so on, with n

increas-

ing in units of one up to inﬁnity. Table 6.2 lists these ﬁrst four series for the

hydrogen atom.

218 Chapter 6 Quantum Chemistry: The Strange World of Atoms

Spectral Series in the Hydrogen Atom

Spectral Series n

Wavelength Range

Lyman 1 2,3,4,... X-ray and UV

Balmer 2 3, 4, 5,... Visible

Ritz-Paschen 3 4, 5, 6,... Short-wave infrared

Brackett 4 5,6,7,... Long-wave infrared

TABLE 6.2

EXERCISE 6.3 Using the Balmer Equation

The longest wavelength for the Balmer series is found when n = 3:

= 1.0968 ×10

−2



−



−1

= 0.0015233 nm

−1

λ =

0.0015233 nm

−1

= 656.47 nm

What is the smallest energy of radiation for this series?

Solution

The smallest energy corresponds to the longest wavelength, because energy and

wavelength are inversely proportional. We just calculated the longest wavelength to

be 656.47 nm, and if we put this into SI units as 6.5647 × 10

−7

m, we can convert it

directly into the energy:

smallest

(6.626 ×10

−34

J·s) ×(3.00 × 10

m·s

−1

)

6.5647 ×10

−7

= 3.03 × 10

−19

PRACTICE 6.3

Calculate the wavelength of a photon corresponding to the collapse of an electron

from n = 4 to n = 3; from n =6 to n = 4; from n =6 to n = 5.

See Problems 25, 26, 30–34, and 111.

The equation describing the wavelengths of light found in the hydrogen emis-

sion spectrum has been

empirically derived, which means that it has been derived

from experiments and observations rather than from theory. In other words,

Balmer, Lyman, and other mathematicians and spectroscopists of the late

nineteenth century performed comprehensive measurements on the hydrogen

emission spectrum and ﬁtted a mathematical expression to the data. Empirical ﬁts

to data generally raise more questions than they answer, because scientists want to

know why the equation works.

Why, for example,is the hydrogen emission spectrum

quantized to give only certain wavelengths?

Figure 6.8 shows the visible region of

the emission spectra of several elements. Note that the wavelengths of light emit-

ted by each metal are speciﬁc to that element. For instance, if we focus an instru-

ment at one of those wavelengths and look speciﬁcally for that signal, we will

observe it only if that particular element is present in

the sample. In other words, if we tune our instrument

to one of the wavelengths in the copper spectrum, our

instrument will be able to detect if copper is present.

This process is quite useful to the environmental

chemist. By exciting atoms within a small sample of

river water, the chemist can detect the presence of

copper. In addition, the intensity of the light detected

by the instrument can be related to the quantity of

copper in the river water sample. It is currently possi-

ble to measure nearly two dozen elements in water at

the same time at the parts per billion level (ppb, micro-

grams, µg, of element per liter of solution) or parts per

trillion level (ppt, ng/L) by simultaneously focusing on

one discrete wavelength given off by each element.

Table 6.3 gives several of the elements, the emis-

sion wavelengths typically used for detection, and the

minimum concentration of the element that can be

6.3 Atomic Emission and Absorption Spectroscopy, Chemical Analysis and the Quantum Number 219

Application

HEMICAL ENCOUNTERS:

Simultaneous

Determination of

Elements in Water

Simultaneous Determination of Elements in Water

Wavelength Detection Limit

Element (nm) (ppb = µg/L)

Al 396.152 1.5

As 188.980 3.5

Ba 455.403 0.04

Ca 315.887 1.5

Cd 214.439 0.3

Cr 267.716 0.5

Cu 324.754 0.3

Mo 202.032 0.8

Pb 220.353 3.0

Zn 213.857 0.3

TABLE 6.3

Source: Fast Analysis of Water Samples Comparing Axially and Radially Viewed

CCD Simultaneous ICP-AES, Varian, Inc., http://www.varianinc.com/image/

vimage/docs/products/spectr/icpoes/atworks/ icpes028.pdf (accessed

February 2006).

220 Chapter 6 Quantum Chemistry: The Strange World of Atoms

Wavelength (nm)

400 450 500 550 600 650 700 750

FIGURE 6.8

The emission spectra of different metals.

Each wavelength emitted is unique to

that element, which makes it possible to

analyze the element’s concentration at

very low concentrations.

reliably detected (“detection limit”). Note that the majority of these wavelengths

are in the ultraviolet region of the electromagnetic spectrum. Atomic emission,

absorption, and related spectroscopy techniques are among the most important

in chemistry for the analysis of elements in all manner of samples from brass to

blood. It is through atomic spectra that we learned core ideas about the funda-

mental structure of the atom.

6.4 The Bohr Model of Atomic Structure

In Chapter 2, we saw how the model of the atom evolved from the indivisible

spheres proposed by Dalton all the way to the “nuclear atom” view that emerged

from Rutherford’s discovery of the nucleus. We also discussed the way in which

scientiﬁc models develop and noted that useful information is gleaned even from

models that eventually turn out to be incorrect or only partially correct. Like the

earlier models, the Bohr model we are about to consider provides an incomplete

and oversimpliﬁed view of how atoms are really constructed. But it is useful to

examine this model and to appreciate how it solved certain puzzles of atomic

emission and contributed to scientists’ development of the quantum picture of

the atom. The work is so important that it earned Niels Bohr the Nobel Prize in

physics in 1922.

The simplest atom is that of hydrogen with a single electron and a nucleus of

a single proton. According to the Bohr model of the hydrogen atom, the electron

is found some distance from the nucleus, whirling around it in a circular orbit, as

shown in Figure 6.9. The electron can be in any one of an inﬁnite number of pos-

sible orbits around the nucleus, each orbit at a particular distance, or radius, from

the nucleus. Not just any radius will do, however. The orbits are spatially

quantized. Much like the layers of an onion, they are conﬁned to speciﬁc

locations, in spheres of ever-increasing radii, r, expressed in nanometers,

according to the equation:

r = 0.052917 n

where n, our quantum number for the Bohr model, can be 1, 2, 3, 4,...,∞.

Electrons in orbits close to the nucleus travel much more rapidly than

electrons in the outside orbits. This means that the kinetic energy of an

FIGURE 6.9

Schematic of the Bohr model of

the hydrogen atom.

n = 3

n = 2

Electron

Nucleus

Electron

orbits

n = 1

Video Lesson: CIA

Demonstration: Flame Colors

Video Lesson: The Bohr Model

electron in the orbits close to the nucleus is greater than the kinetic energy of those

in the outer orbits. This is true for satellites that orbit the earth just as it is for elec-

trons. The space shuttle circles the planet Earth many times during a voyage in a

low Earth orbit of between 185 and 650 km; it must travel at a very high speed,

about 27,900 km/h, in order to maintain its orbit.On the other hand, geosynchro-

nous satellites, such as weather and communications satellites, are 35,800 km

above the Earth’s surface and travel at a speed of only 11,300 km/h. They appear to

remain stationary over a speciﬁc place because their movement just matches the

rotation of our planet.

The increase in kinetic energy of the negatively charged electron closest to the

nucleus is offset by a greater decrease in the potential energy due to its position

relative to the positive nucleus. In other words, because opposite charges attract,

the closer the electron and the nucleus are to each other, the lower the potential

energy of the electron. The key outcome is that electrons closer to the nucleus have

lower energy than those farther away.

We call the allowed orbits

energy levels, because the electron in a given orbit

will have a constant total energy according to this equation:

=−

2.1786 ×10

−18

What is especially interesting about this equation is that the quantum number n

is the same quantum number we discovered in the equation for the radius. Note

that this equation calculates the energy of a given orbit as a negative number. This

is because a completely free electron (corresponding to n = ∞) is assigned an energy

of zero. When electrons become bound within atoms, their energy falls, and be-

cause it is falling from zero, it must become negative. This means that as an elec-

tron falls from higher orbits to lower ones, its energy has increasingly larger

negative values relative to zero, the value for the free electron. The lower values of

the quantum number have more negative energies than the higher values. The

most negative value for energy occurs when n = 1. At this orbit (n = 1), the elec-

tron is as strongly bound to the atom as is possible, and it is in its lowest energy

state in the atom. Unless energy is supplied to the hydrogen atom, its electron will

tend to be in the most strongly bound level, closest to the nucleus. As we noted

above, for the hydrogen atom, this is the smallest quantum number, n = 1. Any

atom that contains all its electrons in their lowest possible energy levels is said to

be in the

ground state, like the ground ﬂoor of a building.

Larger quantum numbers, n = 2,3,4,...,∞, represent progressively higher

atomic energy levels, just as the second, third, fourth,...ﬂoors of a building rep-

resent progressively higher gravitational energies. When electrons are moved

from the ground state into these higher energy levels, we say that they occupy

excited states. Atoms and molecules in excited states generally tend to relax back

down to their ground states after a short period of time.

Energy must be supplied to the hydrogen atom to promote its electron from

the ground state (n = 1) to any other state (n > 1). Conversely, the atom must

lose energy when the electron falls from any state to a lower energy state. These

electronic transitions, as they are called, can be explained very well by the quan-

tum picture of the Bohr atom. The electron must gain or lose the exact amount

of energy that separates two energy levels (two orbits) in order to jump between

the levels. The change in energy is expressed mathematically as

 E = E

− E

where E

is the energy of the ﬁnal state of the electron, E

is the energy of the initial

state of the electron, and



E is the difference in energy between these two levels.

6.4 The Bohr Model of Atomic Structure 221

In 1922 Niels Bohr (1885–1962) was

awarded the Nobel Prize in physics for his

work on the structure of the atom and his

explanation of how atoms emit light.

Speed just right

Speed too slow

The speed of a satellite is related to its

distance from the planet it circles.

Slower-moving objects must have

higher orbits or they will crash back

to the surface.

The change in energy made during an electronic transition can be either pos-

itive or negative as long as the energy is conserved. If



E is positive, energy is sup-

plied to the atom to make the electron jump from a lower to a higher state. If



is negative, energy is released from the atom because the electron falls from a

higher to a lower energy level, as shown in Figure 6.10.

One way to supply or release the required energy is for the atom to absorb or

emit electromagnetic radiation. This is where the hydrogen emission spectrum

proves useful. If we write the equation for hydrogen emission in terms of energy

instead of wavelength, we can show that the emission lines are the result of elec-

tronic transitions from the higher energy levels (or states) of the excited hydrogen

atom to lower energy states. We call the quantum number of the initial energy

level, n

and that of ﬁnal energy level n

E =−2.1786 × 10

−18



−



Note that the energy change is a discrete value. In other words, the energy is quan-

tized. For the electron to move from one energy level to a lower energy level, it

must lose this discrete amount of energy. Knowing that the energy is related to

the frequency indicates that h

, the electromagnetic energy, is equal to the energy

lost in the transition,



hν =

= E

This relationship (∆E = h

), although somewhat similar to that proposed by

Einstein, was proposed by Niels Bohr in an effort to correlate the energy emitted

from a hydrogen atom during an electronic transition with the frequency of ob-

served light. The relationship is referred to as the Bohr frequency condition.

If we combine the last two equations to calculate the wavelength of radiation

predicted for the hydrogen emission spectrum and convert from meters to

nanometers (1 m = 10

nm), we get

= 1.097 ×10



−



−1

= 1.097 ×10

−2



−



−1

222 Chapter 6 Quantum Chemistry: The Strange World of Atoms

4th level—high energy

Ground level—low energy

–

Nucleus

Photon

–

Nucleus

Photon

FIGURE 6.10

Energy changes that occur in an atom. An increase in energy results in promotion of an electron to a

higher energy level. A release of energy is observed when the electron returns to its original state.

6.4 The Bohr Model of Atomic Structure 223

This is the equation that Rydberg, Balmer, and others found by analyzing the

hydrogen emission spectrum and empirically ﬁtting an equation to the results.

When n

=1, we get the Lyman series. When n

=2, we get the Balmer series. We

can continue this process until we generate all of the energies of electromagnetic

radiation observed to be emitted from the hydrogen atom. These electronic tran-

sitions in the visible region are depicted schematically in Figure 6.11.

EXERCISE 6.4 Energy-to-Frequency Conversion in the Hydrogen Atom

A hydrogen atom in one of its excited states has an energy of −1.5129 × 10

−20

What frequency of radiation is emitted when the atom relaxes down to its ground

state (n = 1)?

First Thoughts

An important idea here is that an atom moving from its excited state to its ground

state releases energy (



E is −), typically as light, whereas an atom that goes from the

ground state to an excited state absorbs energy (



E is +).

Solution

The energy of the photon emitted must exactly match the energy lost by the hydro-

gen atom as the electron moves from a higher energy level to its ground state:

photon

=−E

electron

hν =−

(

− E

)

ν =

−

(

− E

)

The ﬁnal state energy is given by

=−

2.1786 ×10

−18

=−

2.1786 ×10

−18

=−2.1786 ×10

−18

ν =−

− E

)

=−

(−2.1786 ×10

−18

J) − (−1.5129 × 10

−20

6.626 ×10

−34

J·s

ν =−

(−2.163 ×10

−18

6.626 ×10

−34

J·s

= 3.265 × 10

−1

6 2 5 2 4 2 3 2

n = 6

n = 5

n = 4

n = 3

n = 2

n = 1

FIGURE 6.11

The visible electronic transitions of the

hydrogen emission spectrum. The energy

of the radiation observed must exactly

match the difference between energy of

the excited state and that of the ground

state (most energetically stable state).

Visualization: The Line Spectrum

of Hydrogen

224 Chapter 6 Quantum Chemistry: The Strange World of Atoms

Lasers and Their Applications

Lasing Medium Wavelength Application

TABLE 6.4

Diode

Helium/neon (HeNe)

Argon and krypton

HeCd

Solid state

Nd:YAG

Ruby

ArK excimer

635 nm (red)

670 nm (deep red)

780 nm, 800 nm, 900 nm,

1550 nm (Infrared)

632.5 nm (red)

457.9 nm (violet-blue)

488 nm (blue)

514 (green)

646 (red)

1060 nm (mid-IR)

442 nm (violet blue)

325 nm (UV)

1064 nm (IR)

694 nm (red)

193 nm (UV)

CD players, DVDs, laser printers,

laser pointers, high-speed ﬁber

optics, dental surgery

Alignment, barcode scanners, laser

pointers, light shows, blood cell counting

Forensic medicine, high-performance printing,

general and ophthalmic surgery, holography,

light shows

Industrial metal cutting, welding, surgery

Nondestructive testing and spectroscopy

Materials processing (drilling, cutting,

welding), target designation, surgery,

hair removal

Laser eye surgery

Further Insights

Our discussion is focusing on the hydrogen atom. Helium, with its two electrons,

was discovered in 1868 by the British astronomer Norman Lockyer and, in separate

observations of the emission spectrum from the Sun, Pierre Janssen of France. The

solar spectrum showed lines never before seen, indicating a previously unknown

element. Lockyer named the element helium after the Greek word for Sun, Helios.

The discovery was conﬁrmed in 1895 by the Scottish chemist William Ramsey, who

saw the same spectrum when looking at gases coming from the heating of the

uranium-containing mineral cleveite. Ramsey is known for discovering all of the

noble gases except radon.

PRACTICE 6.4

What wavelength of radiation needs to be supplied to a hydrogen atom in its ground

state to raise it to its excited state that has an energy of −1.5129 × 10

−20

See Problems 35–40 and 43–46.

We make use of the emission of light from excited atoms everyday. For exam-

ple, when electric current is passed through a tube containing a small amount

of neon gas, it causes the neon atoms to become excited. As they return to the

ground state, they emit light.

What is this contraption?

When the tube is bent to spell words, we refer to the contraption as a neon

sign. Another example of light emission from excited atoms is found in

lasers.

Lasers (the acronym stands for “light ampliﬁcation by stimulated emission of

radiation”) provide a powerful light source because they involve the simultaneous

collapse of a large population of excited atoms to the ground state.

The light emitted by each excited atom has the same phase (the peaks and

troughs of the light waves line up), giving rise to a powerful emission of light.

Table 6.4 sheds some light on common lasers and their uses.

224 Chapter 6 Quantum Chemistry: The Strange World of Atoms

Application

HEMICAL

ENCOUNTERS:

The Nature and

Applications of

Lasers

A laser pointer,

one example of

the many uses

of lasers.

A neon sign emits light when an electric

current is passed through the tube.

6.5 Wave–Particle Duality 225

HERE’S WHAT WE KNOW SO FAR

■

Electromagnetic radiation is characterized by wavelength, frequency, and

energy. These values can be interconverted.

■

Electromagnetic radiation is quantized into photons, the smallest unit of

radiation.

■

An atom can gain or lose discrete units of energy when an electron jumps

between the orbits allowed by the Bohr model.

■

Each line in the hydrogen emission spectrum is created by the light released

when an electron falls between two of the allowed orbits in the Bohr model.

■

Each time an electron falls between two levels, one photon of light is released,

carrying exactly the amount of energy equal to the difference between the two

energy levels.

■

The frequencies (and therefore wavelengths) of light emitted are determined by

the energy levels available for electrons to fall between. In other words,the light

coming out of the atom is a consequence of the atom’s electronic structure.

6.5 Wave–Particle Duality

We can dig deeper into the explanations that enable us to make sense of atoms if

we look at some apparent contradictions between the way the world appears on

the basis of everyday experience and the way it behaves at the level of individual

atoms. These two views of the world are known as the classical mechanical and

quantum mechanical views. In

classical mechanics we have particles and we have

waves, each with their own distinct characteristics. The particles do not act like

waves, and the waves do not act like particles. This is the kind of behavior we see

on a macroscopic level, as shown in Figure 6.12. In quantum mechanics, we can

still talk about particle and wave behavior, but the distinction between particles

and waves becomes fuzzier, although it is still of great importance, as we shall see.

According to classical mechanics, particles have exact locations, and their posi-

tions can be deﬁned precisely in space. Classically, a particle can move at any speed

(below the speed of light), provided that it is supplied with the necessary energy.

Also, if a particle is moving, it carries momentum (p), which is deﬁned as mass

multiplied by velocity (v):

p = m ×v

A 100-ton coal car moving at 30 mi/h has considerably more momentum than a

1-ton compact car moving at the same speed. A coal car traveling at 30 mi/h has

more momentum than another coal car that is crawling

along the tracks. Both mass and velocity contribute to

momentum.

One of the main descriptors a particle has is mass. In

addition, because all particles can carry momentum, they

must have mass. This means that we can use momentum to

test whether something is behaving like a classical particle

or not.

The Bohr model of the hydrogen atom deﬁes the classi-

cal logic of the behavior of particles. Although electrons

have momentum and therefore carry mass, they are quan-

tized in the Bohr model both in terms of where they can be

and in terms of how much energy they can have. The quan-

tized nature of space and energy is not the kind of behavior

we expect from particles on a macroscopic level. For example,

FIGURE 6.12

A person is very different from a wave. In

the quantum realm the nature of people

(particles) and waves become intertwined

in strange but very signiﬁcant ways.

The lighter of two cars has much less momentum, even if the

vehicles are traveling at the same speed. Consequently, it usually

suffers the greater amount of damage in a collision.

226 Chapter 6 Quantum Chemistry: The Strange World of Atoms

FIGURE 6.13

A tuned violin string plays a note when it vibrates

at a speciﬁc frequency, when a standing wave is set up

between the end points of the string. A different note

can be played when the player’s ﬁnger is used to vary

the length of the string. But only certain lengths—

certain positions of the player’s ﬁnger—will allow a

proper note to be played when a standing wave of

a different frequency occurs. The notes are a set of

quantized frequencies.

when we think of a moving race car, we envision it as capable of possessing any

velocity or mass within a continuous range. This is not true for the quantum me-

chanical view of the atom.

However, quantized energy states and locations have long been known in

everyday life. For example, musicians who play stringed instruments quantize the

energy of the vibration of the strings by shortening or lengthening the string

being played. This changes the frequency of the string’s vibration by setting the

wavelength of the vibration, as shown in Figure 6.13. A wave vibrating between

two ﬁxed endpoints is called a

standing wave.

The crucial word here is wave. All of the cases that were known to create quan-

tized states prior to the Bohr atom were systems that could be described by

“waves,” oscillating forms of energy that had a speciﬁc wavelength and frequency.

To clarify the distinction between the ideas of particles and waves, we can ﬁrst list

the characteristic classical properties of particles as follows:

In the Classical Mechanical View of the World:

■

Particles have a continuous range of available energies and positions (such as

a person going out for a walk).

■

Particles have a single value of energy and position measured at any one time

(such as a person seated in a chair).

■

Particles carry momentum (have mass, such as the mass of the person).

■

Particles are quantized (exist as individual particles, such as the person himself

or herself).

We can now build up a similar list of criteria for the classical mechanical view of

waves. In the view of classical mechanics, waves are neither composed of single

units nor located at exactly one place in space as are individual particles. As we

can see in our picture of the violin string in Figure 6.13, the wave exists every-

where between the bridge and the player’s ﬁnger. The amplitude changes between

these two sites, but the wave exists along the whole vibrating part of the violin

string. It is smeared out over a pretty large volume of space.

Waves can also be quantized by conﬁning them to a speciﬁc region in space,

as we do when we tie the two ends of the violin string down with the bridge and

our ﬁnger. In order for the amplitude of the wave to go to zero at the ends, the

number of wavelengths that must ﬁt between the two ends must be some half-

integral number of the wavelength:

Length of string = nλ

In this example, the length of the string, l, sets the possible wavelengths that the

vibration can have to λ = 2l, l,(2/3)l,(1/2)l,(2/n)l,...,0.This is illustrated in

Figure 6.14, where we can see the ﬁrst few allowed standing waves. Note that

6.6 Why Treating Things as “Waves” Allows Us to Quantize Their Behavior 227

whereas 2l (n = 1) and l (n = 2) are permissible values, no wavelengths between

these two values are allowed.

We can now list the characteristic classical mechanical properties of waves:

In the Classical Mechanical View of the World:

■

Waves have speciﬁc energy values (such as the speciﬁc notes on a violin

string).

■

Waves occupy a range of spatial positions all at the same time (such as the

positions of the vibrating violin string).

■

Waves do not carry momentum (have no mass; the sound waves from the

violin don’t have a mass).

■

Waves have a continuous range of amplitudes (smallest, single wave units;

such as the loudness of each note from the violin).

In our everyday (macroscopic) world, the characteristics of particles and

waves are separate. But here is a truly strange thing: If we investigate matter and

energy on a quantum level, where dimensions are on the order of atomic spacing,

things we once regarded as particles, such as electrons, can also behave like waves,

and things that were traditionally thought of as waves, such as light, can also behave

like particles. The electrons will still behave like particles, and the electromagnetic

radiation will still retain the properties of waves. But the distinction between par-

ticles and waves is no longer absolute. All quantum “things”—electrons, protons,

photons of electromagnetic radiation, whatever—have the properties of both waves

and particles simultaneously.

This schizophrenic behavior is called the

wave–particle duality, which pro-

poses that at very small dimensions, things behave like both waves and particles,

regardless of how they behave on the macroscopic level. The earliest experiments

performed on the electron investigated particle behavior by measuring the

electron mass (the Millikan oil drop experiment, Chapter 2) and position (J. J.

Thomson and the cathode ray tube, Chapter 2). However, we can show that the

energy quantization observed in the hydrogen emission spectrum or predicted by

the Bohr model results from the wave nature of the electron. The electron be-

haves as both a particle and a wave. It has mass, carries momentum, and is found

as a discrete single unit, which are properties expected of a classical particle, but

when contained in an atom or molecule it is spatially and energetically quantized,

which are properties expected of a classical wave.

6.6 Why Treating Things as “Waves”

Allows Us to Quantize Their Behavior

To see why the wave nature of the electron allows its energy levels to be quantized,

look at Figure 6.15, where we have drawn a “cross section” of several possible

orbits for the Bohr model. The Bohr model is three-dimensional, with spherical

orbits, but that would be hard to show schematically, so we will show a circular

cross section of the orbit.

Much like the standing wave vibrating on the violin string, only certain wave-

lengths will ﬁt on a circle. If the radius of the circle is r, we can quantize the

allowed wavelengths, λ, to be integral multiples of the circumference, which you

might recall from your high school geometry class has a length of 2

πr:

Circumference = 2πr = nλ

In other words, we can ﬁt n waves of wavelength λ around the circumference of

the circle, as shown in Figure 6.16.

1 half-wavelength

Node

2 half-wavelengths

3 half-wavelengths

Node Node

FIGURE 6.14

Overtones of a violin string. In addition

to the half-wavelength standing wave,

the violin also produces integral multi-

ples of the half-wavelength. These

vibrations are more energetic than the

half-wavelength vibration and are not

deliberately played by a violinist.