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CHAPTER 7

QUANTUM THEORY AND ATOMIC STRUCTURE

The Nature of LightElectromagnetic radiation (also called electromagnetic energy or radiant energy) – consists of energy propagated by electric and magnetic fields that increase and decrease in intensity as they move through space

This classical wave model explains why rainbows form, how magnifying glasses work, and many other familiar observations.

The Wave Nature of LightThe wave properties of electromagnetic radiation are described by three variables and one constant…

Frequency – the number of cycles the wave undergoes per second [s-1, or hertz (Hz)]

Wavelength – the distance between any point on a wave and the corresponding point on the next crest or trough of a wave (the distance the wave travels in one cycle) [m, nm, pm, angstroms (A)]

Speed – the distance it moves per unit time (meters/second). Electromagnetic radiation moves at 2.99792458x108 m/s (3.00x108 m/s) in a vacuum. This is a physical constant and is referred to as the speed of light.

Amplitude – the height of the crest (or depth of the trough) For an electromagnetic wave, the amplitude is related to the intensity of the radiation. Light of a particular color has a specific frequency (and thus, wavelength) but it can be dimmer (lower amplitude,

less intense) or brighter (higher amplitude, more intense)

The Electromagnetic Spectrum

Visible light is a small region of the electromagnetic spectrum

All waves in the spectrum travel at the same speed through a vacuum but differ in frequency and wavelength

Long-wavelength, low frequency radiation…

Used in microwave ovens, radios, and cell phones

Electromagnetic emissions are everywhere…

Human artifacts such as light bulbs, x-ray equipment, and car motors

Natural sources such as the Sun, lightning, radioactivity, and even the glow of fireflies!

Sample Problem 7.1 p. 263Interconverting Wavelength and Frequency1.00 A 1.00 m = 1.00x10-10 m

1x1010 A

(x-ray)

Speed(c) = frequency(v) x wavelength(λ)

3.00x108 m/s = frequency(v) x 1.00x10-10 m

frequency(v) = 3.00x1018 s-1

(radio signal)

325 cm 1 m = 3.25 m100 cm


3.00x108 m/s = frequency(v) x 3.25 m


(Blue sky)

473 nm 1.00 m = 4.73x10-7 m1x109 nm


3.00x108 m/s = frequency(v) x 4.73x10-7 m


Follow-Up Problem 7.1 p. 263Speed(c) = frequency(v) x wavelength(λ)

3.00x108 m/s = 7.23x1014 Hz x wavelength(λ)

wavelength(λ) = 4.15x10-7 m

4.15x10-7 m 1x1010 A = 4150 A1 m

4.15x10-7 m 1x109 nm = 415 nm1 m

The Classical Distinction Between Energy and Matter

Some distinctions between the behavior of waves and matter…

Refraction and dispersiono Light of a given wavelength travels at different speeds

through various transparent media – vacuum, air (even different temperatures of air), water, quartz, etc.

When a light wave passes from one medium into another, the speed of the wave changes

Refraction – change in speed causes a change in direction (figure below)

Dispersion – white light separates (disperses) into its component colors when it passes through a prism (or other refracting object) – rainbow is an example

In contrast to a wave of light, a particle of matter, like a pebble (above figure) does not undergo refraction…

Diffraction and interferenceo Diffraction – the bending of a wave when it strikes

the edge of an object If a wave passes through a slit about as wide as

its wavelength, it bends around both edges of the slit and forms a semicircle on the other side of the opening

In contrast… if you throw a collection of particles (a hand full of sand for example) at a small opening, some hit the edge of the opening, while others go through the opening and continue in a narrower group.

o Interference – when waves of light pass through two adjacent slits (see above diagram), the nearby emerging semicircular waves interact through the process of interference

If the waves’ crests coincide (in phase), they interfere constructively (amplitudes add together to form brighter region)

If the waves’ troughs coincide (out of phase), they interfere destructively (amplitudes cancel to form a darker region)

In contrast… particles do not exhibit interference

The Particle Nature of LightThree observations involving matter and light confused physicists at the turn of the 20th century…

Blackbody Radiation and the Quantum Theory of Energy

Observation When an object is heated to about 1000K, it begins to emit visible light (as you can see in the photo of the smoldering coal).

Observation When an object is heated to about 1500K, it begins to emit light that is brighter and more orange (as you can see in the photo of the electric heating element)

Observation When an object is heated to about 2000K, it emits light that is still brighter and whiter (as you can see in the light bulb filament)

Explanation In 1900, Max Planck (German physicist) assumed that the hot glowing object could emit (or absorb) only certain quantities of energy (E = nhv)

E = energy of the radiation n = is a positive integer (1, 2, 3, and so on) quantum

number h = Planck’s constant (6.626x10-34 J∙s) v = frequency

Conclusions

hot objects emit only certain quantities of energy and that the energy must be emitted from the object’s atomso this means that each atom emits only certain quantities

of energy

this would mean that each atom has only certain quantities of energy to start with

thus, the energy of an atom is quantizedo energy is emitted in fixed quantities

Each change in an atom’s energy occurs when the atom absorbs or emits one or more “packets”, or definite amounts, of energy.o Each energy packet is called a quantum (fixed

quantity or quanta) A quantum of energy is equal to hv

An atom changes its energy state by emitting (or absorbing) one or more quanta

The energy of the emitted (or absorbed) radiation is equal to the difference in the atom’s energy stateso ∆Eatom = Eemitted or absorbed

o ∆E = hv h = Planck’s constant v = frequency

Photoelectric Effect and the Photon Theory of Light

Despite the idea of quantization, physicist still pictured energy traveling in waves…

the wave model could not explain the second confusing observation (the flow of current when light strikes a metal)

Observation the photoelectric effect (when monochromatic light of sufficient frequency shines on a metal plate, a current flows)

Possible conclusion the current arises because the light transfers energy that frees electrons from the metal surface (this conclusions has two confusing features)

1. Presence of a threshold… for current to flow, the light shining on the metal must have a minimum, or threshold,

frequency, and different metals have different minimum frequencies. The wave theory associates the energy of light with the amplitude (intensity), not its frequency (color). This theory predicts that an electron would break free when it absorbs enough energy from light of any color.

2. Absence of a time lag…current flows the moment light of the minimum frequency shines on the metal, regardless of the light’s intensity. (The wave theory predicts that with dim light there would be a time lag before the current flows, because the electrons would have to absorb enough energy to break free)

Explanation the photon theory… Einstein proposed that light itself is particulate, quantized into tiny “bundles” of energy, called photons.

Each atom changes its energy, ∆Eatom, when it absorbs or emits one photon, one particle of light, whose energy is related to its frequency, not its amplitude.

How does the photon theory explain the two features of the photoelectric effect?

Why is there a frequency threshold?o A beam of light consists of an enormous number of

photons. The intensity (brightness) is related to the number of photons, but not the energy of each. Therefore, a photon of a certain minimum energy must be absorbed to free an electron from the surface. Since energy depends on frequency (hv), the theory predicts a threshold frequency.

Why is there no time lag?o An electron breaks free when it absorbs a photon of

enough energy It cannot break free by “saving up” energy from

several photons, each having less than the minimum energy.

Example… ping-pong photons (if one ping-pong ball doesn’t have enough energy to knock a book of a shelf, neither does a series of ping-pong balls, because the book can’t store the energy from the individual impacts. But, one baseball traveling at the same speed does have enough energy)

The current is weak in dim light because fewer photons of enough energy can free fewer electrons per unit time, but some current flows as

soon as light of sufficient energy (frequency) strikes the metal.

Sample Problem 7.2 p. 267Calculating the Energy of Radiation from its Wavelength∆E = hv

Step 1

Convert wavelength to meters

1.20 cm 1 m = 0.0120 m100 cm

Step 2

Calculate frequency from wavelength


3.00x108 m/s = frequency(v) x 0.0120 m


Step 3

Calculate energy of one photon

∆E = hv

∆E = (6.626x10-34 J∙s) (2.50x1010 s-1)

∆E = 1.66x10-23 J

Follow-Up Problem 7.2 p. 267(a)


3.00x108 m/s = frequency(v) x 1x10-8 m

frequency(v) = 3x1016 s-1

∆E = hv

∆E = (6.626x10-34 J∙s) (3x1016 s-1)

∆E = 2x10-17 J

(b)




∆E = hv

∆E = (6.626x10-34 J∙s) (6x1014 s-1)

∆E = 4x10-19 J

(c)




∆E = hv

∆E = (6.626x10-34 J∙s) (3x1012 s-1)

∆E = 2x10-21 J

As wavelength increases (frequency decreases) energy decreases

Atomic SpectraThe third confusing observation about matter and energy involved the light emitted when an element is vaporized and then excited electrically.

Line Spectra and the Rydberg EquationWhen light from electrically excited gaseous atoms passes through a slit and is refracted by a prism, it does not create a continuous spectrum or rainbow, as sunlight does.

Instead, it creates a line spectrum, a series of lines at specific frequencies separated by black spaces.o Each line spectrum is characteristic of the element

producing it. (Figure 7.8 below)

Features of the Rydberg Equation

Johannes Rydberg (Swedish physicist) developed a relationship, called the Rydberg equation that predicted the position and wavelength of any line in a given series.

1/λ = R (1/n21 - 1/n2

2)o λ = wavelengtho n1 and n2 are positive integers with n2 > n1

o R = Rydberg constant (1.096776x107 m-1)

Problems with Rutherford’s Nuclear Model

Almost as soon as Rutherford proposed his nuclear model, a major problem arose…

A positive nucleus and a negative electron attract each other, and for them to stay apart, the kinetic energy of the electron’s motion must counterbalance the potential energy of the attraction.

o Laws of classical physics say that a negative particle moving in a curved path around a positive particle must emit radiation and thus lose energy.

If electrons behave in this way, they would spiral into the nucleus and atoms would collapse!

Laws of classical physics would also suggest that the emitted radiation would create a continuous spectrum, not a line spectrum.

The Bohr Model of the Hydrogen AtomNiels Bohr (Danish physicist) suggested a model for the hydrogen atom that did predict the existence of line spectra…

Postulates of the model…o The hydrogen atom has only certain energy levels

Bohr referred to these as stationary states Each state is associated with a fixed circular

orbit around the nucleuso The higher the energy level, the further

the orbit is from the nucleuso The atom does not radiate energy while in one of its

stationary states Although this violates classical physics, the atom

does not change energy while the electron moves within an orbit

o The atom changes to another stationary state (the electron moves to another orbit) only by absorbing or emitting a photon. The energy of the photon (hv) equals the difference in the energies of the two states.

Ephoton = ∆Eatom = Efinal – Einitial = hv

Features of the Model

Quantum numbers and the electron orbit… the quantum number n is a positive integer (1, 2, 3, …) associated with radius of an electron orbit, which is directly related to the electron’s energy. (the lower the n value, the smaller the

radius of the orbit, and the lower the energy level

Ground state… when the electron is in its first orbit (n=1), it is closest to the nucleus, and the hydrogen atom is in its lowest energy level, called the ground state

Excited states… if the electron is in any orbit farther from the nucleus, the atom is in its excited state. With the electron in the second orbit (n=2), the atom is in the first excited state; when it is in the third orbit (n=3), the atom is in its second excited state, and so forth.

Absorption… if a hydrogen atom absorbs a photon whose energy equals the difference between lower and higher energy levels, the electron moves to the outer (higher energy) orbit.

Emission… if a hydrogen atom in a higher energy level (electron in a farther orbit) returns to a lower energy level

(electron in closer orbit), the atom emits a photon whose energy equals the difference between the two levels.

How the Model Explains Line Spectra

A spectral line results because a photon of specific energy (and thus frequency) is emitted. The emission occurs when the electron moves to an orbit closer to the nucleus as the atom’s energy changes from a higher state to a lower one.

Key point…an atomic spectrum is not continuous because the atom’s energy is not continuous, but rather has only certain states.

Bohr’s model accounts for three series of spectral lines of hydrogen… When a sample of hydrogen is excited, the atoms absorb different quantities of energy. Because there are so many atoms in the whole sample, all the energy levels (orbits) have electrons.

Infrared series… when electrons drop from an outer energy level to the n = 3

Visible series… when electrons drop from an outer energy level to the n = 2

Ultraviolet series… when electrons drop from an outer energy level to the n = 1

Limitations of the Model (Bohr’s)

Although it fails to predict the spectrum of any other atom (mainly because hydrogen has only one electron and we now know that electrons do not move in fixed, defined orbits), we still use the terms ground state and excited state and retain the central idea that the energy of an atom occurs in discrete levels and it changes energy by absorbing or emitting a photon of specific energy.

The Energy Levels of the Hydrogen AtomApplying Bohr’s Equation for the Energy Levels of an Atom

Finding the difference in energy between two levels…

∆E = Efinal – Einitial = -2.18x10-18 J (1/n2final – 1/n2

initial)

When the atom emits energy… the electron moves closer to the nucleus (nfinal < ninitial) and the atom’s final energy is a larger negative number and ∆E is negative.

When the atom absorbs energy…the electron moves away from the nucleus (nfinal > ninitial) and the atom’s final energy is a smaller negative number and ∆E is positive.

Finding the energy needed to ionize the H atom…

H(g) H+(g) + e-

∆E = Efinal – Einitial = -2.18x10-18 J (1/n2final – 1/n2

initial)

ninitial = 1 and nfinal = 0

so…

∆E = Efinal – Einitial = -2.18x10-18 J (1/02final – 1/12

initial)

∆E = -2.18x10-18 J (0 – 1)

∆E = 2.18x10-18 J

Energy must be absorbed to remove an electron, so ∆E must be positive (which it is in the above example)

Finding the wavelength of a spectral line…

∆E = hv (remember that c = v x λ , so v = c/λ)

∆E = hc/λ

λ = hc/∆E

Sample Problem 7.3 p. 272Determining ∆E and λ of an Electron Transition(a)

∆E = -2.18x10-18 J (1/02final – 1/12

initial)

∆E = -2.18x10-18 J (1/42final – 1/12

initial)

∆E = -2.18x10-18 J (1/16– 1/1)

∆E = 2.04x10-18 J

(b)

∆E = hc/λ

λ = hc/∆E

λ = (6.626x10-34J∙s)(3.00x108 m/s) / 2.04x10-18 J

λ = 9.74x10-8 m

9.74x10-8 m 1x109 nm = 97.4 nm1m

Follow-Up Problem 7.3 p. 272(a)

∆E = -2.18x10-18 J (1/n2final – 1/n2

initial)

∆E = -2.18x10-18 J (1/32 – 1/62)

∆E = -2.18x10-18 J (1/9 – 1/36)

∆E = -1.82x10-19 J

(b)

∆E = hc/λ

λ = hc/∆E

λ = (6.626x10-34J∙s)(3.00x108 m/s) / 1.82x10-19 J

λ = 1.09x10-6 m

1.09x10-6 m 1x1010 A = 1.09x104 A1m

The Wave-Particle Duality of Matter and EnergyOne of the results of Einstein’s work was his discovery that matter and energy are alternate forms of the same entity…

This is embodied in his famous E = mc2 which relates the quantity of energy equivalent to a given mass

Results that showed energy to be particle like had to co-exist with others that showed matter to be wavelike

The Wave Nature of Electrons and the Particle Nature of PhotonsLouis de Broglie – French physics student

Proposed that if energy is particle-like, perhaps matter is wavelikeo He reasoned that if electrons have wavelike motion in

orbits of fixed radii, they would have only certain allowable frequency and energies

Example… because the end of a guitar string is fixed, only certain vibrational frequencies (and wavelengths) are allowable to create a note.

See figure 7.12 belowo Combining the equations for mass-energy equivalence

(E = mc2) and energy of a photon (E = hv = hc/λ) derived an equation for the wavelength of any particle of mass m

λ = h / mu λ = wavelength h = 6.626x10-34 kg∙m2/s m = mass in kg u = speed in m/s

According to this equation, matter behaves as though it moves in a wave

An object’s wavelength is inversely proportional to its mass

Sample Problem 7.4 p. 276Calculating the de Broglie Wavelength of an Electronλ = h / mu

λ = 6.626x10-34 kg∙m2/s / (9.11x10-31kg)(1.00x106 m/s)

λ = 7.27x10-10 m

Follow-Up Problem 7.4 p. 276100. nm 1 m = 1x10-7m

1x109 nm

λ = h / mu

1x10-7 m = (6.626x10-34 kg∙m2/s) / (9.11x10-31 kg)(u)

(1x10-7 m) (9.11x10-31 kg)(u) = (6.626x10-34 kg∙m2/s)

u = 7.27x103 m/s

So…

If electrons travel in waves, they should exhibit diffraction and interference. A fast-moving electron has a wavelength of about 1x10-10 m, so a beam of such electrons should be diffracted by the spaces between atoms in a crystal (about 10-10 m)

C. Davidson and L. Germer guided a beam of x-rays and then a beam of electrons at a nickel crystal and obtained two diffraction patterns (thus particles with mas and charge create diffraction patterns, just as electromagnetic waves do)o The electron microscope is a major application of this

understanding

Electron micrograph of blood cells (x1200)

The Particle Nature of Photons

The de Broglie wave wavelength equation suggests that we can calculate the momentum (p), the product of mass and speed, for a photon.

λ = h / mc = h / p

The inverse relationship between p and λ means that shorter wavelength (higher energy) photons have greater momentum.

Wave-Particle Duality

Classical experiments had shown matter to be particle-like and energy to be wavelike. Results on the atomic scale show electrons moving in waves and photons having momentum. Thus, every property of matter was also a property of energy.

The truth is both matter and energy show both behaviorso Matter is both wave-like and particle likeo Energy is both wave-like and particle like

Heisenberg’s Uncertainty PrincipleIn classical physics, a moving particle has a definite location at any instant, whereas a wave is spread out in space. If an electron has the properties of both a particle and wave, can we determine its position in the atom?

In 1927, Werner Heisenberg (German physicist) postulated the uncertainty principle

o It is impossible to know simultaneously the position and momentum of a particle.

The Quantum-Mechanical Model of the AtomAcceptance of the dual nature of matter and energy (both are wave-like as well as particle like) and the uncertainty principle culminated in the field of quantum mechanics

Examines the wave nature of objects on the atomic scale Erwin Schrӧdinger (1926) derived an equation that is the

basis for the quantum-mechanical model of the hydrogen atomo The model describes quantities of energy that result

from allowed frequencies of its electron’s wavelike motion

The electron’s position can only be known within a certain probability

The Atomic Orbital and the Probable Location of the ElectronTwo central aspects…

The Schrӧdinger Equation and the Atomic Orbital

The electron’s matter-wave occupies the space near the nucleus and is continuously influenced by it.

Hψ = Eψ

ψ = (Greek psi, pronounced “sigh”) is a wave function (or atomic orbital), a mathematical description of the electron’s matter-wave in three dimension

H = Hamiltonian operator (represents a set of mathematical operations that, when carried out with a particular ψ, yields one of the allowed energy states of the atom

Each solution of the equation gives an energy state associated with a given atomic orbital

Important point…

An “orbital” in the quantum-mechanical model bears no resemblance to an “orbit” in the Bohr modelo An orbit is an electron’s actual path around the

nucleuso An orbital is a mathematical function that describes

the electron’s matter-wave but has no physical meaning

The Probable Location of the Electron

While we cannot know exactly where the electron is at any moment, we know where it probably is (where it spends most of its time)

We can determine this by squaring the wave function ψ2

o This is called the probability density A probability of finding the electron in some tiny

volume of the atom

We depict the electron’s probable location in several ways, which we will look at first of the hydrogen atom’s ground state

Probability of the electron being in some tiny volume of the atom…

For each energy level, we can create an electron probability density diagram, or more simply an electron density diagramo The value of ψ2 for a given volume is shown with

dots: the greater the density of dots, the higher probability of finding the electron in that volume

For the ground state of hydrogen, the electron probability density decrease with distance from the nucleus along a line, r

These diagrams are also called electron cloud depictions because, if we could take a time-exposure photograph of the electron’s wavelike motion around the nucleus, it would appear as a “cloud” of positions.o The electron cloud is an imaginary picture of the

electron changing its position rapidly over time; it does not mean that an electron is a diffuse cloud of charge

Total probability density at some distance from the nucleus…

To find the radical probability distribution (the total probability of finding the electron at some distance r from the nucleus, we first mentally divide the volume around the nucleus into thin, concentric, spherical layers, like the layers of an onion (Figure 7.16 above)o The falloff in probability density with distance has

important effect… Near the nucleus, the volume of each layer

increases faster than its density of dots decreases The result…

o The total probability peaks in a layer near, but not at, the nucleus

Probability contour and the size of the atom…

We can visualize an atom with a 90% probability contouro The electron is somewhere within that volume 90% of

the time (Figure 7.16 E above)

Quantum Numbers of an Atomic OrbitalAn atomic orbital is specified by three quantum numbers…

Are part of the solution of the Schrӧdinger equation Indicate the size, shape, and orientation in space

Principle quantum number ( n )

Positive integer (1, 2, 3, and so forth) It indicates the relative size of the orbital (the relative

distance from the nucleus ) Specifies the energy level of the hydrogen atom The higher the n value, the higher the energy level When the electron occupies an orbital n = 1, the hydrogen

atom is in its ground stateo When the electron occupies an orbital with n = 2 (first

excited state), the atom has more energy

Angular momentum quantum number ( l )

Is an integer from 0 to n-1 Is related to the shape of the orbital The principle quantum number sets a limit on the angular

momentum numbero n = 1, l can only equal 0o n = 2, l can equal 0 or 1

o n = 3, l can equal 0, 1, or 2

Magnetic Quantum Number ( m l)

Is an integer from -1 through 0 to +1 Prescribes the three dimensional orientation of the orbital in

the space around the nucleus l limits ml

o l = 0 can only have ml = 0o l = 1 can have three ml values, -1, 0, +1

The number of ml values equals 2l + 1 The total number or orbitals, for a given n value

is n2

Sample Problem 7.6 p. 282Determining Quantum Numbers for an Energy Leveln = 3, l = 0, 1, 2

l = 0, ml = 0

l = 1, ml = -1, 0, +1

l = 2, m2 = -2, -1, 0, +1, +2

There are nine ml values, so there are nine (9) orbitals with n = 3

Follow-Up Problem 7.6 p. 282n = 4, l = 0, 1, 2, 3

l = 0, ml = 0

l = 1, ml = -1, 0, +1

l = 2, ml = -2, -1, 0, +1, +2

l = 3, ml = -3, -2, -1, 0, +1, +2, +3

Quantum Numbers and Energy LevelsLevel

The atom’s energy levels, or shells, are given by the n valueo The smaller the n value, the lower the energy level

and the greater the probability that the electron is closer to the nucleus

Sublevel

The atom’s levels are divided into sublevels, or subshells, that are given by the l value. Each designates the orbital shape with a lettero l = 0 is the s sublevelo l = 1 is the p sublevelo l = 2 is the d sublevelo l = 3 is the f sublevel

sublevels with l values greater than 3 are designated by consecutive letters after f: g sublevel, h sublevel, and so on…

A sublevel is named with its n value and letter designationo n = 2 and l = 0 is called 2s

Orbital

Each combination of n, l, and ml specifies the size (energy), shape, and spatial orientation of one of the atom’s orbitals.

Sample Problem 7.7Determining Sublevel Names and Orbital Quantum Numbers(a)

n = 3, l = 2 3d -2, -1, 0, +1, +2 5 orbitals

(b)

n = 2, l = 0 2s 0 1 orbital

(c)

n = 5, l = 1 5p -1, 0, +1 3 orbitals

(d)

n = 4, l = 3 4f -3,-2, -1, 0, +1, +2, +3 7 orbitals

Follow-Up Problem 7.7 p. 2832p n = 2, l = 1, ml = -1, 0, +1

5f n = 5, l = 3, ml = -3, -2, -1, 0, +1, +2, +3

Sample Problem 7.8 p. 283Identifying Incorrect Quantum Numbers

n l ml name(a) 1 1 0 1p(b) 4 3 +1 4d(c) 3 1 -2 3p

(a) A sublevel with n = 1 can have only l = 0, not 1 = 1 The only possible sublevel name would be 1s

(b) A sublevel with l = 3 is an f sublevel. The name should be 4f

(c) A sublevel with 1 = 1 can only have -1, 0, or +1 for ml

Follow-Up Problem 7.8 p. 283n l ml name

(a) 4 1 0 4p(b) 2 1 0 2p(c) 3 2 -2 3d(d) 2 0 0 2s

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