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Chapter 7Quantum Theory of AtomBushra Javed

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Contents

1. The Wave Nature of Light2. Quantum Effects and Photons3. The Bohr Theory of the Hydrogen Atom4. Quantum Mechanics 5. Quantum Numbers and Atomic Orbitals6. Quantum Numbers and Atomic Orbitals

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A Theory that Explains Electron Behavior• The quantum-mechanical model explains the manner

in which electrons exist and behave in atoms

• It helps us understand and predict the properties of atoms that are directly related to the behavior of the electrons

The Nature of Light:Its Wave Nature

• Light is a form of electromagnetic radiation– composed of perpendicular oscillating waves, one

for the electric field and one for the magnetic field• an electric field is a region where an electrically charged

particle experiences a force• a magnetic field is a region where a magnetized particle

experiences a force

• All electromagnetic waves move through space at the same, constant speed– 3.00 x 108 m/s in a vacuum = the speed of light, c

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Electromagnetic Radiation

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Electromagnetic Spectrum

The range of frequencies and wavelengths of electromagnetic radiation is called the electromagnetic spectrum.

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Characterizing Waves• The amplitude is the height of the wave

– the distance from node to crest • or node to trough

– the amplitude is a measure of how intense the light is – the larger the amplitude, the brighter the light

• The wavelength (l) is a measure of the distance covered by the wave– the distance from one crest to the next

Wave Characteristics

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Characterizing Waves• The frequency (n) is the number of waves that pass a

point in a given period of time

– the number of waves = number of cycles– units are hertz (Hz) or cycles/s = s−1

• 1 Hz = 1 s−1

• The total energy is proportional to the amplitude of the waves and the frequency– the larger the amplitude, the more force it has

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Characterizing Waves• The longer the wavelength of light, the lower

the frequency.

• The shorter the wavelength of light, the higher the frequency.

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Color• The color of light is determined by

its wavelength or frequency• White light is a mixture of all the

colors of visible light– a spectrumRedOrangeYellowGreenBlueViolet

• When an object absorbs some of the wavelengths of white light and reflects others, it appears colored

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Wavelength, Frequency & Speed of Light

Wavelength and Frequency are related as:

λ α 1/ v

c = λ v

Where c is the speed of light

• Light waves always move through empty space at the same speed.

• Therefore speed of light is a fundamental constant.

c = 3.00 x 108 m/s

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The Wave Nature of LightExample 1Which of the following statements is incorrect?

a. The product of wavelength and frequency of electromagnetic radiation is a constant.

b. As the energy of a photon increases, its frequency decreases.

c. As the wavelength of a photon increases, its frequency decreases.

d. As the frequency of a photon increases, its wavelength decreases.

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Units of Wavelength & Frequency• Units of Wavelength:

nanometers, micrometers, meters and Angstrom

1 nm = 10-9m

1A° = 10-10m

• Units of Frequency:

1/second (per second) or 1s-1

Hertz = Hz, MHz and GHz

1-1s = 1 Hz

s

110x6.4

s

m10x3.00

λ14

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c

Wavelength & Frequency Example 2What is the wavelength of blue light with a frequency of 6.4 1014/s?

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n = 6.4 1014/sc = 3.00 108 m/s

c = nl so l = c/n

l = 4.7 10−7 m

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Wavelength & Frequency Example 3What is the frequency of light having a wavelength of 681 nm?

l = 681 nm = 6.81 10−7 mc = 3.00 108 m/s

c = nl so n = c/l

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m3.00 10

s6.81 10 m

c

l = 4.41 1014 /s

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Electromagnetic SpectrumExample 4Which type of electromagnetic radiation has the highest energy?

a. radio wavesb. gamma raysc. blue lightd. red light

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Wave Theory of light

One property of waves is that they can be diffracted—that is, they spread out when they encounter an obstacle about the size of their wavelength.

In 1801, Thomas Young, a British physicist, showed that light could be diffracted. By the early 1900s, the wave theory of light was well established.

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Diffraction

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The Wave/Particle Nature of Light

• Max Planck worked on the light of various frequencies emitted by the hot solids at different temperatures.

• In 1900, Max Planck proposed that radiant energy is not continuous, but is emitted in small bundles.

• Planck theorized that the atoms of an heated object vibrate with a definite frequency.

• An individual unit of light energy is a photon.

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Dual Nature of Light• Einstein expanded on Planck’s work.

• Through Photoelectric Effect ,he showed that when light of particular frequency shines on a metal surface it knocks out electrons.

• This knocking out of electrons is dependent only on the frequency of the shining light.

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The Photoelectric Effect

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Einstein’s Explanation• Einstein proposed that the light energy was

delivered to the atoms in packets, called quanta or photons

• The energy of a photon of light is directly proportional to its frequency– inversely proportional to its wavelength– the proportionality constant is called Planck’s

Constant, (h) and has the value 6.626 x 10−34 J s∙

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Einstein’s Explanation

Light, therefore, has properties of both waves and matter.

Neither understanding is sufficient alone. This is called the particle–wave duality of light.

Ejected Electrons• One photon at the threshold frequency gives the

electron just enough energy for it to escape the atom– binding energy, f

• When irradiated with a shorter wavelength photon, the electron absorbs more energy than is necessary to escape

• This excess energy becomes kinetic energy of the ejected electron

Kinetic Energy = Ephoton – Ebinding

KE = hn − f25

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The Photoelectric EffectExample 5:Suppose a metal will eject electrons from its surface when struck by yellow light. What will happen if the surface is struck with ultraviolet light?

a. No electrons would be ejected.b. Electrons would be ejected, and they would have the

same kinetic energy as those ejected by yellow light.c. Electrons would be ejected, and they would have greater kinetic energy than those ejected by yellow light. d. Electrons would be ejected, and they would have lower kinetic energy than those ejected by yellow light.

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Photoelectric EffectExample 6The blue–green line of the hydrogen atom spectrum has a wavelength of 486 nm. What is the energy of a photon of this light?

l = 486 nm = 4.86 10−7 mc = 3.00 108 m/sh = 6.63 10−34 J s

E = hn andc = nl so E = hc/l

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m6.63 10 Js 3.00 10

sλ 4.86 10 m

hcE

E = 4.09 10−19 J

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Problems with Rutherford’s Nuclear Model of the Atom

• Electrons are moving charged particles• According to classical physics, moving charged

particles give off energy• Therefore electrons should constantly be

giving off energy• The electrons should lose energy, crash into

the nucleus, and the atom should collapse!!– but it doesn’t

Continuous Vs Line SpectraIn addition, this understanding could not explain the observation of line spectra of atoms.A continuous spectrum contains all wavelengths of light. A line spectrum shows only certain colors or specific wavelengths of light. When atoms are heated, they emit light. This process produces a line spectrum that is specific to that atom.

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Spectra• When atoms or molecules absorb energy, that

energy is often released as light energy– fireworks, neon lights, etc.

• When that emitted light is passed through a prism, a pattern of particular wavelengths of light is seen that is unique to that type of atom or molecule – the pattern is called an emission spectrum– non-continuous– can be used to identify the material

• flame tests30

Exciting Gas Atoms to Emit Light with Electrical Energy

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Identifying Elements with Flame Tests

Na K Li Ba

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Emission Spectra

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Emission Line Spectra• When an electrical voltage is passed across a

gas in a sealed tube, a series of narrow lines is seen.

• These lines are the emission line spectrum. The emission line spectrum for hydrogen gas shows three lines: 434 nm, 486 nm, and 656 nm.

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The Bohr Model of the Atom• The nuclear model of the atom does not explain what

structural changes occur when the atom gains or loses energy

• Bohr developed a model of the atom to explain how the structure of the atom changes when it undergoes energy transitions

• Bohr’s major idea was that the energy of the atom was quantized, and that the amount of energy in the atom was related to the electron’s position in the atom– quantized means that the atom could only have very

specific amounts of energy

Bohr’s Model• The electrons travel in orbits that are at a fixed

distance from the nucleus– Energy Levels– therefore the energy of the electron was

proportional to the distance the orbit was from the nucleus

• Electrons emit radiation when they “jump” from an orbit with higher energy down to an orbit with lower energy– the emitted radiation was a photon of light– the distance between the orbits determined the

energy of the photon of light produced37

38Atomic Line Spectra

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Bohr’s Model of AtomExample 7Which of the following energy level changes in a hydrogen atom produces a visible spectral line?

a. 3 → 1b. 4 → 2c. 5 → 3d. all of the above

Rydberg’s Spectrum Analysis

• Rydberg analyzed the spectrum of hydrogen and found that it could be described with an equation that involved an inverse square of integers

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Bohr’s Model of AtomEnergy-Level PostulateAn electron can have only certain energy values, called energy levels. Energy levels are quantized.

For an electron in a hydrogen atom, the energy is given by the following equation:

RH = 2.179 10−18 J

n = principal quantum number7 | 41

2H

n

RE

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Bohr’s Model of Atom

The energy of the emitted or absorbed photon is related to DE:

We can now combine these two equations:

2i

2f

H

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nnRh

constant sPlanck'

Δ electron

h

hEEphoton

Bohr’s Model of Atom

Light is absorbed by an atom when the electron transition is from lower n to higher n (nf > ni). In this case, DE will be positive.

Light is emitted from an atom when the electron transition is from higher n to lower n (nf < ni). In this case, DE will be negative.

An electron is ejected when nf = ∞. 7 | 43

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Electron transitions for an electron in the hydrogen atom.

Example 8.What is the wavelength of the light emitted when the electron in a hydrogen atom undergoes a transition from n = 6 to n = 3?ni = 6nf = 3RH = 2.179 10−18 J

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m6.626 10 J s 3.00 10

sλ

1.816 10 J

= −1.816 10−19 J

soλ

Δhc

E E

hc

Δλ

18 1 1Δ 2.179 10 J

3 62 2E

2i

2f

H

11Δ

nnRE

1.094 10−6 m

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Bohr’s Model of AtomExample 9What is the frequency (in kHz ) of light emitted when the electron in a hydrogen atom undergoes a transition from level n = 6 to level n = 2? h = 6.63 × 10-34 J . s), RH = 2.179 × 10-18 J)

a. 5.49 × 105 kHzb. 7. 31× 1011 kHzc. 7. 31 × 1014 kHzd. 3.64 × 10–28 kHz

Wave properties of Matter

In 1923, Louis de Broglie, a French physicist, reasoned that particles (matter) might also have wave properties.

The wavelength of a particle of mass, m (kg), and velocity, v (m/s), is given by the de Broglie relation:

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λ

6.626 10 J s

h

mv

h

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Wave properties of MatterExample 10Compare the wavelengths of (a) an electron traveling at a speed that is one-hundredth the speed of light and (b) a baseball of mass 0.145 kg having a speed of 26.8 m/s (60 mph).

mv

hλ

Electronme = 9.11 10−31 kgv = 3.00 106 m/s

Baseballm = 0.145 kgv = 26.8 m/s

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Electronme = 9.11 10−31 kgv = 3.00 106 m/s

Baseballm = 0.145 kgv = 26.8 m/s

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6.63 10 J sλ

m9.11 10 kg 3.00 10

s

2.43 10−10 m

346.63 10 J sλ

m0.145 kg 26.8

s

1.71 10−34 m

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Quantum (or Wave) mechanics

Building on de Broglie’s work, Erwin Schrodinger devised a theory that could be used to explain the wave properties of electrons in atoms and molecules.

The branch of physics that mathematically describes the wave properties of submicroscopic particles is called or wave mechanics.

Uncertainty Principle

• Heisenberg stated that the product of the uncertainties in both the position and speed of a particle was inversely proportional to its mass– x = position, Dx = uncertainty in position– v = velocity, Dv = uncertainty in velocity– m = mass

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Uncertainty Principle

• This means that the more accurately you know the position of a small particle, such as an electron, the less you know about its speed– and vice-versa

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Schrodinger’s Equation

• Schodinger’s Equation allows us to calculate the probability of finding an electron with a particular amount of energy at a particular location in the atom

• Solutions to Schödinger’s Equation produce many wave functions, A plot of distance vs. Y2 represents an orbital, a probability distribution map of a region where the electron is likely to be found

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Solutions to the Wave Function, Y• Calculations show that the size, shape, and

orientation in space of an orbital are determined to be three integer terms in the wave function– added to quantize the energy of the electron

• These integers are called quantum numbers– principal quantum number, n– angular momentum quantum number, l– magnetic quantum number, ml

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Principal Quantum Number, n• Characterizes the energy of the electron in a particular

orbital– corresponds to Bohr’s energy level

• n can be any integer 1• The larger the value of n, the more energy the orbital

has• Energies are defined as being negative

– an electron would have E = 0 when it just escapes the atom

• The larger the value of n, the larger the orbital• As n gets larger, the amount of energy between orbitals

gets smaller

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Angular Momentum Quantum Number, l

• The angular momentum quantum number determines the shape of the orbital

• l can have integer values from 0 to (n – 1)• Each value of l is called by a particular letter that

designates the shape of the orbital

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Angular Momentum Quantum Number,

s orbitals are spherical

– p orbitals are like two balloons tied at the knots

– d orbitals are mainly like four balloons tied at the – Knot

– f orbitals are mainly like eight balloons tied at the – knot

Magnetic Quantum Number, ml

• The magnetic quantum number is an integer that specifies the orientation of the orbital– the direction in space the orbital is aligned relative

to the other orbitals• Values are integers from −l to +l

– including zero– gives the number of orbitals of a particular shape

• when l = 2, the values of ml are • −2, −1, 0, +1,+2• which means there are five orbitals with l = 2

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Describing an Orbital

• Each set of n, l, and ml describes one orbital• Orbitals with the same value of n are in the

same principal energy level– aka principal shell

• Orbitals with the same values of n and l are said to be in the same sublevel

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Energy Shells and Subshells

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Quantum Numbers

Spin Quantum Number, ms

This quantum number refers to the two possible orientations of the spin axis of an electron.

It may have a value of either +1/2 or -1/2.

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The Quantum Numbers

Example 11

Give the notation (using letter designations for 1) of

subshells for the following quantum numbers.

n. = 3, l = 1

n. = 4, l = 0

n. = 5,1 = 3

n. = 6,1 = 2

n. = 2,1 = 1

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The Quantum NumbersExample 12

Determine how many valid sets of quantum numbers

exist for a 3p orbital ; Give two examples.

Strategy:

Write the the number of possible values for each

quantum number.

Quantum number n 1 m ms

Possible values3 1 1, 0, -1 +1/2, -1/2

# of possible values 11 3 2

Possible sets of values for a 3p electron: (1)(1)(3)(2) = 6

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The Quantum Numbers

There are six sets of quantum numbers that describe

a 3p electron.

 

Examples:

 

n = 3 1=1 m1= 0 ms = +1/2

 

n = 3 1=1 m1 = -1 ms = +1/2

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The Quantum Numbers

Example 13Which of the following sets of quantum numbers (n, l, ml, ms) is not permissible?

a. 1 0 0 +½b. 4 0 0 -½c. 2 2 1 +½d. 3 1 0 -½

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The Quantum NumbersExample 14

Determine how many valid sets of quantum

numbers exist for 4d orbitals ;Give three examples.

 

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The Quantum Numbers

Example 15

Determine how many valid sets of quantum

numbers exist for 5f orbitals ;Give three

examples.

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The Shapes of Orbitalss Orbitls :1. All s orbitals are spherical 2. The probability of finding an s electron depends only

on distance from the nucleus, not on direction3. There is only one orientation of a sphere in space Therefore S orbital has ml = 0

4. The value of ψ2 for an s-orbital is greatest near the nucleus

5. The size of the s orbital increases as the number of shell increases i.e. 3s > 2s>1s

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S Orbitals

Probability Density FunctionThe probability density function represents the total probability of finding an electron at a particular point in space

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p Orbitals

1. p orbitlas are dumbbell –shaped.2. The electron distribution is concentrated in identical

lobes on either side of the nucleus and separated by a planar node cutting through the nucleus

3. Probability of finding a p electron near the nucleus is zero.

4. The two lobes of a p orbital have different phases of a wave (different algebraic signs)

5. The size of p orbitals increases in the order 4p>3p>2p

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d Orbitals 1. The third and higher shells contain each contain five

d orbitals2. d orbitals have two shapes 3. Four of the five d orbitals are cloverleaf –shaped and

have four lobes of maximum electron probability4. The fifth d orbital is similar in shape to a pz orbital

but has an additional donut –shaped region of electron probability centered in the xy plane

5. All five d orbitals have the same energy6. Alternating lobes of the d orbitals have different

phases

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d Orbitals

f orbitals

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