Chapter 7 The Quantum-Mechanical Model of the Atom

Chapter 7The Quantum-

Mechanical Model of the

Atom

2007, Prentice Hall

Chemistry: A Molecular Approach, 1st Ed.Nivaldo Tro

Roy KennedyMassachusetts Bay Community College

Wellesley Hills, MA

Tro, Chemistry: A Molecular Approach 2

The Behavior of the Very Small

• electrons are incredibly smalla single speck of dust has more electrons than the

number of people who have ever lived on earth• electron behavior determines much of the

behavior of atoms• directly observing electrons in the atom is

impossible, the electron is so small that observing it changes its behavior


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

electrons exist and behave in atoms• helps us understand and predict the properties of atoms

that are directly related to the behavior of the electronswhy some elements are metals while others are nonmetalswhy some elements gain 1 electron when forming an anion,

while others gain 2why some elements are very reactive while others are

practically inertand other Periodic patterns we see in the properties of the

elements


The Nature of Lightits Wave Nature

• light is a form of electromagnetic radiationcomposed of perpendicular oscillating waves, one for the

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

experiences a forcea magnetic field is a region where an magnetized particle experiences

a force

• all electromagnetic waves move through space at the same, constant speed3.00 x 108 m/s in a vacuum = the speed of light, c


Speed of Energy Transmission


Electromagnetic Radiation


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, () is a measure of the distance covered by the wavethe distance from one crest to the next

or the distance from one trough to the next, or the distance between alternate nodes


Wave Characteristics


Characterizing Waves• the frequency, () is the number of waves that

pass a point in a given period of timethe number of waves = number of cyclesunits are hertz, (Hz) or cycles/s = s-1

1 Hz = 1 s-1

• the total energy is proportional to the amplitude and frequency of the wavesthe larger the wave amplitude, the more force it hasthe more frequently the waves strike, the more total

force there is


The Relationship Between Wavelength and Frequency

• for waves traveling at the same speed, the shorter the wavelength, the more frequently they pass

• this means that the wavelength and frequency of electromagnetic waves are inversely proportionalsince the speed of light is constant, if we know

wavelength we can find the frequency, and visa versa

m

cs s

m1-


Example 7.1- Calculate the wavelength of red light with a frequency of 4.62 x 1014 s-1

the unit is correct, the wavelength is appropriate for red light

Check:

Solve:∙ = c, 1 nm = 10-9 m

Concept Plan:

Relationships:

= 4.62 x 1014 s-1

, (nm)Given:

Find:

s-1) (m) (nm)

c

m 10

nm 19

m 1049.6 1062.4

1000.3c 71-s14

-1sm8

nm 1049.6m 10

nm 1m 1049.6 29

7


Practice – Calculate the wavelength of a radio signal with a frequency of 100.7 MHz


m 98.2 10007.1

1000.3c1-s8

-1sm8

1-8-16

s 10007.1MHz 1

s 10MHz 00.71

Practice – Calculate the wavelength of a radio signal with a frequency of 100.7 MHz

the unit is correct, the wavelength is appropriate for radiowaves

Check:

Solve:∙ = c, 1 MHz = 106 s-1

Concept Plan:

Relationships:

= 100.7 MHz , (m)

Given:Find:

Hz) (s-1) (m)

c

MHz 1s 10 -16


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 while reflecting others, it appears colored the observed color is predominantly the colors reflected

15

Amplitude & Wavelength


Electromagnetic Spectrum


Continuous Spectrum


The Electromagnetic Spectrum• visible light comprises only a small fraction of

all the wavelengths of light – called the electromagnetic spectrum

• short wavelength (high frequency) light has high energyradiowave light has the lowest energygamma ray light has the highest energy

• high energy electromagnetic radiation can potentially damage biological moleculesionizing radiation


Thermal Imaging using Infrared Light


Using High Energy Radiationto Kill Cancer Cells


Interference

• the interaction between waves is called interference

• when waves interact so that they add to make a larger wave it is called constructive interferencewaves are in-phase

• when waves interact so they cancel each other it is called destructive interferencewaves are out-of-phase


Interference


Diffraction• when traveling waves encounter an obstacle or opening

in a barrier that is about the same size as the wavelength, they bend around it – this is called diffraction traveling particles do not diffract

• the diffraction of light through two slits separated by a distance comparable to the wavelength results in an interference pattern of the diffracted waves

• an interference pattern is a characteristic of all light waves


Diffraction


2-Slit Interference


The Photoelectric Effect• it was observed that many metals emit electrons when a

light shines on their surface this is called the Photoelectric Effect

• classic wave theory attributed this effect to the light energy being transferred to the electron

• according to this theory, if the wavelength of light is made shorter, or the light waves intensity made brighter, more electrons should be ejected remember: the energy of a wave is directly proportional to its

amplitude and its frequency if a dim light was used there would be a lag time before

electrons were emitted to give the electrons time to absorb enough energy


The Photoelectric Effect


The Photoelectric EffectThe Problem

• in experiments with the photoelectric effect, it was observed that there was a maximum wavelength for electrons to be emittedcalled the threshold frequencyregardless of the intensity

• it was also observed that high frequency light with a dim source caused electron emission without any lag time


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 was directly proportional to its frequencyinversely proportional to it wavelengththe proportionality constant is called Planck’s

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

chhE


Example 7.2- Calculate the number of photons in a laser pulse with wavelength 337 nm and total energy 3.83 mJ

Solve:E=hc/, 1 nm = 10-9 m, 1 mJ = 10-3 J, Epulse/Ephoton = # photons

Concept Plan:

Relationships:

= 337 nm, Epulse = 3.83 mJ

number of photonsGiven:

Find:

hcEphoton

nm 1m 10 9

J 108598.5 1037.3

1000.3 10626.6hcE 19

m7

-1sm8sJ34

photon

m 1037.3nm 1

m 10nm 1037.3 79

2

nm) (m) Ephotonnumberphotons

photon

pulse

EE

J 1083.3mJ 1

J 10mJ 83.3 33

photons 1049.6

J 108598.5J 1083.3photons ofnumber

15

3

3


Practice – What is the frequency of radiation required to supply 1.0 x 102 J of energy from

8.5 x 1027 photons?


1-7

sJ34J26

photon s 108.1 10626.6 10761.1

hE

What is the frequency of radiation required to supply 1.0 x 102 J of energy from 8.5 x 1027 photons?

Solve:E=h, Etotal = Ephoton∙# photons

Concept Plan:

Relationships:

Etotal = 1.0 x 102 J, number of photons = 8.5 x 1027

Given:Find:

h

Ephoton

(s-1)Ephotonnumberphotons

photons ofnumber E total

J 10761.1105.8

J 100.1E 2627

2

photon


Ejected Electrons• 1 photon at the threshold frequency has just

enough energy for an electron to escape the atombinding energy,

• for higher frequencies, 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 = h -


Spectra• when atoms or molecules absorb energy, that energy is

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

• when that light is passed through a prism, a pattern is seen that is unique to that type of atom or molecule – the pattern is called an emission spectrumnon-continuouscan be used to identify the material

flame tests

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

2

22

1

1-7

n1

n1m 10097.11


Emission Spectra


Exciting Gas Atoms to Emit Light with Electrical Energy

Hg He H


Examples of Spectra

Oxygen spectrum

Neon spectrum


Identifying Elements with Flame Tests

Na K Li Ba


Emission vs. Absorption Spectra

Spectra of Mercury


Bohr’s Model• Neils Bohr proposed that the electrons could only have

very specific amounts of energy fixed amounts = quantized

• the electrons traveled in orbits that were a fixed distance from the nucleusstationary states therefore the energy of the electron was proportional the

distance the orbital was from the nucleus• electrons emitted radiation when they “jumped” from

an orbit with higher energy down to an orbit with lower energy the distance between the orbits determined the energy of the

photon of light produced


Bohr Model of H Atoms


Wave Behavior of Electrons• de Broglie proposed that particles could have wave-like

character• because it is so small, the wave character of electrons is

significant• electron beams shot at slits show an interference

pattern the electron interferes with its own wave

• de Broglie predicted that the wavelength of a particle was inversely proportional to its momentum

)s(m(kg)s

mkg

m1-

2

2

velocitymass

h


however, electrons actually present an interference pattern, demonstrating the behave like waves

Electron Diffraction

if electrons behave like particles, there should only be two bright spots on the target


m 1074.2

1065.2kg 10.119

10626.6

mvh

10

m631-

mk342

2

s

s

g

Solve:

Example 7.3- Calculate the wavelength of an electron traveling at 2.65 x 106 m/s

=h/mv

Concept Plan:

Relationships:

v = 2.65 x 106 m/s, m = 9.11 x 10-31 kg (back leaf) m

Given:Find:

vmh

m, v (m)


Practice - Determine the wavelength of a neutron traveling at 1.00 x 102 m/s

(Massneutron = 1.675 x 10-24 g)


m 1096.3

1000.1kg 10675.1

10626.6

mvh

9

m227-

mk342

2

s

s

g

Solve:

Practice - Determine the wavelength of a neutron traveling at 1.00 x 102 m/s

=h/mv, 1 kg = 103 g

Concept Plan:

Relationships:

v = 1.00 x 102 m/s, m = 1.675 x 10-24 g m

Given:Find:

vmh

m (kg), v (m)m(g)

g 10kg 13

kg 10675.1

g 10kg 1g 10675.1

27

324


Complimentary Properties• when you try to observe the wave nature of the

electron, you cannot observe its particle nature – and visa versawave nature = interference patternparticle nature = position, which slit it is passing

through• the wave and particle nature of nature of the

electron are complimentary propertiesas you know more about one you know less about

the other


Uncertainty Principle• Heisenberg stated that the product of the uncertainties

in both the position and speed of a particle was inversely proportional to its massx = position, x = uncertainty in positionv = velocity, v = uncertainty in velocitym = mass

• the means that the more accurately you know the position of a small particle, like an electron, the less you know about its speedand visa-versa

m1

4hv

x


Uncertainty Principle Demonstration

any experiment designed to observe the electron results in detection of a single electron particle and no interference pattern


Determinacy vs. Indeterminacy• according to classical physics, particles move in a path

determined by the particle’s velocity, position, and forces acting on itdeterminacy = definite, predictable future

• because we cannot know both the position and velocity of an electron, we cannot predict the path it will follow indeterminacy = indefinite future, can only predict

probability• the best we can do is to describe the probability an

electron will be found in a particular region using statistical functions

51

Trajectory vs. Probability


Electron Energy• electron energy and position are complimentary

because KE = ½mv2

• for an electron with a given energy, the best we can do is describe a region in the atom of high probability of finding it – called an orbitala probability distribution map of a region where the

electron is likely to be founddistance vs. 2

• many of the properties of atoms are related to the energies of the electrons

Eψψ Η


Wave Function, • calculations show that the size, shape and

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

• these integers are called quantum numbersprincipal quantum number, nangular momentum quantum number, lmagnetic quantum number, ml


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

orbitalcorresponds 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

2

18- 1J 10-2.18Enn for an electron in H


Principal Energy Levels in Hydrogen

56

Electron Transitions• in order to transition to a higher energy state, the

electron must gain the correct amount of energy corresponding to the difference in energy between the final and initial states

• electrons in high energy states are unstable and tend to lose energy and transition to lower energy statesenergy released as a photon of light

• each line in the emission spectrum corresponds to the difference in energy between two energy states


Predicting the Spectrum of Hydrogen• the wavelengths of lines in the emission spectrum of

hydrogen can be predicted by calculating the difference in energy between any two states

• for an electron in energy state n, there are (n – 1) energy states it can transition to, therefore (n – 1) lines it can generate

• both the Bohr and Quantum Mechanical Models can predict these lines very accurately

2

182

18

electronhydrogen releasedphoton

1J 1018.21J 1018.2hch

EEEE

initialfinal

initialfinal

nn

58

Hydrogen Energy Transitions

m 1046.7

104466.2

1000.3 10626.6Ehc 6

J20-

sm8

sJ34

Solve:

Example 7.7- Calculate the wavelength of light emitted when the hydrogen electron transitions from n = 6 to n = 5

E=hc/En = -2.18 x 10-18 J (1/n2)

Concept Plan:

Relationships:

ni = 6, nf = 5

mGiven:

Find:

2H

1REn

ni, nf Eatom Ephoton

Ech

Eatom = -Ephoton

J 104466.261

51J 1018.2E 20

2218

atom

Ephoton = -(-2.6644 x 10-20 J) = 2.6644 x 10-20 J

the unit is correct, the wavelength is in the infrared, which is appropriate because less energy than 4→2 (in the visible)

Check:


Practice – Calculate the wavelength of light emitted when the hydrogen electron transitions from n = 2 to n = 1

m 1021.1

1064.1

1000.3 10626.6Ehc 7

J18-

sm8

sJ34

Solve:

Calculate the wavelength of light emitted when the hydrogen electron transitions from n = 2 to n = 1

E=hc/En = -2.18 x 10-18 J (1/n2)

Concept Plan:

Relationships:

ni = 2, nf = 1

mGiven:

Find:

2H

1REn

ni, nf Eatom Ephoton

Ech

Eatom = -Ephoton

J 1064.121

11J 1018.2E 18

2218

atom

Ephoton = -(-1.64 x 10-18 J) = 1.64 x 10-18 J

the unit is correct, the wavelength is in the UV, which is appropriate because more energy than 3→2 (in the visible)

Check:

62

Probability & Radial Distribution Functions

• 2 is the probability density the probability of finding an electron at a particular point in

space for s orbital maximum at the nucleus?decreases as you move away from the nucleus

• the Radial Distribution function represents the total probability at a certain distance from the nucleusmaximum at most probable radius

• nodes in the functions are where the probability drops to 0


Probability Density Function


Radial Distribution Function


The Shapes of Atomic Orbitals• the l quantum number primarily 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 orbitals orbitals are sphericalp orbitals are like two balloons tied at the knotsd orbitals are mainly like 4 balloons tied at the knotf orbitals are mainly like 8 balloons tied at the knot


l = 0, the s orbital• each principal energy

state has 1 s orbital• lowest energy orbital in a

principal energy state• spherical• number of nodes = (n – 1)

67

2s and 3s2s

n = 2,l = 0

3sn = 3,l = 0


l = 1, p orbitals

• each principal energy state above n = 1 has 3 p orbitalsml = -1, 0, +1

• each of the 3 orbitals point along a different axispx, py, pz

• 2nd lowest energy orbitals in a principal energy state• two-lobed• node at the nucleus, total of n nodes


p orbitals


l = 2, d orbitals• each principal energy state above n = 2 has 5 d orbitals

ml = -2, -1, 0, +1, +2• 4 of the 5 orbitals are aligned in a different plane

the fifth is aligned with the z axis, dz squareddxy, dyz, dxz, dx squared – y squared

• 3rd lowest energy orbitals in a principal energy state• mainly 4-lobed

one is two-lobed with a toroid• planar nodes

higher principal levels also have spherical nodes


d orbitals


l = 3, f orbitals

• each principal energy state above n = 3 has 7 d orbitalsml = -3, -2, -1, 0, +1, +2, +3

• 4th lowest energy orbitals in a principal energy state• mainly 8-lobed

some 2-lobed with a toroid

• planar nodeshigher principal levels also have spherical nodes


f orbitals


Why are Atoms Spherical?


Energy Shells and Subshells

Chapter 7 The Quantum-Mechanical Model of the Atom

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