CHAPTER 7 7.1 From Classical Physics to Quantum Theory Waves ...
7.1 From Classical Physics to Quantum Theory CHAPTER 7 Quantum Theory and the Electronic Structure of Atoms
• Classical Physics viewed energy as continuous; i.e., any quantity of energy could be released. • Max Planck found that atoms and molecules (under some conditions) emit energy only in discrete quantities called quanta, founding quantum theory
Waves • Wave – a vibrating disturbance by which energy is transmitted • Familiar examples – water waves – radio waves – microwaves
•Wavelength, λ – distance between identical points on successive waves Unit = length (m) Top wave has 3λ of bottom. •Frequency, ν – # waves that pass through a point per s Unit = cycles/s, Hz, s -1 Bottom wave has 3ν of top. •Amplitude – distance from midpoint of wave to peak or trough Top and bottom waves have same amplitude.
Speed of Waves
Electromagnetic Radiation
• Speed (u) = λν • Units of λν are length/time, usually m/s • Speed of wave depends on type and nature of medium in which it travels
• Energy is transmitted in the form of electromagnetic waves. • Can be emitted or absorbed by atoms. • Electric and magnetic field components w/ same λ, ν and u, but traveling in mutually perpendicular planes • Speed of light in vacuum (c) = 3.0 x 108 m/s
Magnetic field component ⊥ electric field component
Electromagnetic radiation
c = λν speed of light frequency wavelength
Quantum Theory • Problem –Heated solids emit electromagnetic radiation. –Amount of radiant energy is related to its wavelength. –Classical physics cannot explain.
• Planck’s solution = quantum theory
Planck’s Quantum Theory • Planck’s assumption – atoms and molecules emit or absorb energy only in discrete quantities. • “Bundles” of energy are quanta –smallest quantity of energy that can be emitted or absorbed
What is the frequency of green light with wavelength 500 nm?
E = hν Energy of a photon
Frequency of light
c = λν ν = c/λ ν = 3.00 x 108 m•s-1 /500 x 10-9 m ν = 6.00 x 10 14 s -1 What is the energy of a photon of this light?
Planck’s Constant = 6.63 x 10-34 J. s
E = hν = (6.63 x 10-34 J.s)(6.00 x 10 14 s -1) E = 3.80 x 10-19 J
7.2 Photoelectric Effect • Light strikes the metal, which ejects electrons • If light were continuous waves, sufficiently intense light of any frequency would eject electrons.
Einstein’s Explanation • Using Planck’s quantum theory – Light is made up of particles called photons – Each photon has energy = hν – Electrons are held in metal with certain binding energy, BE – Photon must have energy > BE – Photon energy above BE shows up as kinetic energy (KE) of photoelectron
• Emission of electrons from certain metals exposed to light • Number, not energy, of photoelectrons depends on intensity of light • Light must be above a certain threshold frequency to eject an electron
Photoelectric Effect
hν = KE + BE Energy of incident photon
Binding energy of emitted electron
Kinetic energy of emitted electron
Wave-Particle Duality • Light can behave as 1. Wave – e.g., diffraction by grating 2. Particle – e.g., photoelectric effect
• Depending on experiment, light can behave one way or the other • Particles of matter also have dual nature
or diffraction grating
Characteristic emission spectra are used to identify elements. Spectra can be in infrared, visible, ultraviolet, etc. regions.
Bohr’s explanation of H line spectrum • Electrons orbit nucleus like a planet around the sun, held electrostatically • Hydrogen atom consists of 1 electron orbiting 1 proton • Electron can only be located in certain stable orbits • Moving from one orbit to another causes atom to absorb or emit a photon of energy
7.3 The Bohr Hydrogen Atom •Atomic emission spectra – continuous or line spectra of radiation emitted by excited substances •Familiar “red hot” metal is continuous emission (all wavelengths) in the visible •Energized gas-phase atoms show line spectra –emission at specific wavelengths –Why?
( n1 - n1 )
∆E = hν = R H
2 i
2
f
Rydberg Constant = Final energy 2.18 x 10-18 J Initial energy level level • Energy of transition (∆E) = energy of photon (hν) • E levels farther from nucleus are closer together • E = -RH/n2, quantum number n = 1, 2, 3...
•Equation applies only to hydrogen atom •E absorption has same value, opposite sign
7.4 Dual Nature of Electron • Louis de Broglie proposed that electron acts like a standing wave in its orbit • [Demo: standing wave in rubber tubing] • Integral number of wavelengths must fit exactly into its orbit • Electron’s wave – particle duality
λ= wavelength
Nonallowed orbit: circumference of orbit ? integral # wavelengths
de Broglie example
h mv
mass
Allowed orbit: circumference of orbit = integral # wavelengths
•
velocity
•h = Planck’s Constant = 6.63 x 10-34 J.s •Remember: J = kg•m2•s-2 •Wave properties only observable for submicroscopic objects!
What is the de Broglie wavelength of an electron moving at a velocity of 1.00 m/s? λ = h/mv = (6.62 x 10-34 J•s)/(9.11 x 10-31 kg)(1.0 m/s) = 7.27 x 10-4 m • What is the de Broglie wavelength of a 1.0 kg ball moving at a velocity of 1.0 m/s? λ = h/mv = (6.62 x 10-34 J•s)/(1.00 kg)(1.00 m/s) = 6.62 x 10-34 m – Is this an observable wavelength?
7.5 Quantum Mechanics • Problems with Bohr model
Heisenberg Uncertainty Principle
– Works only for hydrogen-like (1 e-) atoms – Can’t explain extra lines in H spectrum in magnetic field – One can’t specify location of a wavelike e-
• It is impossible to know simultaneously both the momentum and the position of a particle with certainty. • Electrons do not move around the nucleus in well defined orbits
Heisenberg Uncertainty Principle
Schrödinger equation (1926)
∆x∆p = h 4π uncertainty in position
uncertainty in Planck’s constant momentum (mv)
Degree of uncertainty becomes significant for very small objects such as an electron.
Atomic orbitals • Wave function of an electron in an atom • Plot of Ψ 2 describes the shape of the probability distribution of an electron • Each atomic orbital has a characteristic energy • Assume that multi-electron atoms have orbitals that resemble 1-e- atom
• Relates a wave function, Ψ, to a mathematical operator and an energy • Ψ 2 describes the probability of finding an electron in a certain region of space (i.e., electron density) • Launched quantum mechanics • Can be solved exactly only for hydrogen-like (1 e-) atoms
7.6 Quantum Numbers • Derived from the Schrödinger Equation • Set of 4 [n, l, ml, ms ,] describes an electron in an orbital • First 3 define the distribution of electron density (atomic orbital) • 4th defines the spin of an electron
Angular Momentum Quantum Number, l
Principal Quantum Number, n • n = 1, 2, 3, 4...
• defines energy of e• related to average distance of efrom nucleus • larger n ⇒ larger distance ⇒ higher energy • e-s with same n are in same “shell”
Magnetic Quantum Number, ml • ml = -l, ….0…..+l • describes orientation in space of orbital. • e-s with same n, l and ml are in same orbital
Experimental Observation of ms -
• Deflection of atoms with 1 unpaired e in 2 directions in inhomogeneous magnetic field
• • • •
l = 0, 1, ….(n-1) describes shape of orbital e-s with same n and l are in same “subshell” Conventional names of orbitals
0
1
2
3
4
5
Name s
p
d
f
g
h
l
Historically named for appearance of spectroscopic lines from these orbitals: sharp, principal, diffuse, fundamental
Electron Spin Quantum Number, ms • ms = -1/2, +1/2 • e-s spin “clockwise” or “counterclockwise” • Spin makes them behave as magnets
7.7 Shapes and Orientations of Atomic Orbitals • Described by l (shape) and ml (orientation) • Illustrated as – electron density diagram – boundary surface diagram that encloses 90% of total e- density in an orbital
Electron density diagram
Boundary surface diagram Boundary surface diagrams
n=1 l=0 ml = 0
The Three 2p Orbitals
n=2 l=1 ml = -1, 0, +1 •No simple relationship of specific orbital to specific m l
n=2 l=0 ml = 0
n=3 l=0 ml = 0
The Five 3d Orbitals
n=3 l=2 ml = -2, -1, 0, +1, +2 •No simple relationship of specific orbital to specific ml •Despite different shapes, all are “degenerate” (same energy)
Shielding in multi-electron atoms
Shielding in multi-electron atoms
This 1s e - spends very little time between the other 1s e - and nucleus.
This 1s e - spends most of its time between this 2p e - and nucleus.
Ineffective shielding: Each 1s e - feels nearly full nuclear charge
Effective shielding: 1s e - feels nearly full nuclear charge; 2p e - feels diminished nuclear charge
Mnemonic for order of orbitals from lowest to highest energy
Pauli Exclusion Principle
7.8 Electron Configuration • How are electrons distributed among the atomic orbitals? • Electron configurations for “ground state” atoms – Atoms in their lowest energy state – Several possible “excited state” configurations
•
No two electrons in an atom can have the same four quantum numbers –
•
(One can’t have two identical solutions to the Schrödinger equation.)
Implications 1. Each orbital can hold no more than two electrons 2. e-s must spin in opposite directions (i.e., ms = +1/2, -1/2)
Element
Orbital Diagram Electron Config.
H He Li Be B C N O F Ne
Hund’s Rule
Hund’s Rule • “Maximum multiplicity” • Most stable arrangement of e- in subshells has greatest number of parallel spins • Least interelectronic repulsion • Place one e- in each subshell before adding second to any subshell
•Placing one e- in each subshell keeps negative e- farthest apart •2px1 2p y1 2pz1 is spherically symmetric
Hund’s Rule
Hund’s Rule
Carbon atom has 6 electrons, 4 with n = 2 Best arrangement?
Carbon atom has 6 electrons, 4 with n = 2 Paired arrangement?
2s
2px
2py
2pz
2s
2px
2py
2pz
Hund’s Rule
Hund’s Rule
Carbon atom has 6 electrons, 4 with n = 2 Unpaired, opposite arrangement?
Carbon atom has 6 electrons, 4 with n = 2 Unpaired, parallel arrangement!
2s
2px
2py
2pz
2s
2px
2py
2pz
Choice of px, p y or pz does not matter
Magnetism and unpaired e• Diamagnetic substance – Weakly repelled by magnetic field – All electrons are paired
• Paramagnetic substance – Attracted into magnetic field – Unpaired electrons are in random orientations
• Ferromagnetic substance – Strongly attracted into magnetic field – Unpaired electrons are parallel
Element
Orbital Diagram Electron Config.
H He Li Be B C N O F Ne
7.9 Aufbau Principle • Building-up Principle • Fill from the bottom up! • Add one e- at a time into lowest energy orbital • Structure of periodic table results from Aufbau • Use structure of periodic table to easily derive electron configurations Structure of the Periodic Table results from “Aufbau” filling of atomic orbitals with electrons.
Writing Electron Configurations • Find the noble gas that comes before the element • Noble gas core in square brackets – e.g., [He]
• Use the order of filling of atomic orbitals and Hund’s rule for the remaining electrons – e.g., ground state Sc is
[Ar] 4s 2 3d1
Transition Metals (d-block) • Transition metals have partially filled d subshells or readily give rise to cations with partially filled d subshells. – Scandium to copper – Yttrium to silver – Lanthanum to Gold
• ns fills before (n-1)d levels
Anomalous e- configurations • When one e- short of a half-full or full d subshell, one s e- from a pair may be “promoted” to fill or half-fill it • Why? – Typical explanation • “half -filled or filled subshells have special stability”
– Alternative (better) explanation • lose interelectronic repulsion of ns 2 e-
• maximizes number of parallel spins • resulting ns 1 (n-1)d5 or ns 1 d10 configuration has e-s well distributed in space, with spherical symmetry
f-Block Transition Metals • Lanthanides (rare earths) – Partially filled 4f subshell in atom or ion – Cerium to lutetium
• Actinides – Partially filled 5f subshell in atom or ion – Thorium to lawrencium – All radioactive, many synthetic
Notable Exceptions • Chromium [Ar] 4s 2 3d4 [Ar] 4s 1 3d5
WRONG! CORRECT
• Silver [Kr]5s 2 4d9 [Kr]5s 1 4d10
WRONG! CORRECT
• Challenges – Why is carbon [He] 2s 2 2p2? – Rationalize Ni, Pd, Pt!
• Classical Physics viewed energy as continuous; i.e., any quantity of energy could be released. • Max Planck found that atoms and molecules (under some conditions) emit energy only in discrete quantities called quanta, founding quantum theory
Waves • Wave – a vibrating disturbance by which energy is transmitted • Familiar examples – water waves – radio waves – microwaves
•Wavelength, λ – distance between identical points on successive waves Unit = length (m) Top wave has 3λ of bottom. •Frequency, ν – # waves that pass through a point per s Unit = cycles/s, Hz, s -1 Bottom wave has 3ν of top. •Amplitude – distance from midpoint of wave to peak or trough Top and bottom waves have same amplitude.
Speed of Waves
Electromagnetic Radiation
• Speed (u) = λν • Units of λν are length/time, usually m/s • Speed of wave depends on type and nature of medium in which it travels
• Energy is transmitted in the form of electromagnetic waves. • Can be emitted or absorbed by atoms. • Electric and magnetic field components w/ same λ, ν and u, but traveling in mutually perpendicular planes • Speed of light in vacuum (c) = 3.0 x 108 m/s
Magnetic field component ⊥ electric field component
Electromagnetic radiation
c = λν speed of light frequency wavelength
Quantum Theory • Problem –Heated solids emit electromagnetic radiation. –Amount of radiant energy is related to its wavelength. –Classical physics cannot explain.
• Planck’s solution = quantum theory
Planck’s Quantum Theory • Planck’s assumption – atoms and molecules emit or absorb energy only in discrete quantities. • “Bundles” of energy are quanta –smallest quantity of energy that can be emitted or absorbed
What is the frequency of green light with wavelength 500 nm?
E = hν Energy of a photon
Frequency of light
c = λν ν = c/λ ν = 3.00 x 108 m•s-1 /500 x 10-9 m ν = 6.00 x 10 14 s -1 What is the energy of a photon of this light?
Planck’s Constant = 6.63 x 10-34 J. s
E = hν = (6.63 x 10-34 J.s)(6.00 x 10 14 s -1) E = 3.80 x 10-19 J
7.2 Photoelectric Effect • Light strikes the metal, which ejects electrons • If light were continuous waves, sufficiently intense light of any frequency would eject electrons.
Einstein’s Explanation • Using Planck’s quantum theory – Light is made up of particles called photons – Each photon has energy = hν – Electrons are held in metal with certain binding energy, BE – Photon must have energy > BE – Photon energy above BE shows up as kinetic energy (KE) of photoelectron
• Emission of electrons from certain metals exposed to light • Number, not energy, of photoelectrons depends on intensity of light • Light must be above a certain threshold frequency to eject an electron
Photoelectric Effect
hν = KE + BE Energy of incident photon
Binding energy of emitted electron
Kinetic energy of emitted electron
Wave-Particle Duality • Light can behave as 1. Wave – e.g., diffraction by grating 2. Particle – e.g., photoelectric effect
• Depending on experiment, light can behave one way or the other • Particles of matter also have dual nature
or diffraction grating
Characteristic emission spectra are used to identify elements. Spectra can be in infrared, visible, ultraviolet, etc. regions.
Bohr’s explanation of H line spectrum • Electrons orbit nucleus like a planet around the sun, held electrostatically • Hydrogen atom consists of 1 electron orbiting 1 proton • Electron can only be located in certain stable orbits • Moving from one orbit to another causes atom to absorb or emit a photon of energy
7.3 The Bohr Hydrogen Atom •Atomic emission spectra – continuous or line spectra of radiation emitted by excited substances •Familiar “red hot” metal is continuous emission (all wavelengths) in the visible •Energized gas-phase atoms show line spectra –emission at specific wavelengths –Why?
( n1 - n1 )
∆E = hν = R H
2 i
2
f
Rydberg Constant = Final energy 2.18 x 10-18 J Initial energy level level • Energy of transition (∆E) = energy of photon (hν) • E levels farther from nucleus are closer together • E = -RH/n2, quantum number n = 1, 2, 3...
•Equation applies only to hydrogen atom •E absorption has same value, opposite sign
7.4 Dual Nature of Electron • Louis de Broglie proposed that electron acts like a standing wave in its orbit • [Demo: standing wave in rubber tubing] • Integral number of wavelengths must fit exactly into its orbit • Electron’s wave – particle duality
λ= wavelength
Nonallowed orbit: circumference of orbit ? integral # wavelengths
de Broglie example
h mv
mass
Allowed orbit: circumference of orbit = integral # wavelengths
•
velocity
•h = Planck’s Constant = 6.63 x 10-34 J.s •Remember: J = kg•m2•s-2 •Wave properties only observable for submicroscopic objects!
What is the de Broglie wavelength of an electron moving at a velocity of 1.00 m/s? λ = h/mv = (6.62 x 10-34 J•s)/(9.11 x 10-31 kg)(1.0 m/s) = 7.27 x 10-4 m • What is the de Broglie wavelength of a 1.0 kg ball moving at a velocity of 1.0 m/s? λ = h/mv = (6.62 x 10-34 J•s)/(1.00 kg)(1.00 m/s) = 6.62 x 10-34 m – Is this an observable wavelength?
7.5 Quantum Mechanics • Problems with Bohr model
Heisenberg Uncertainty Principle
– Works only for hydrogen-like (1 e-) atoms – Can’t explain extra lines in H spectrum in magnetic field – One can’t specify location of a wavelike e-
• It is impossible to know simultaneously both the momentum and the position of a particle with certainty. • Electrons do not move around the nucleus in well defined orbits
Heisenberg Uncertainty Principle
Schrödinger equation (1926)
∆x∆p = h 4π uncertainty in position
uncertainty in Planck’s constant momentum (mv)
Degree of uncertainty becomes significant for very small objects such as an electron.
Atomic orbitals • Wave function of an electron in an atom • Plot of Ψ 2 describes the shape of the probability distribution of an electron • Each atomic orbital has a characteristic energy • Assume that multi-electron atoms have orbitals that resemble 1-e- atom
• Relates a wave function, Ψ, to a mathematical operator and an energy • Ψ 2 describes the probability of finding an electron in a certain region of space (i.e., electron density) • Launched quantum mechanics • Can be solved exactly only for hydrogen-like (1 e-) atoms
7.6 Quantum Numbers • Derived from the Schrödinger Equation • Set of 4 [n, l, ml, ms ,] describes an electron in an orbital • First 3 define the distribution of electron density (atomic orbital) • 4th defines the spin of an electron
Angular Momentum Quantum Number, l
Principal Quantum Number, n • n = 1, 2, 3, 4...
• defines energy of e• related to average distance of efrom nucleus • larger n ⇒ larger distance ⇒ higher energy • e-s with same n are in same “shell”
Magnetic Quantum Number, ml • ml = -l, ….0…..+l • describes orientation in space of orbital. • e-s with same n, l and ml are in same orbital
Experimental Observation of ms -
• Deflection of atoms with 1 unpaired e in 2 directions in inhomogeneous magnetic field
• • • •
l = 0, 1, ….(n-1) describes shape of orbital e-s with same n and l are in same “subshell” Conventional names of orbitals
0
1
2
3
4
5
Name s
p
d
f
g
h
l
Historically named for appearance of spectroscopic lines from these orbitals: sharp, principal, diffuse, fundamental
Electron Spin Quantum Number, ms • ms = -1/2, +1/2 • e-s spin “clockwise” or “counterclockwise” • Spin makes them behave as magnets
7.7 Shapes and Orientations of Atomic Orbitals • Described by l (shape) and ml (orientation) • Illustrated as – electron density diagram – boundary surface diagram that encloses 90% of total e- density in an orbital
Electron density diagram
Boundary surface diagram Boundary surface diagrams
n=1 l=0 ml = 0
The Three 2p Orbitals
n=2 l=1 ml = -1, 0, +1 •No simple relationship of specific orbital to specific m l
n=2 l=0 ml = 0
n=3 l=0 ml = 0
The Five 3d Orbitals
n=3 l=2 ml = -2, -1, 0, +1, +2 •No simple relationship of specific orbital to specific ml •Despite different shapes, all are “degenerate” (same energy)
Shielding in multi-electron atoms
Shielding in multi-electron atoms
This 1s e - spends very little time between the other 1s e - and nucleus.
This 1s e - spends most of its time between this 2p e - and nucleus.
Ineffective shielding: Each 1s e - feels nearly full nuclear charge
Effective shielding: 1s e - feels nearly full nuclear charge; 2p e - feels diminished nuclear charge
Mnemonic for order of orbitals from lowest to highest energy
Pauli Exclusion Principle
7.8 Electron Configuration • How are electrons distributed among the atomic orbitals? • Electron configurations for “ground state” atoms – Atoms in their lowest energy state – Several possible “excited state” configurations
•
No two electrons in an atom can have the same four quantum numbers –
•
(One can’t have two identical solutions to the Schrödinger equation.)
Implications 1. Each orbital can hold no more than two electrons 2. e-s must spin in opposite directions (i.e., ms = +1/2, -1/2)
Element
Orbital Diagram Electron Config.
H He Li Be B C N O F Ne
Hund’s Rule
Hund’s Rule • “Maximum multiplicity” • Most stable arrangement of e- in subshells has greatest number of parallel spins • Least interelectronic repulsion • Place one e- in each subshell before adding second to any subshell
•Placing one e- in each subshell keeps negative e- farthest apart •2px1 2p y1 2pz1 is spherically symmetric
Hund’s Rule
Hund’s Rule
Carbon atom has 6 electrons, 4 with n = 2 Best arrangement?
Carbon atom has 6 electrons, 4 with n = 2 Paired arrangement?
2s
2px
2py
2pz
2s
2px
2py
2pz
Hund’s Rule
Hund’s Rule
Carbon atom has 6 electrons, 4 with n = 2 Unpaired, opposite arrangement?
Carbon atom has 6 electrons, 4 with n = 2 Unpaired, parallel arrangement!
2s
2px
2py
2pz
2s
2px
2py
2pz
Choice of px, p y or pz does not matter
Magnetism and unpaired e• Diamagnetic substance – Weakly repelled by magnetic field – All electrons are paired
• Paramagnetic substance – Attracted into magnetic field – Unpaired electrons are in random orientations
• Ferromagnetic substance – Strongly attracted into magnetic field – Unpaired electrons are parallel
Element
Orbital Diagram Electron Config.
H He Li Be B C N O F Ne
7.9 Aufbau Principle • Building-up Principle • Fill from the bottom up! • Add one e- at a time into lowest energy orbital • Structure of periodic table results from Aufbau • Use structure of periodic table to easily derive electron configurations Structure of the Periodic Table results from “Aufbau” filling of atomic orbitals with electrons.
Writing Electron Configurations • Find the noble gas that comes before the element • Noble gas core in square brackets – e.g., [He]
• Use the order of filling of atomic orbitals and Hund’s rule for the remaining electrons – e.g., ground state Sc is
[Ar] 4s 2 3d1
Transition Metals (d-block) • Transition metals have partially filled d subshells or readily give rise to cations with partially filled d subshells. – Scandium to copper – Yttrium to silver – Lanthanum to Gold
• ns fills before (n-1)d levels
Anomalous e- configurations • When one e- short of a half-full or full d subshell, one s e- from a pair may be “promoted” to fill or half-fill it • Why? – Typical explanation • “half -filled or filled subshells have special stability”
– Alternative (better) explanation • lose interelectronic repulsion of ns 2 e-
• maximizes number of parallel spins • resulting ns 1 (n-1)d5 or ns 1 d10 configuration has e-s well distributed in space, with spherical symmetry
f-Block Transition Metals • Lanthanides (rare earths) – Partially filled 4f subshell in atom or ion – Cerium to lutetium
• Actinides – Partially filled 5f subshell in atom or ion – Thorium to lawrencium – All radioactive, many synthetic
Notable Exceptions • Chromium [Ar] 4s 2 3d4 [Ar] 4s 1 3d5
WRONG! CORRECT
• Silver [Kr]5s 2 4d9 [Kr]5s 1 4d10
WRONG! CORRECT
• Challenges – Why is carbon [He] 2s 2 2p2? – Rationalize Ni, Pd, Pt!
