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Chapter 3
Quantum Theory and the Electronic Structure of Atoms
Quantum Theory and the Electronic Structure of Atoms
3
3.1 Energy and Energy Changes
Forms of Energy
Units of Energy
3.2 The Nature of Light
Properties of Waves
The Electromagnetic Spectrum
The Double-Slit Experiment
3.3 Quantum Theory
Quantization of Energy
Photons and the Photoelectric Effect
3.4 Bohr’s Theory of the Hydrogen Atom
Atomic Line Spectra
The Line Spectrum of Hydrogen
3.5 Wave Properties of Matter
The de Broglie Hypothesis
Diffraction of Electrons
3.6 Quantum Mechanics
The Uncertainty Principle
The Schr dinger Equation
The Quantum Mechanical Description of the Hydrogen Atom
Quantum Theory and the Electronic Structure of Atoms
3
3.7 Quantum Numbers
Principal Quantum Number (n)
Angular Momentum Quantum Number (l)
Magnetic Quantum Number (ml)
Electron Spin Quantum Number (ms)
3.8 Atomic Orbitals
s Orbitals
p Orbitals
d Orbitals and other High-Energy Orbitals
Energies of Orbitals
3.9 Electron Configuration
Energies of Atomic Orbitals in Many-Electron Systems
The Pauli Exclusion Principle
Aufbau Principle
Hund’s Rule
General Rules for Writing Electron Configurations
3.10 Electron Configurations and the Periodic Table
Energy and Energy Changes
Energy is the capacity to do work or transfer heat.
All forms of energy are either kinetic or potential.
Kinetic energy (Ek) is the energy of motion.
m is the mass of the object
u is its velocity
One form of kinetic energy of particular interest to chemists is thermal energy, which is the energy associated with the random motion of atoms and molecules.
3.1
Forms of Energy
Potential energy is the energy possessed by an object by virtue of its position.
There are two forms of potential energy of great interest to chemists:
Chemical energy is energy stored within the structural units of chemical substances.
Electrostatic energy is potential energy that results from the interaction of charged particles.
Q1 and Q2 represent two charges separated by the distance, d.
Energy and Energy Changes
Kinetic and potential energy are interconvertible – one can be converted to the other.
Although energy can assume many forms, the total energy of the universe is constant.
Energy can neither be created nor destroyed.
When energy of one form disappears, the same amount of energy reappears in another form or forms.
This is known as the law of conservation of energy.
Units of Energy
The SI unit of energy is the joule (J), named for the English physicist James Joule.
It is the amount of energy possessed by a 2-kg mass moving at a speed of 1 m/s.
Ek = mu2 = (2 kg)(1 m/s)2 = 1 kg m2/s2 = 1 J
The joule can also be defined as the amount of energy exerted when a force of 1 newton (N) is applied over a distance of 1 meter.
1 J = 1 N · m
Because the magnitude of a joule is so small, we often express large amounts of energy using the unit kilojoule (kJ).
1 kJ = 1000 J
Classical Physics
Principal assumptions
The physical state of any system can be described by
a set of quantities called dynamical variables that take on well defined values at any instant in time.
2. The future state of any system is completely determined if the initial state of the system is known,
3. The energy of a system can be varied in a continuous manner over the allowed range.
Properties of Waves
All forms of electromagnetic radiation travel in waves.
Waves are characterized by:
Wavelength (λ; lambda) – the distance between identical points on successive waves
Frequency (ν; nu) – the number of waves that pass through a particular point in 1 second.
Amplitude – the vertical distance from the midline of a wave to the top of the peak or the bottom of the trough.
Wavelength (l) is the distance between identical points on successive waves.
Amplitude is the vertical distance from the midline of a wave to the peak or trough.
Properties of Waves
Frequency (n) is the number of waves that pass through a particular point in 1 second (Hz = 1 s-1).
wavelength (m) frequency (s-1) = velocity (m/s)
= v
= wavelength = frequency v = velocity
Note: speed of light c is a constant
The longer the wavelength,
the lower the frequency.
Amplitude (intensity) of a wave.
light
Wave Function
k: angular wave number
T: period
: angular frequency
sinusoidal wave
superposition
constructive interference
destructive interference
Standing Wave 驻波
y=(2A sin kx) cos t
Two Dimensional Wave
Different behaviors of waves and particles.
The Nature of Light
The speed of light (c) through a vacuum is a constant:
c = 2.99792458×108 m/s
Normally rounded to, c = 3.00×108 m/s.
Speed of light, frequency and wavelength are related:
λ is expressed in meters
v is expressed in reciprocal seconds (s 1)
s-1 is also known as hertz (Hz)
The Nature of Light
Visible light is only a small component of the continuum of radiant energy known as the electromagnetic spectrum.
3.2
The Electromagnetic Spectrum
An electromagnetic wave has both an electric field component and a magnetic component.
The electric and magnetic components have the same frequency and wavelength.
When light passes through two closely spaced slits, an interference pattern is produced.
The Double-Slit Experiment
Constructive interference is a result of adding waves that are in phase.
Destructive interference is a result of adding waves that are out of phase.
This type of interference is typical of waves and demonstrates the wave nature of light.
When a solid is heated, it emits electromagnetic radiation, known as blackbody radiation, over a wide range of wavelengths.
The amount of energy given off at a certain temperature depends on the wavelength.
Classical physics failed to completely explain the phenomenon.
Assumed that radiant energy was continuous; that is, could be emitted or absorbed in any amount.
Max Planck suggested that radiant energy is only emitted or absorbed in discrete quantities, like small packages or bundles.
A quantum of energy is the smallest quantity of energy that can be emitted (or absorbed).
Quantum Theory
3.3
The energy E of a single quantum of energy is
h is called Planck’s constant: 6.63×10 34 J s
The idea that energy is quantized rather than continuous is like walking up a staircase or playing the piano
You cannot step or play anywhere (continuous), you can only step on a stair or play on a key (quantized).
Quantum Theory
Albert Einstein used Planck’s theory to explain the photoelectric effect.
Electrons are ejected from the surface of a metal exposed to light of a certain minimum frequency, called the threshold frequency.
The number of electrons ejected is proportional to the intensity.
Below the threshold frequency no electrons were ejected, no matter how bright (or intense) the light.
Photons and the Photoelectric Effect
Einstein proposed that the beam of light is really a stream of particles.
These particles of light are now called photons.
Each photon (of the incident light) must posses the energy given by the equation:
Photons and the Photoelectric Effect
Shining light onto a metal surface can be thought of as shooting a beam of particles – photons – at the metal atoms.
If the ν of the photons equals the energy the binds the electrons in the metal, then the light will have enough energy to knock the electrons loose.
If we use light of a higher ν, then not only will the electrons be knocked loose, but they will also acquire some kinetic energy.
Photons and the Photoelectric Effect
This is summarized by the equation
KE is the kinetic energy of the ejected electron
W is the binding energy of the electron
Photons and the Photoelectric Effect
Behavior of Light
As wave
diffraction
As particle
Photoelectric effect
Duality 二象性
Sunlight is composed of various color components that can be recombined to produce white light.
The emission spectrum of a substance can be seen by energizing a sample of material with some form of energy.
The “red hot” or “white hot” glow of an iron bar removed from a fire is the visible portion of its emission spectrum.
The emission spectrum of both sunlight and a heated solid are continuous; all wavelengths of visible light are present.
Bohr’s Theory of the Hydrogen Atom
3.4
Line spectra are the emission of light only at specific wavelengths.
Atomic Line Spectra
The main components of a typical spectrophotometer.
Monochromator (wavelength selector) disperses incoming radiation into continuum of component wavelengths that are scanned or individually selected.
Sample in compartment absorbs characteristic amount of each incoming wavelength.
Computer converts signal into displayed data.
Source produces radiation in region of interest. Must be stable and reproducible. In most cases, the source emits many wavelengths.
Lenses/slits/collimaters narrow and align beam.
Detector converts transmitted radiation into amplified electrical signal.
Emission and absorption spectra of sodium atoms.
Flame tests.
strontium 38Sr
copper 29Cu
Bohr’s Theory of the Hydrogen Atom
Every element has its own unique emission spectrum.
Bohr’s Theory of the Hydrogen Atom
The Rydberg equation can be used to calculate the wavelengths of the four visible lines in the emission spectrum of hydrogen.
R∞ is the Rydberg constant (1.09737317 x 107 m 1)
λ the wavelength of a line in the spectrum
n1 and n2 are positive integers where n2 > n1.
The Line Spectrum of Hydrogen
Neils Bohr attributed the emission of radiation by an energized hydrogen atom to the electron dropping from a higher-energy orbit to a lower one.
As the electron dropped, it gave up a quantum of energy in the form of light.
Bohr showed that the energies of the electron in a hydrogen atom are given by the equation:
En is the energy
n is a positive integer
As an electron gets closer to the nucleus, n decreases.
En becomes larger in absolute value (but more negative) as n gets smaller.
En is most negative when n = 1.
Called the ground state, the lowest energy state of the atom
For hydrogen, this is the most stable state
The stability of the electron decreases as n increases.
Each energy state in which n > 1 is called an excited state.
The Line Spectrum of Hydrogen
The Line Spectrum of Hydrogen
Bohr’s theory explains the line spectrum of the hydrogen atom.
Radiant energy absorbed by the atom causes the electron to move from the ground state (n = 1) to an excited state (n > 1).
Conversely, radiant energy is emitted when the electron moves from a higher-energy state to a lower-energy excited state or the ground state.
The quantized movement of the electron from one
energy state to another is analogous to a ball moving
and down steps.
nf is the final state
ni is the initial state
Quantum staircase.
Bohr’s Theory of the Hydrogen Atom
Suppose an electron is initially in an excited state, ni.
During emission, the electron drops to a lower energy state, nf.
The energy difference between the initial and final states is
Bohr’s Theory of the Hydrogen Atom
To calculate wavelength, substitute c/λ for ν and rearrange:
Bohr’s Theory of the Hydrogen Atom
nf is the final state
ni is the initial state
The Bohr explanation of the three series of spectral lines.
Calculate the wavelength (in nm) of the photon emitted when an electron transitions from the n = 4 state to the n = 2 state in a hydrogen atom.
Worked Example 3.5
Solution
Setup
h = 6.63×10-34 J s and c = 3.00×108 m/s
2.18×10-18 J
(6.63×10-34 J s )(3.00×108 m/s)
1
λ
=
1
22
1
42
-
= 2.055×106 m-1
λ = 4.87×10-7 m
1 nm
1×10-9 m
= 487 nm
×
Think About It Look again at the line spectrum of hydrogen and make sure your result matches one of them. Note that for an emission, ni, is always greater than nf, and the equation gives a positive result.
Wave Properties of Matter
Louis de Broglie reasoned that if light can behave like a stream of particles (photons), then electrons could exhibit wavelike properties.
According to de Broglie, electrons behave like standing waves.
Only certain wavelengths are allowed.
At a node, the amplitude of the wave is zero.
3.5
Wave Properties of Matter
De Broglie deduced that the particle and wave properties are related by the following expression:
λ is the wavelength associated with the particle
m is the mass (in kg)
u is the velocity (in m/s)
The wavelength calculated from this equation is known as the de Broglie wavelength.
Calculate the de Broglie wavelength of the “particle” in the following two cases: (a) a 25-g bullet traveling at 612 m/s and (b) an electron (m = 9.109×10-31 kg) moving at 63.0 m/s.
Worked Example 3.6
Solution
25 g ×
Setup
h = 6.63×10-34 J s, or 6.63×10-34 kg m2/s; Remember m must be expressed in kg.
= 0.025 kg
λ =
1 kg
1000 g
h
mu
6.63×10-34 kg m2/s
(0.025 kg)(612 m/s)
=
= 4.3×10-35 m
λ =
h
mu
= 1.16×10-5 m
6.63×10-34 kg m2/s
(9.109×10-31 kg)(63.0 m/s)
=
Think About It While you are new at solving these problems, always write out the units of Planck’s constant (J s) as kg m2/s. This will enable you to check your unit cancellations and detect common errors such as expressing mass in grams rather than kilograms. Note that the calculated wavelength of a macroscopic object, even one as small as a bullet, is extremely small. An object must be at least as small as a subatomic particle in order for its wavelength to be large enough for us to observe.
Diffraction of Electrons
Experiments have shown that electrons do indeed possess wavelike properties:
X-ray diffraction pattern of aluminum foil
Electron diffraction pattern of aluminum foil.
The Heisenberg uncertainty principle states that it is impossible to know simultaneously both the momentum p and the position x of a particle with certainty.
Δx is the uncertainty in position in meters
Δp is the uncertainty in momentum
Δu is the uncertainty in velocity in m/s
m is the mass in kg
Quantum Mechanics
3.6
Worked Example 3.7
Strategy The uncertainty in the velocity, 1 percent of 5×106 m/s, is Δu. Calculate Δx and compare it with the diameter of they hydrogen atom.
An electron in a hydrogen atom is known to have a velocity of 5×106 m/s + 1 percent. Using the uncertainty principle, calculate the minimum uncertainty in the position of the electron and, given that the diameter of the hydrogen atom is less than 1 angstrom ( ), comment on the magnitude of this uncertainty compared to the size of the atom.
Setup The mass of an electron is 9.11×10-31 kg. Planck’s constant, h, is 6.63×10-34 kg m2/s.
Worked Example 3.7
Solution
Δu = 0.01 × 5×106 m/s = 5×104 m/s
Δx =
Δx =
h
4π mΔu
6.63×10-34 kg m2/s
4π(9.11×10-31 kg)(5×104 m/s)
> 1×10-9 m
An electron in a hydrogen atom is known to have a velocity of 5×106 m/s + 1 percent. Using the uncertainty principle, calculate the minimum uncertainty in the position of the electron and, given that the diameter of the hydrogen atom is less than 1 angstrom ( ), comment on the magnitude of this uncertainty compared to the size of the atom.
The minimum uncertainty in the position x is 1×10-9 m = 10 . The uncertainty is 10 times larger than the atom!
Think About It A common error is expressing the mass of the particle in grams instead of kilograms, but you should discover this inconsistency if you check your unit cancellation carefully. Remember that if one uncertainty is small, the other must be large. The uncertainty principle applies in a practical way only to submicroscopic particles. In the case of a macroscopic object, where the mass is much larger than that of an electron, small uncertainties, relative to the size of the object, are possible for both position and velocity.
Quantum (wave) Mechanics
Time-independent Schrodinger wave equation with solutions called stationary-state functions.
The wave function must satisfy
1. y must be single-valued at all points.
2. The total area under y2(x) must be equal to unity or
3. y must be smooth or dy/dx must be continuous at all points.
Qualitative Aspects of the Wavefunction
proportional to d2y/dx2
describes curvature of wavefunction
potential energy portion
kinetic energy portion
Ground-state wave function is a compromise to minimize each term.
The Schr dinger Equation
HY = EY
d2Y
dy2
d2Y
dx2
d2Y
dz2
+
+
8p2mQ
h2
(E-V(x,y,z)Y(x,y,z) = 0
+
how y changes in space
mass of electron
total quantized energy of the atomic system
potential energy at x,y,z
wave function
The wavefunction y contains all the dynamical information about the system it describes.
The trick is to determine what y is.
And to figure out how to extract the desired information.
The Schr dinger equation is a secular equation(久期方程)!
(operator) (eigenfunction) = (eigenvalue) (same eigenfunction)
Solving Schr dinger Equation
算符
本征函数
本征值
Exact solution in polar spherical coordinates (r, q, f ) results in three quantum numbers that indicate the allowed quantum states.
Schr dinger Equation for Hydrogen
principal quantum number, n: n = 1, 2, 3, …
angular momentum quantum number, l: l = 0,1, …,n-1
magnetic quantum number, ml: ml = -l,...-1, 0, +1, …,+l
atomic orbital: wavefunction for a single electron which describes the position of the electron
Hydrogen and hydrogen-like atoms orbital energy depends only on n.
n>1 multiple orbitals exist corresponding to different combination of n and l. They are
collectively called an energy shell
degenerate: have the same energy
subshell: Within an energy shell, a given set of distinct orbitals exist with the same value of l.
l 0 1 2 3 4 5
Name of Subshell s p d f g h
Radial and Angular Parts of the Wavefunction
radial part
Solving Schr dinger Equation
It is easier to solve Schr dinger equation in spherical coordinate (r, , ) than in Cartesian coordinate (x, y, z).
= R(r) ( ) ( )
R: radial part, related with two parameters, n & l.
Y( , ) = ( ) ( ): angular part, related with two parameters, l & m.
Secular Equation
H = E
The result of solving this equation is getting a series of eigenfunction ’s ( 1, 2…..), with corresponding eigenvalues of E (E1, E2,…).
Each eigenfunction is represented with several parameters, n, l, ml.
But what does these eigenfunctions mean
Quantum Numbers and Atomic Orbitals
An atomic orbital is specified by three quantum numbers.
n the principal quantum number - a positive integer
l the angular momentum quantum number - an integer from 0 to n-1
ml the magnetic moment quantum number - an integer from -l to +l
Quantum Numbers
Quantum numbers are required to describe the distribution of electron density in an atom.
There are three quantum numbers necessary to describe an atomic orbital.
The principal quantum number (n) – designates size
The angular moment quantum number (l) – describes shape
The magnetic quantum number (ml) – specifies orientation
3.7
Quantum Numbers
The principal quantum number (n) designates the size of the orbital.
Larger values of n correspond to larger orbitals.
The allowed values of n are integral numbers: 1, 2, 3 and so forth.
The value of n corresponds to the value of n in Bohr’s model of the hydrogen atom.
A collection of orbitals with the same value of n is frequently called a shell.
Quantum Numbers
The angular moment quantum number (l) describes the shape of the orbital.
The values of l are integers that depend on the value of the principal quantum number
The allowed values of l range from 0 to n – 1.
Example: If n = 2, l can be 0 or 1.
A collection of orbitals with the same value of n and l is referred to as a subshell.
l 0 1 2 3
Orbital designation s p d f
Quantum Numbers
The magnetic quantum number (ml) describes the orientation of the orbital in space.
The values of ml are integers that depend on the value of the angular moment quantum number:
– l,…0,…+l
Quantum Numbers
Quantum numbers designate shells, subshells, and orbitals.
Worked Example 3.8
Strategy Recall that the possible values of ml depend on the value of l, not on the value of n.
What are the possible values for the magnetic quantum number (ml) when the principal quantum number (n) is 3 and the angular quantum number (l) is 1
Solution The possible values of ml are -1, 0, and +1.
Setup The possible values of ml are – l,…0,…+l.
Think About It Consult Table 3.2 to make sure your answer is correct. Table 3.2 confirms that it is the value of l, not the value of n, that determines the possible values of ml.
Interpreting the wavefunction
Born interpretation of y
According to the wave theory of light, the square of the amplitude of an EM wave is proportional to the intensity of light.
But since light behaves as a particle, the intensity must be a measure of the probability density of photons in a volume of space.
Applying this same idea to particles indicates that the value of |y|2 at a point is proportional to the probability density of finding the particle at that point.
Born Interpretation of y
The probability of finding a particle between x and x+dx is proportional to |y|2dx.
|y|2 probability density (real and never negative).
y probability amplitude
Can be complex
In 3-D, the probability of finding a particle in an infinitesimal volume dt = dxdydz is proportional to |Y|2 dt
Nickel (110)
Cesium & Iodine on Copper (111)
a molecule assembled from 8 cesium and 8 iodine atoms
Quantum Numbers
The electron spin quantum number (ms ) is used to specify an electron’s spin.
There are two possible directions of
spin.
Allowed values of ms are + and .
Quantum Numbers
A beam of atoms is split by a magnetic field.
Statistically, half of the electrons spin clockwise, the other half spin counterclockwise.
Quantum Numbers
To summarize quantum numbers:
principal (n) – size
angular (l) – shape
magnetic (ml) – orientation
electron spin (ms) direction of spin
Required to describe an atomic orbital
Required to describe an electron in an atomic orbital
2px
principal (n = 2)
angular momentum (l = 1)
related to the magnetic quantum number (ml )
Atomic Orbitals
All s orbitals are spherical in shape but differ in size:
1s < 2s < 3s
2s
angular momentum quantum number (l = 0)
ml = 0; only 1 orientation possible
principal quantum number (n = 2)
3.8
Atomic Orbitals
The p orbitals:
Three orientations:
l = 1 (as required for a p orbital)
ml = –1, 0, +1
Atomic Orbitals
The d orbitals:
Five orientations:
l = 2 (as required for a d orbital)
ml = –2, –1, 0, +1, +2
Energies of Orbitals
The energies of orbitals in the hydrogen atom depend only on the principal quantum number.
2nd shell (n = 2)
3d subshell (n = 3; l = 2)
2p subshell (n = 2; l = 1)
3rd shell (n = 3)
2s subshell
(n = 2; l = 0)
3p subshell (n = 3; l = 1)
3s subshell (n = 3; l = 0)
Worked Example 3.9
Strategy Consider the significance of the number and the letter in the 4d designation and determine the values of n and l. There are multiple values for ml, which will have to be deduced from the value of l.
List the values of n, l, and ml for each of the orbitals in a 4d subshell.
Solution 4d
Possible ml are -2, -1, 0, +1, +2.
Setup The integer at the beginning of the orbital designation is the principal quantum number (n). The letter in an orbital designation gives the value of the angular momentum quantum number (l). The magnetic quantum number (ml) can have integral values of – l,…0,…+l.
principal quantum number, n = 4
angular momentum quantum number, l = 2
Think About It Consult the following figure to verify your answers.
Wavefunctions for Many Electron Atoms
For helium (He): function of six position variables, x1, y1, and z1 for electron 1 and x2, y2, and z2 for electron 2.
For an atom with N electrons
Schrodinger equation for helium (He)
potential energy term
kinetic energy term
Potential energy term:
electron-electron repulsion
electron-nuclear attraction
Electron-electron repulsions not present in hydrogen.
Without electron-electron repulsions
where f denotes an orbital for an individual electron
leads to unsatisfactory results.
Solution: self-consistent field (Hartree/SCF) method
Self-consistent field (Hartee/SCF) method
For an atom with N electrons
Schematic representation of SCF method
SCF orbitals can be described using the same set of quantum numbers (n, l, ml).
The four quantum number (n, l, ml, ms ) completely label an electron in any orbital in any atom.
Computationally intensive accomplished by sophisticated computer programs.
electron configuration – how electrons are distributed among the various atomic orbitals
orbital diagram – pictorial representation of the electron configuration which shows the spin of the electron
Pauli Exclusion Principle
No two electrons in an atom can have the same four quantum numbers (n, l, ml, ms).
Same values of ms
Effect of radial probability function for Ar
3 distinct shells
Correct representation
Paramagnetic
unpaired electrons
2p
Diamagnetic
all electrons paired
2p
Diamagnetism and Paramagnetic
Paramagnetic - attracted by a magnet
Diamagnetic – slightly repelled by a magnet
Gouy balance - provides direct evidence of electron configurations
Aufbau (building-up) Principle
orbital energy – negative amount of energy required to remove an electron from a given orbital
An electron configuration is constructed according to the Pauli exclusion principle, so that the total energy of the configuration is a minimum.
Energy of orbitals in a single electron atom
Energy only depends on principal quantum number n
En = -RH
( )
1
n2
n=1
n=2
n=3
Factors Affecting Atomic Orbital Energies
Additional electron in the same orbital
An additional electron raises the orbital energy through electron-electron repulsions.
Additional electrons in inner orbitals
Inner electrons shield outer electrons more effectively than do electrons in the same sublevel.
Higher nuclear charge lowers orbital energy (stabilizes the system) by increasing nucleus-electron attractions.
The Effect of Nuclear Charge (Zeffective)
The Effect of Electron Repulsions (Shielding)
The effect of nuclear charge on orbital energy.
Shielding
Electron density varies with distance from nucleus as shown by radial probability plots.
For same n: sThe effect of orbital shape
Energy of orbitals in a multi-electron atom
Energy depends on n and l
n=1 l = 0
n=2 l = 0
n=2 l = 1
n=3 l = 0
n=3 l = 1
n=3 l = 2
Order of orbitals (filling) in multi-electron atom
1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s
Graphical representation of Madelung’s rule
Electron Configurations
The electron configuration describes how the electrons are distributed in the various atomic orbitals.
In a ground state hydrogen atom, the electron is found in the 1s orbital.
1s1
principal (n = 1)
angular momentum (l = 0)
number of electrons in the orbital or subshell
1s
2s
2p
2p
2p
The use of an up arrow indicates an electron with ms = +
Ground state electron configuration of hydrogen
3.9
Electron Configurations
If hydrogen’s electron is found in a higher energy orbital, the atom is in an excited state.
2s1
1s
2s
2p
2p
2p
A possible excited state electron configuration of hydrogen
Electron Configurations
The helium emission spectrum is more complex than the hydrogen spectrum.
There are more possible energy transitions in a helium atom because helium has two electrons.
Electron Configurations
In a multi-electron atoms, the energies of the atomic orbitals are split.
Splitting of energy levels refers to the splitting of a shell (n=3) into subshells of different energies (3s, 3p, 3d)
Electron Configurations
According to the Pauli exclusion principle, no two electrons in an atom can have the same four quantum numbers.
1s2
1s
2s
2p
2p
2p
The ground state electron configuration of helium
Quantum number
Principal (n)
Angular moment (l)
Magnetic (ml)
Electron spin (ms)
1
0
0
+
1
0
0

describes the 1s orbital
describes the electrons in the 1s orbital
Electron Configurations
The Aufbau principle states that electrons are added to the lowest energy orbitals first before moving to higher energy orbitals.
1s22s1
1s
2s
2p
2p
2p
The ground state electron configuration of Li
The 1s orbital can only accommodate 2 electrons (Pauli exclusion principle)
The third electron must go in the next available orbital with the lowest possible energy.
Li has a total of 3 electrons
Electron Configurations
The Aufbau principle states that electrons are added to the lowest energy orbitals first before moving to higher energy orbitals.
1s
2s
2p
2p
2p
1s22s2
The ground state electron configuration of Be
Be has a total of 4 electrons
Electron Configurations
The Aufbau principle states that electrons are added to the lowest energy orbitals first before moving to higher energy orbitals.
1s
2s
2p
2p
2p
The ground state electron configuration of B
1s22s22p1
B has a total of 5 electrons
Electron Configurations
According to Hund’s rule, the most stable arrangement of electrons is the one in which the number of electrons with the same spin is maximized.
1s22s22p2
1s
2s
2p
2p
2p
The ground state electron configuration of C
The 2p orbitals are of equal energy, or degenerate.
Put 1 electron in each before pairing (Hund’s rule).
C has a total of 6 electrons
Electron Configurations
According to Hund’s rule, the most stable arrangement of electrons is the one in which the number of electrons with the same spin is maximized.
1s22s22p3
1s
2s
2p
2p
2p
The ground state electron configuration of N
The 2p orbitals are of equal energy, or degenerate.
Put 1 electron in each before pairing (Hund’s rule).
N has a total of 7 electrons
Electron Configurations
According to Hund’s rule, the most stable arrangement of electrons is the one in which the number of electrons with the same spin is maximized.
1s22s22p4
1s
2s
2p
2p
2p
The ground state electron configuration of O
O has a total of 8 electrons
Once all the 2p orbitals are singly occupied, additional electrons will have to pair with those already in the orbitals.
Electron Configurations
According to Hund’s rule, the most stable arrangement of electrons is the one in which the number of electrons with the same spin is maximized.
1s22s22p5
1s
2s
2p
2p
2p
The ground state electron configuration of F
F has a total of 9 electrons
When there are one or more unpaired electrons, as in the case of oxygen and fluorine, the atom is called paramagnetic.
Electron Configurations
According to Hund’s rule, the most stable arrangement of electrons is the one in which the number of electrons with the same spin is maximized.
1s22s22p6
1s
2s
2p
2p
2p
The ground state electron configuration of Ne
Ne has a total of 10 electrons
When all of the electrons in an atom are paired, as in neon, it is called diamagnetic.
Electron Configurations
General rules for writing electron configurations:
Electrons will reside in the available orbitals of the lowest possible energy.
Each orbital can accommodate a maximum of two electrons.
Electrons will not pair in degenerate orbitals if an empty orbital is available.
Orbitals will fill in the order indicated in the figure.
Worked Example 3.10
Write the electron configuration and give the orbital diagram of a calcium (Ca) atom (Z = 20).
Solution
Ca 1s22s22p63s23p64s2
Setup Because Z = 20, Ca has 20 electrons. They will fill in according to the diagram at right. Each s subshell can contain a maximum of two electrons, whereas each p subshell can contain a maximum of six electrons.
1s2
2s2
2p6
3s2
3p6
4s2
Think About It Look at the figure again to make sure you have filled the orbitals in the right order and that the sum of electrons is 20. Remember that the 4s orbital fills before the 3d orbitals.
Electron Configurations and the Periodic Table
The electron configurations of all elements except hydrogen and helium can be represented using a noble gas core.
The electron configuration of potassium (Z = 19) is 1s22s22p63s23p64s1.
Because 1s22s22p63s23p6 is the electron configuration of argon, we can simplify potassium’s to [Ar]4s1.
1s22s22p63s23p64s1
The ground state electron configuration of K:
[Ar]
[Ar]4s1
3.10
1s22s22p63s23p64s1
Electron Configurations and the Periodic Table
Elements in Group 3B through Group 1B are the transition metals.
Following lanthanum (La), there is a gap where the lanthanide (rare earth) series belongs.
Electron Configurations and the Periodic Table
After actinum (Ac) comes the actinide series.
Electron Configurations and the Periodic Table
Electron Configurations and the Periodic Table
There are several notable exceptions to the order of electron filling for some of the transition metals.
Chromium (Z = 24) is [Ar]4s13d5 and not [Ar]4s23d4 as expected.
Copper (Z = 29) is [Ar]4s13d10 and not [Ar]4s23d9 as expected.
The reason for these anomalies is the slightly greater stability of d subshells that are either half-filled (d5) or completely filled (d10).
4s
3d
3d
3d
3d
3d
[Ar]
Cr
Greater stability with half-filled 3d subshell
Electron Configurations and the Periodic Table
There are several notable exceptions to the order of electron filling for some of the transition metals.
Chromium (Z = 24) is [Ar]4s13d5 and not [Ar]4s23d4 as expected.
Copper (Z = 29) is [Ar]4s13d10 and not [Ar]4s23d9 as expected.
The reason for these anomalies is the slightly greater stability of d subshells that are either half-filled (d5) or completely filled (d10).
Electron Configurations and the Periodic Table
4s
3d
3d
3d
3d
3d
[Ar]
Cu
Greater stability with filled 3d subshell
Worked Example 3.11
Write the electron configuration for an arsenic atom (Z = 33) in the ground state.
Solution
As [Ar]4s23d104p3
Setup The noble gas core for As is [Ar], where Z = 18 for Ar.
The order of filling beyond the noble gas core is 4s, 3d, and 4p. Fifteen electrons go into these subshells because there are 33 – 18 = 15 electrons in As beyond its noble gas core.
2
2
2
2
6
6
3
10
Think About It Arsenic is a p-block element; therefore, we should expect its outermost electrons to reside in a p subshell.
Chapter Summary: Key Points
3
Forms of Energy
The Nature of Light
Properties of Waves
The Electromagnetic Spectrum
The Double-Slit Experiment
Quantum Theory
Quantization of Energy
Photons and the Photoelectric Effect
Bohr’s Theory of the Hydrogen Atom
Atomic Line Spectra
The Line Spectrum of Hydrogen
Wave Properties of Matter
The de Broglie Hypothesis
Diffraction of Electrons
Quantum Mechanics
The Uncertainty Principle
The Schr dinger Equation
The Quantum Mechanical Description of the Hydrogen Atom
Quantum Numbers (n,l,ml,ms)
Atomic Orbitals
s orbitals, p orbitals, d orbitals and other High-Energy Orbitals
Chapter Summary: Key Points
3
Energies of Orbitals
Electron Configuration
Energies of Atomic Orbitals in Many-Electron Systems
The Pauli Exclusion Principle
The Aufbau Principle
Hund’s Rule
General Rules for Writing Electron Configurations
Electron Configurations and the Periodic Table

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