Download Chapter 7 The Quantum-Mechanical Model of the Atom

11/9/2011
Principles of Chemistry: A Molecular Approach, 1st Ed.
Nivaldo Tro
Chapter 7
The QuantumMechanical
Model of the
Atom
Roy Kennedy
Massachusetts Bay Community College
Wellesley Hills, MA
Edited by K.M. Hattenhauer
The Behavior of the Very Small
- electrons are incredibly small
 A 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
(Heisenberg Uncertainty Principle)
quantum-mechanical model
- explains how electrons exist and behave in atoms.
- help to understand and predict the properties of atoms that
are directly related to the behavior of the electrons
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Quantum-Mechanical Theory
Schrodinger equation
orbital
- for an electron with a given energy, the best we can do is
describe a region of the atom with a high probability of
finding it
- a probability distribution map of a region where the
electron is likely to be found where distance vs. 2
- many of the properties of atoms are related to the energies of
the electrons.
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Wave Function, 
- calculations show that the size, shape, and orientation in
space of an orbital are determined by three integer terms in
the wave function
quantum numbers
- integer solutions of the wave function
i.) principal quantum number, n
ii.) angular momentum quantum number, l
iii.) magnetic quantum number, ml
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Quantum Numbers
Principal Quantum Number, n
- characterizes the energy of the electron in a particular orbital
- can have values of any integer of n 1
- the larger the value of n, the more energy the orbital has and
the larger the orbital
- energies are defined as being negative.
 An electron has E = 0 when it just escapes the atom.
- as n gets larger,
larger the amount of energy between orbitals gets
smaller
Principal Energy Levels in Hydrogen
En = -2.18 x 10-18 J (1/n2)
(for the electron in H)
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Quantum Numbers
electron transitions
- in order to transition to a higher energy state, the electron
must gain the exact 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 states.
 energy released as a photon of light
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The Electromagnetic Spectrum
electromagnetic spectrum
- the range of wavelengths of all possible electromagnetic
radiation
- visible light comprises only a small fraction of all the
wavelengths
Note: wavelength/frequency relationship
- short-wavelength (high-frequency) light has high
energy
low frequency
and energy
high frequency
and energy
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Spectra
- when atoms or molecules absorb energy, that energy is often
released as light energy
emission spectrum
- when that emitted light is
passed through a prism,
pattern of particular
wavelengths of light is
seen that is unique to that
type of atom or molecule
 not continuous
 can be used to identify the
material (similar to flame tests)
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Quantum Numbers
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, and therefore (n – 1) lines it can
generate
- both the Bohr and quantum mechanical models can predict
these lines very accurately for a one-electron system.
- the energy of a photon released is equal to the difference in
energy between the two levels the electron is jumping
between and can be calculated by subtracting the energy of
the initial state from the energy of the final state
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Quantum Numbers
Energy Transitions in Hydrogen
Eelectron = Efinal state − Einitial state
Eemitted photon = −Eelectron
Emission spectrum of Hydrogen
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Quantum Numbers
Angular Momentum Quantum Number
- primarily determines the shape of the orbital
- can have values of any integer from l = 0 …. (n – 1)
- each value of l is called by a particular letter that designates
the shape of the orbital
i.) if l=0, called s orbitals and are spherical.
ii.) if l=1, called p orbitals and are like two balloons tied at
the knots (dumbbell)
iii.) if l=2, called d orbitals and are mainly like four balloons
tied at the knots (double dumbbell)
iv.) if l=3, called f orbitals and are mainly like eight balloons
tied at the knots
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Quantum Numbers
Angular Momentum Quantum Number, l
i.) l = 0, s orbital (spherical)
- each principal energy state has 1 s orbital
- lowest energy orbital in a principal energy
state
for 2s
n = 2,
l=0
Tro, Principles of Chemistry: A Molecular Approach
for 3s
n = 3,
l=0
Note:
- the number of s orbitals
in each principal energy
level is given by value
of ml where ml = -l ….+l
- for l = 0; ml = 0 which
means that there will be
one s orbital for a
particular energy level
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Quantum Numbers
Angular Momentum Quantum Number, l
ii.) l = 1, p orbitals (dumbbell – two lobed)
- each principal energy state above n = 1 has 3 p orbitals
given by the related value of ml where ml = -l ….+l and for
l = 1 then ml = −1, 0, +1
- each of the three orbitals
points along a different
axis and designated as
p x , py , pz
- second lowest energy
orbitals in a principal
energy state
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Quantum Numbers
Angular Momentum Quantum Number, l
iii.) l = 2, d orbitals (double dumbbell – mainly four lobed)
- each principal energy state above n = 2 has 5 d orbitals
given by the related value of ml where ml = -l ….+l and for
l = 2 then ml = −2, −1, 0, +1, +2
- four of the five orbitals are aligned in a different plane
(dxy, dyz, dxz, dx squared – y squared) with the fifth is aligned with
the z axis, dz squared.
- third lowest energy orbitals in a principal energy state
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Quantum Numbers
iv.) l = 3, f orbitals (mainly eight-lobed)
- each principal energy state above n = 3 has 7 d orbitals
given by the related value of ml where ml = -l ….+l and for
l = 3 then ml = -3, −2, −1, 0, +1, +2, +3
- fourth lowest energy orbitals in a principal energy state
Summary of
Energy Shells
and Subshells
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