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Chapter 7: Quantum Mechanical Model of the Atom

For Ms.Hubbell / October 24, 2012
by

sarah wong

on 3 April 2014

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Transcript of Chapter 7: Quantum Mechanical Model of the Atom

Quantum Mechanic Model of the Atom
by: Sarah Wong // SCH3U3 // Mrs.Hubbell // 30 October 2012
1894 - 1900
Max Planck
The History of Quantum Mechanics
German physicist who was awarded the 1918 Nobel Prize in Physics for his quantum theory.
Planck was researching
black-body
radiation.
Thus he adopted a new approach, and worked backwards based on his own experiments and calculations.
This was a topic that stumped many physicists before Planck's time. Initially, Planck too could not develop a theory or formula to describe the emission of radiation from black bodies
using classical physics.
A black-body is a non-reflective, opaque object that absorbs all electromagnetic radiation, or all frequencies of light, directed at it.
Also known as cavity radiation, black-body radiation is when an object absorbs all radiation directed at it (becomes heated) and re-radiates energy which is unique to the object itself, not the radiation that was directed at it. The radiation changes as the temperature of the object changes.
Classical physics stated that light travels in continuous waves, and atoms and molecules can emit/absorb arbitrary amounts of energy.
Planck stated that light can only be emitted/absorbed in discrete amounts, or "packets", which he called
quanta

(sing. quantum)
, sparking the development of his quantum theory.
The smallest amount of energy that can be emitted or absorbed in the form of electromagnetic radiation (or light).
A single quantum of energy can be expressed as:
energy of a single quantum
Planck's constant
frequency
frequency
wavelength
speed of light in a vacuum
8
(6.63 x 10 J s )
.
-34
(3.0 x 10 m/s)
wavelength
speed of light in a vacuum
8
(3.0 x 10 m/s)
Planck's constant
(6.63 x 10 J s )
.
-34
energy of a single quantum
or...
1905
Quantum theory
states that energy is always emitted in integral multiples of .
e.g. vs.
A quantum in energy can be compared to electrons in atoms. Atoms can only have whole number values of these, not 0.5 electrons or 1.4 electrons. Likewise, pennies cannot be further split into smaller units of money. 0.2 pennies, or 1.67 pennies do not exist.
Albert Einstein
German-born American physicist who was awarded the 1921 Nobel Prize in Physics for his explanation of the photoelectric effect. Considered one of the greatest physicists that ever lived.
Photoelectric Effect
A piece of metal placed in a vacuum ejects electrons from its surface in the presence of incident light of at least a certain frequency (called the
threshold frequency
).
clean piece of metal
positive electrode
attracted to
incident light
ammeter
battery
vacuum
An apparatus used to explain the photoelectric effect involves light (visible light waves) shining at a clean piece of metal in a vacuum. At some frequencies of light, electrons will be liberated from the atoms in the metal and will jump to the opposite side where the positively charged electrode is.
Classical physics, which uses the wave theory of matter, expects that:
more intense light = more energy, so more electrons liberated from metal
However, this was not the case. At any frequency below a particular frequency called the
the threshold frequency
, electrons would no longer jump to the electrode. Therefore, this is another occasion when classical physics has failed to explain the behaviour of energy and light.
Planck's constant
frequency
(6.63 x 10 J s )
.
-34
work function
maximum KE of the ejected electrons
(
The threshold frequency is dependent on the metal. Thus, Na would have a certain threshold frequency different from that of Mg.
)
how strongly the electrons are held in the metal
To explain the behaviour of light in this experiment, Einstein proposed that perhaps light travels as a stream of particles called
photons.
particles of light
After experimentation with the apparatus, Einstein was able to derive the following equation.
Again, this activity of light defies classical physics and rather seems to take on a new perspective that treats light as particles. Thus came the establishment of the
particle-wave duality
.
Energy is made of individual units called quanta.
Radiation is also quantized in the same way.
Electrons surround the nucleus in an atom in specific, fixed orbits, however they can jump to a higher level orbit if they are energized with electricity, heat, or any other kind of energy.
When an electron in a higher-energy orbital returns to a lower energy orbital, a packet of energy
(a photon)
is emitted with specific characteristics (i.e. frequency, colour) that match the distance the electron has jumped.
How do light sources emit photons?
1913
Niels Bohr
Danish physicist who was awarded the 1922 Nobel Prize in Physics for his theory of the hydrogen atom spectrum. One of the fathers of modern physics.
A substance, when energized with some form of energy (e.g. thermal energy, high-voltage electric discharge), emits a unique glow of visible light.
It is known that when visible white light is passed through a prism, it spreads out into a rainbow, or rather the full visible spectrum.
If the characteristic light emitted by the energized substance is passed through a prism, unless it is white light, the visible light spectrum will only show the different wavelengths of visible light that the substance emitted. This is called the
line spectrum.
This is known as the
emission spectrum
. The one above illustrates the emission spectrum for an energized sample of hydrogen. It can be seen that:
the absorption of visible light + emission of visible light = full spectrum
Each element has a different emission spectrum, thus they can be described as "element fingerprints".
Either continuous or line spectrum of radiation emitted by substances.
Bohr figured out how this works:
In a hydrogen atom, there is 1 electron that is naturally located in the orbital closest to the nucleus. When it is in this orbital

(n = 1), it is in its
ground state.
The light emission only at specific wavelengths.
(aka ground level) The lowest energy state of a system.
When energy is put into the atom (thermal, electrical, etc.), the electrons absorb the energy and thus jump to higher energy levels. The hydrogen atom is now in the
excited state.
(aka excited level) Being higher in energy than the ground state.
The energy is eventually released, thus the electrons lose energy as well. As a result, they must move back down energy levels and emit radiant energy in the form of photons.
If the electrons return to the ground state (n=1), the atom emits UV rays. This is called the Lyman Series.
If
If the electrons jump down to n=2, the atom emits visible rays. This is called the Balmer Series.
If the electrons jump down to n=3, the atom emits infrared rays. This is called the Paschen Series.
These jumps down to lower energy levels are called
transitions.
Each spectral line in the light spectrum corresponds with different transitions.
greater distance between the initial energy level and the final energy level
= more energy released
= more photons released
= brighter spectral line
Light possesses properties of both waves and particles, and depending on the experiment, behaves as one or the other.
http://upload.wikimedia.org/wikipedia/commons/thumb/b/b2/Hydrogen_transitions.svg/400px-Hydrogen_transitions.svg.png
http://www.teylersmuseum.eu/afbeeldingen/personen/fs/bohr.jpg
http://ircamera.as.arizona.edu/astr_250/images/spectra.jpg
http://www.lowdensitylifestyle.com/media/uploads/2011/01/AlbertEinstein.jpg
http://adam.humanisti.sk/wp-content/2007/10/max_planck.jpg
Every element emits a unique amount of radiation when heated, indicated by its line spectrum.
Rutherford's model of the atom showed electrons surrounding the nucleus in circular orbits. However, this didn't make sense because the electron would be attracted to the positive nucleus, and thus spiral inward. Bohr discovered that electrons orbit the nucleus in specific, fixed energy orbitals. His theory helped to explain the emission spectrum of hydrogen.
Bohr's Model of the Atom
http://www.splung.com/nuclear/images/atoms10.gif
threshold frequency
The amount of energy absorbed by the electron is determined only by the frequency of the light, not how much is given off or how bright it is.
Planck's constant multiplied by the frequency of the light
must be greater than
Planck's constant multiplied by the threshold frequency
, otherwise, there will be no movement of electrons (no current).

This does not make sense according to the wave theory of light.
(energy given off by light / energy absorbed by electron)
(amount of energy required to liberate electron from metal atom)
Again, Bohr theorized that electrons are located in specific energy levels at fixed distances to the nucleus. However, he could not explain why.
1926
Planck's constant
(6.63 x 10 J s )
.
-34
Louis de Broglie
French physicist and prince. He was awarded the 1929 Nobel Prize in Physics for his proposal of the idea that matter and radiation both have properties of waves and particles.
If matter possesses properties of waves and particles, that would mean electrons possess wave-like particles as well.
De Broglie stated that electrons bound to the nucleus behave like standing waves, also known as stationary waves, such as the sound wave produced when a guitar string is plucked. The length of the wave must fit the circumference of the orbit exactly (circumference = integral number of wavelengths). Eventually the amplitude would become zero and the wave would cease to exist, forming a circular shape of the electron orbit.
Points on a wave that have amplitude of zero are called
nodes
.
greater frequency of vibrations = shorter wavelength, more nodes
http://scientificsentence.net/Waves/images/de_Broglie.jpg
http://www.biografiasyvidas.com/biografia/b/fotos/broglie.jpg
circumference of orbit
energy level of electron
radius of orbit
electron wave's wavelength
Mathematically, this can be represented as:
De Broglie's idea became known as the
particle-wave duality.
Radiation and matter possess properties of both waves and particles. They behave as one or the other depending on the activity.
1927
http://upload.wikimedia.org/wikipedia/commons/b/b0/Heisenberg_10.jpg
Werner Heisenberg
German physicist awarded the 1932 Nobel Prize in Physics. One of the founders of modern quantum theory.
As one of the founders of quantum mechanics, Heisenberg developed one of the fundamental principles of the subject: the Heisenberg Uncertainty Principle.
It states that one cannot accurately measure the position and velocity of an electron at the same time. There is always uncertainty because the simple act of observing requires one to shine a light on the electron, which changes its behaviour.
If you were to measure position more precisely, velocity would have to be measured more imprecisely, and vice versa. This is not due to instrumental or procedural error, but rather the nature of the quantum world itself.
http://www.anthroposophie.net/bilder/schroedinger.jpg
Erwin Schrodinger
Austrian physicist awarded the 1933 Nobel Prize in Physics for laying the foundation of the modern quantum theory and the quantum mechanical model of the atom.
Schrodinger created a complicated mathematical equation to describe the likelihood of an electron being in a specific location in an atom at a particular moment in time.
:
:
2
It deals with wave function (the amplitude of a wave), Ψ, in which Ψ is proportional to the intensity of light. The value of Ψ at a given position and time determines the probability of the particle being there at the time.



Wherever the light is most intense is where a photon is likely to be found. Similarly the place where the light is most intense is likely where an electron would be found in the electron cloud around the nucleus.
Schrodinger's equation launched the new field known as
quantum mechanics
(aka wave mechanics). As a result, the ideas developed between 1913 and 1926 is known as the "old quantum theory".
:
Quantum Numbers: An Electron's Address
Describes an electron's distribution in atoms.
principal quantum number
angular momentum number
magnetic quantum number
electron spin quantum number
http://en.citizendium.org/images/thumb/7/7d/Oxygen_atom_electron_configuration.jpg/600px-Oxygen_atom_electron_configuration.jpg
(2, 0, 0, -1/2)
(2, 1, -1, -1/2)
Atomic Orbitals
Though there aren’t definitive shapes to orbitals (since this would violate the concept of wave function which states that the orbital extends from the nucleus and beyond), it is convenient and useful to assign different shapes to atomic orbitals.
Orbital shapes can be illustrated using
boundary surface diagrams.
Enclose roughly 90% of the total electron density in an orbital, meaning it shows where most of the electrons in an atom are located and draws a shape around them.
s orbitals
d orbitals
p orbitals
f orbitals
g orbitals, h orbitals, etc.
S orbitals are the lowest energy orbitals. They exist in all energy levels (all values of n).
They are spherical in shape.
greater energy level = larger sphere
http://3.bp.blogspot.com/-8-wY6hnnluM/TcpTaMkgxnI/AAAAAAAAAeA/M0w15d0Hb3g/s640/s-orbitals_3-up.jpg
P orbitals exist starting in the principle energy level of
n = 2.
Their shape is like two inflated balloons attached at the knots, where the nucleus would be. There are 3 different p orbitals: 2p , 2p , and 2p . The subscripts indicate which axis the orbital is oriented on. Other than their orientation, all 3 orbitals are identical in shape and size.
Again,
greater energy level = larger size orbital
http://3.bp.blogspot.com/-eUkUGuMAwAc/TcpTZ8GemuI/AAAAAAAAAd8/iEI9kIlV0Fc/s1600/porbital.gif
x
y
z
D orbitals exist starting in the principle energy level of
n = 3.
They have a four-leaf clover shape, like 2 p orbitals on two different axes. There are 5 different d orbitals:
3d , 3d , 3d , 3d , and 3d .
They have similar shapes in all energy levels they exist in.
http://x3f.xanga.com/721c904073133207432985/m161436131.gif
F orbitals exist starting in the principle energy level of
n = 4.
F orbitals have very complicated shapes that tend to be difficult to visually represent.
xy
yz
xz
x -y
z
2
2
2
http://www.goiit.com/upload/2007/10/1/04b357f5d24bc7ca72ad640115bab8c4_16678.gif_thumb
F orbitals are significant in understanding the behaviour of elements with atomic number 57 or greater.
Some theorize that there are higher sublevels of orbitals, but they are not pertinent to general chemistry at the moment.
http://www.webelements.com/shop/shopimages/products/normal/POS0007-A2-orbitron-2010.jpg
Electron Configuration
How electrons are arranged different orbitals in an atom.
http://media-1.web.britannica.com/eb-media/06/96906-004-FB4A8411.jpg
Atomic orbitals are filled in order from lowest energy to highest energy:
In atoms, atomic orbitals containing electrons are included in the electron configuration of the element, written as “principle quantum number n, angular momentum number l” with superscripts of the number of electrons in that orbital.
e.g. the electron configuration for Beryllium is 1s 2s
2
2
The sum of the superscripts is equal to the total number of electrons in an atom of the element.
Process for Writing Electron Configuration of Elements
Identify the atomic number of the element. This is the number of electrons in a neutral atom of the element.
e.g. Oxygen O (Z=8)
Fill in the orbitals going in the order indicated by the image above. The order is simple to remember. The principal quantum numbers (or energy levels) are listed out in numerical order in a column, with the various orbitals existing in that energy level written in the rows.
e.g. 1s 2s 2p
2
2
4
Rules When Writing Electron Configurations
Pauli Exclusion Principle
No two electrons in an atom can have the same set of four quantum numbers.
Since a maximum of 2 electrons can occupy each orbital, 2 electrons in an atom can have the same first 3 quantum numbers, but the m values must be different. In other words, the electrons must "spin" in opposite directions.
The direction of spin indicates the orientation of the magnetic field the electron creates. If both spins were in the same direction in an orbital, the magnetic fields created would reinforce each other.
http://quantumanthropology2.files.wordpress.com/2012/04/picture-paul-40.png
s
Diamagnetism and Paramagnetism
Do not contain net unpaired spins and are slightly repelled by a magnet.
Contain net unpaired spins and are attracted by a magnet.
e.g. He (Z = 2)
Further and further down the periodic table, the elements acquire increasingly high numbers of electrons, to the point where writing out the regular electron configuration becomes inconvenient and tedious. An easier way to write them out is called
noble gas configuration.
Shows the electron configuration of elements after Hydrogen and Helium. The closest preceding noble gas' symbol is written between a pair of square brackets, with any extra electrons following after in their appropriate orbitals.
Process for Writing Noble Gas Configuration of Elements
Identify the atomic number of the element. This is the number of electrons in a neutral atom of the element.
e.g. Oxygen O (Z=8)
Identify the nearest preceding noble gas' symbol. Write this symbol between a pair of square brackets.
e.g. [He]
To know which orbitals the remaining electrons fill up, identify the element's period number. This is the valence orbital's energy level. Note that the transition metals begin with an energy level of 3 because that is the first energy level that contains d orbitals.

Then, identify what sublevel block (s, p, d, or f) the element is located in, as shown in the diagram to the right. This is the valence orbital's shape. Now you know what the valence orbital is and can fill the orbitals with the remaining electrons. Remember to follow the order of lowest energy to highest energy. In the periodic table with the sublevel blocks, the order of the remaining electrons can be seen as you travel from the the noble gas horizontally to the element you are writing the noble gas configuration for. Do not include the valence orbital of the noble gas.
e.g. Oxygen's is in the 2nd period, and it is in the p block, therefore its valence orbital is the 2p orbital. The remaining electrons are written as 2s 2p .
http://upload.wikimedia.org/wikipedia/commons/thumb/e/e5/Periodic_Table_structure.svg/500px-Periodic_Table_structure.svg.png
2
4
The final noble gas configuration of Oxygen is: [He] 2s 2p .
2
4
Shielding Effect
The shielding effect is greater for the elements found at the bottom of the periodic table because those elements contain more electrons, which means a stronger shielding effect.
The difference in the strength of attraction between electrons and the nucleus in an atom.
Electrons are negatively charged, thus they are strongly attracted to the positive nucleus and slightly repel each other. Their repulsion decreases the strength of the electrostatic attraction of electrons to the nucleus. A hydrogen atom has no shielding effect because it has only one electron.
In an atom, the electron density near the nucleus of 2s electrons is greater than that of the 2p electrons, and is thus thought of as more “penetrating”. The 2s electrons are less shielded by the 1s electrons than 2p electrons are, therefore they are more strongly attracted to the nucleus. For the same principal quantum number n (energy level), the penetrating power (electron density near the nucleus) of the different sublevels decreases as the shape of the orbitals change: s > p > d > f > ...
Electrons located in the inner orbitals are much harder to remove than electrons located in the valence orbital because they have to get past many electrons upon removal (the electrostatic attraction of the nucleus is stronger for electrons located in the inner orbitals than the outer orbitals). For example, it is easier to remove a 2p electron from an atom (ionization energy) than a 2s electron because its electrostatic attraction to the nucleus is weaker, and thus requires less energy.
http://course1.winona.edu/sberg/ILLUST/orbitals.gif
Hund's Rule
States that the most stable arrangement of electrons in sublevels is the one with the greatest number of parallel spins.
When assigning electrons to atomic orbitals, each orbital within a suborbital (e.g. 1s, 2s, 2p, etc.) must be filled before they can be paired in the same orbital.
This is because the negative charge of the electrons causes them to repel each other, and thus they will “want” to stay as far away from each other as possible. As a result, they would try to fill the other orbitals before having to pair up.
Unpaired electrons must have the same direction of spin.
In a sublevel, all electrons must be spinning up until there are enough electrons to begin pairing electrons.
e.g. Li (Z=3)
1s
2
# of spins = # of spins
1s
2
2s
2
# of spins > # of spins
Odd number of electrons: at least one unpaired spin because an even number is required to completely pair all electrons.

Even number of electrons: may or may not have net unpaired spins. Depends on orbitals the electrons fill up.
Aufbau Principle
As protons are added to the nucleus one by one to build up elements, electrons are similarly added to the atomic orbitals. Comes from German word “aufbauen” meaning “to build”.
Electrons must fill up lower energy orbitals before high energy orbitals.
When writing the electron configuration of elements, as the number of electrons increase (moving right through the periodic table), the atomic orbitals will be “built up” with more electrons.
Transition Metals
Have incompletely filled d orbitals or readily create cations with incompletely filled d orbitals.
First period of transition metals have 3d as their highest energy, occupied orbital. There are some irregularities in the noble gas configuration of select transition metals.

e.g. Chromium (Z=24) -- expected configuration: [Ar]4s 3d ; actual configuration: [Ar]4s 3d
1
5
2
4
Atoms tend to be more stable when the valence orbital is either half-filled or completely filled. In this case, chromium favours a half-filled 3d orbital with 5 electrons, rather than a completely filled 4s orbital and an almost filled 3d orbital.

Also, the shielding effect of the electrons in the same suborbital is relatively small, and the electrons are more strongly attracted to the nucleus.
Lanthanum
(Z=57) last transition metal before lanthanide series
expected configuration: [Xe]6s 5d
actual configuration: [Xe]6s 4f
2
1
2
1
This strange configuration is due to an irregularity in order of energy levels for this particular element. The energies of the 5d and 4f orbitals in lanthanum are very close. In fact, the 4f orbital has a slightly lower energy than 5d, so it is occupied first.
Actinide Series
Last period of elements. Most are synthesized.
These elements follow regular electron configuration patters with a few exceptions.
Matter possesses properties of waves and particles.
The exact location of an electron can never be known with utmost accuracy.
It is impossible to simultaneously measure the position and velocity of an electron with no degree of uncertainty.
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