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Introduction to Quantum Mechanics in Chemistry

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Ryan Leung

on 30 October 2012

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Transcript of Introduction to Quantum Mechanics in Chemistry

Derived from the German word for "building up", this principle is used to derive the correct electron configuration of an atom
In order to "build up" the electron cloud around an atom, according to the Aufbau Principle, orbitals with lower energy levels (lower principle quantum numbers) are filled before higher-energy shells
Electrons are placed in lower-energy orbitals (accounting for both the principal quantum number, and angular momentum quantum number), filling lower-energy orbitals before filling higher-energy ones. However, in order to satisfy Hund's rule, one should arrange the electrons such that they have as many parallel spins as possible, spreading out electrons among orbitals of equal energy as much as possible (since electrons in the same orbital must have opposite spins) Quantum Mechanics Quanta Certain physical properties such as energy and charge are quantized.
This means that they cannot take any arbitrary value, but must be a multiple of a fundamental discrete unit, or quantum. Wave-Particle Duality Quantum Numbers • Quantum numbers are 4 numbers that describe the shapes, properties and ranges (how far from the nucleus an electron will likely be) of atomic orbitals
• The principal quantum number (n) represents the energy level of an orbital (in a single-electron atom, and can be used to calculate the energy level of the orbital in a multi-electron atom), and determines the average distance of the electron from the nucleus. A shell is generally the set of all the orbitals with the same principal quantum number (energy level). Pauli Exclusion Principle Photons The photon is the quantum of light (electromagnetism)
Light was classically thought of as a continuous wave, but it can also be thought of as coming in discrete energy packets known as photons
The energy of a photon is given by: E=hv, where E is the photon's energy, h is the Planck Constant, and v is the frequency
The energy of any electromagnetic ray must therefore be an integral multiple of E Shielding effect In multi-electron atoms, the farther away from the nucleus an atom is (the higher its principal quantum number), the lower its attraction to the nucleus is (less energy is required to free it)
This is because the negatively charged electrons that are closer to the nucleus partially cancel out the nucleus' positive charge, thus shielding the further-out electrons from the nucleus' full attractive influence
The effective nuclear charge on an electron is the given by this formula: effective nuclear charge=(number of protons in the nucleus)-(average number of electrons between electron of interest and nucleus) Hund's Rule When atoms can have more than 1 possible electron configuration allowed by the Pauli Exclusion Principle, they prefer the one that is most stable
The most stable electron configuration is the one with the greatest number of parallel spins
In electron configurations with many parallel spins, they complement each other, while configurations with opposite spins tend to cancel each other out, reducing the net stability of the configuration.
Atoms therefore prefer to have as many unpaired electrons as possible, or as many unfilled orbitals as possible
As such, atoms prefer to be paramagnetic Electron Configuration Diamagnetism/paramagnetism Electrons with opposite spins ( and - ) have magnetic fields that cancel each other out
These electrons are called paired electrons
In diamagnetic atom, all electrons are paired, and they have no net magnetic field. They are slightly repelled by a magnet
Paramagnetic atoms contain net unpaired electrons (electrons whose magnetic fields are not canceled out by another electron with opposite spin). meaning that they have a net magnetic field. They are attracted by magnets. Aufbau Principle Emission Spectra All electromagnetic radiation is made up of different components, with their own frequency
An emission spectrum allows one to characterize the radiation emitted by a glowing object
Emission spectra can either be continuous ranges, or line spectra, or a combination of the above
Line spectra denote a specific wavelength and frequency of electromagnetic radiation that an object emits, while continuous spectra denote a range of frequencies and wavelengths
All elements have emission spectra that can be seen when they are made to glow, by energizing them
Elements glow when their electrons are excited by input energy and jump to a higher energy level, or excited state, before returning to their more stable ground state, releasing the energy in the form of photons
The energy (and therefore frequency) of the photons they release depends on the excited states of the electrons and the ground states (stable, low-energy states, since electrons tend to take the lowest possible energy level) they fall to (the higher the difference, the higher the energy released). Therefore, emission spectra can give some information on the electron configuration of an atom. Comparing the emission spectrum of an unknown substance to that of known elements can determine its constituent elements as well (it is the sum of those of all the constituent elements). Atomic Orbitals Atomic orbitals are based on the wavefunctions of electrons around an atom
They are boundary surfaces, that describe the outer limits of the volume where one would likely find the electron (the electron will be in that volume ~90% of the time)
There are many types of orbitals, each denoted with a different letter, and with a different shape Objects in quantum mechanics cannot entirely be thought of as particles occupying a definite point in space
Instead, objects in quantum mechanics are described by both a particle, and an associated wave, described by the particle's wave function
The particle's existence is "spread out" over the wave function that, when squared, gives the probability that one might find the particle at a given point
Due to the Uncertainty Principle, while the object's final position is probabilistic, it behaves like a wave (exhibiting interference), or a particle (having a position that is observable to a certain extent), depending on which property is being observed (the interference pattern or the position). Neither behavior is a full description of the quantum object, and we must look at both in combination to fully understand it.
In an electron wave function, its wavelength, when multiplied by an integer n, gives the circumference of the electron's orbital (if the wavelength were any greater, the wave would cancel itself out) Above is a chart showing the first 3 types of orbitals, with their respective name (s, p, and d). Note that while the orientations and sizes of the orbitals may change, the shapes of a certain type of orbital generally remain the same. Niels Bohr

Pioneer in the development of the model of the atom Above is an example of an allowed electron configuration as well an example of an unallowed electron configuration, under my Exclusion Principle. The second example is unallowed, since 2 electrons sharing the same orbital have the same spin, making all of their quantum numbers equal, and violating the Exclusion Principle. It turns out that in quantum mechanics, the world works much like a digital signal. Note how the digital signal (red) makes sharp jumps between allowed values (in this case, the gridlines), while the analog signal continuously varies. This is also how a quantized value behaves; it must always be an integral multiple of some discrete fundamental unit, or quantum. Anatomy of an electron configuration The emission spectra of various elements Max Planck Founder of quantum mechanics, he conceived the idea that electromagnetic radiation was emitted in quantized "packets", solving the black body problem that irked classical physicists. All-around awesome person The Black Body Problem A black body is a body that absorbs all incoming radiation and energy
It was used as a model for studying how an object would generally radiate energy when excited or energized, since one did not have to account for reflection or rejection of incoming energy in a specific object.
The black body problem was the problem of describing how a black body of a certain temperature emitted electromagnetic radiation (black body radiation)
classical models failed since they predicted that black body radiation would carry higher and higher energies into infinity, as frequency increased
This means that for any hot body (a body of a greater temperature than its surroundings), all the energy it carried would be emitted by higher-frequency (lower-wavelength) radiation, instantly cooling any hot object in an intense "flash" of high-frequency light. This became known as the "ultraviolet catastrophe."
This flaw was fixed by assuming light was emitted in quantized packets (photons), each carrying an amount of energy proportional to its frequency (see previous slide)
Since electromagnetic radiation was not emitted continuously, and since higher-frequency photons carried more energy, more energy was therefore required to stimulate the emission of a high-frequency photon (since discrete values operated on an "all-or-nothing" system, high-frequency photons that were emitted with a low energy were not allowed), making the emission of high-frequency photons less likely, adjusting for the increased energy carried by each such photon, and causing the net power of the emitted electromagnetic radiation to be finite. Wolfgang Pauli

One of the key contributors to quantum mechanics, through his Exclusion Principle Uncertainty Principle Certain physical quantities have a special relationship in which the greater precision to which one quantities is known, the lower precision to which the other can be known
An example is position and momentum: in order to observe the position of a particle, one must bombard it with other particles or collect it with a detector, thus changing its momentum, as well as "knocking" the particle from its previous position, and introducing uncertainty into the position measurement.
It is therefore impossible to know the exact position and momentum of a particle at any point in time My uncertainty principle has important implications to physics. For example, it is not longer useful for physics to try to predict the future position of a particle, since one can never have the required information for such as prediction anyway (where it was and where it was headed). Instead, quantum mechanics takes a statistical approach to physics; in this example, it predicts the probability that one might find the particle at a given point. A simple illustration of a quantum wave function Werner Heisenberg

Pioneer in quantum mechanics, key contributor to the uncertainty principle Visualization of the Aufbau Principle.The chain of arrows shows the order in which orbitals should be filled. Above are several possible electron configurations for Carbon. However, only the third one is correct, since it has the maximum number of parallel electron spins. Note how electrons are spread out amongst equal-energy orbitals (the 3 2p orbitals) to accommodate for this. This piece of pyrolytic carbon levitates above the magnets (golden bars) because it is diamagnetic, and is repelled by magnets In this illustration of a carbon atom, the valence electrons are "shielded" from the positive influence of the nucleus by the core electrons (the ones in the lower-energy shells), and are easier to remove A graph illustrating the ultraviolet catastrophe. Note how the classical theory (black) predicts that the energy emitted by electromagnetic radiation tends to infinity for short wavelengths (high frequency), while Planck's theory (colored) correctly predicts that energy emitted falls off to zero with higher frequencies. It is useful to think of light as being emitted in discrete "packets" of energy, known as photons Electrons in orbitals behave much like balls on a staircase, in that their energy levels may occupy one of the allowed discrete energy levels, but may not be between them, like how the balls may only rest on a step on the staircase, and not between steps
• The angular momentum quantum number (l) decides the shape of an orbital. It can take an integral value within the range 0 to (n-1). Therefore, the number of values it can take is n. It denotes the type of an orbital, with every value corresponding to a different orbital type and name (see chart below). It also denotes the energy "sub-level" of an orbital (within a certain shell, there can be many smaller, but still discrete energy levels "jumps" or differences). Finally, after nearly a century of development... The collaboration of countless scientists from all over the world... Using the principles you have just learned... We have a complete model of the atom. • The magnetic quantum number ( ) describes the orientation of the orbital. There are (2l+1) possible values for , which can take any integral value ranging from -l to l (including 0, which accounts for the extra possible value).
• The electron spin quantum number ( ), in conjunction with the Pauli Exclusion Principle, explains why only a certain number of electron will fit into a certain shell (energy level). It can take 2 values: and - (up and down). Electrons can be thought of as "spinning" around their own axes, as described by their spin quantum number.
• The spin of the electron influences the magnetic field associated with it, like it would for any moving, charged body
• Opposite spins cancel each other out, while parallel (equal) spins reinforce and complement each other

• The first 3 quantum numbers can completely specify and define an atomic orbital, and the fourth can identify a specific electron in an orbital Within a certain atom, or quantum system, no 2 electrons (or, more generally, fermions) may have all the same quantum numbers
This means that in a certain orbital, there may only be 2 electron, since can only take 2 values, and the other 3 values must be equal, since they specify that the electrons are in the same orbital
It explains why certain shells will only accept a certain number of electrons, since every shell is actually a combination of atomic orbitals, with the same principal quantum number What is Quantum Mechanics? Quantum mechanics is the study of what happens at extremely small scales... Like, on the scale of atoms and subatomic particles type small, and it turns out that in this realm, nothing seems to work like things do on larger scales (described by classical physics). And with the help of a few notable scientists in the development of quantum mechanics, we will see how quantum mechanics provides us with a complete picture of the atom. With that in mind, let's dive right in! By: Ryan Leung
Teacher: Ms. Hubbell
Date: October 29, 2012
Course: SCH3U3 Bibliography Information Images Atom Under Magnifying Glass. N.d. n.p. Web. 28 Oct 2012. <http://www.thefusiongroup.com/blogs/cornerstones/wp-content/uploads/2012/03/Atom-magnifying-glass1-e1330643299783.jpg>.

Carbon Bohr-Rutherford diagram. N.d. n.p. Web. 28 Oct 2012. <http://upload.wikimedia.org/wikipedia/commons/thumb/b/b3/Electron_shell_006_Carbon_-_no_label.svg/480px-Electron_shell_006_Carbon_-_no_label.svg.png >.

Chang, Raymond. Chemistry. 11. McGraw-Hill, 2011. Print.

Diamagnetic pyrolytic carbon levitation. N.d. n.p. Web. 28 Oct 2012. <http://upload.wikimedia.org/wikipedia/commons/c/c9/Diamagnetic_graphite_levitation.jpg>.

Discrete vs. continuous signal. N.d. n.p. Web. 28 Oct 2012. <http://upload.wikimedia.org/wikipedia/commons/thumb/9/9a/Digital.signal.svg/1000px-Digital.signal.svg.png>.

Electron configuration of helium. N.d. Oracle ThinkquestWeb. 28 Oct 2012. <http://library.thinkquest.org/10429/media/eleconfig/helium.gif>.

Helmenstine, Todd. Aufbau Principle Diagram. N.d. About.comWeb. 28 Oct 2012. <http://0.tqn.com/d/chemistry/1/0/j/g/econfiguration.jpg >.

Illustration of the ultraviolet catastrophe. N.d. n.p. Web. 28 Oct 2012. <http://upload.wikimedia.org/wikipedia/commons/a/a1/Blackbody->.

Light as Photons. N.d. n.p. Web. 28 Oct 2012. < http://www7b.biglobe.ne.jp/~kcy05t/zu/Photonpar.gif>.

Max Planck. 1933. n.p. Web. 28 Oct 2012. <http://upload.wikimedia.org/wikipedia/commons/c/c7/Max_Planck_1933.jpg>.

Niels Bohr. N.d. n.p. Web. 28 Oct 2012. <http://upload.wikimedia.org/wikipedia/commons/6/6d/Niels_Bohr.jpg>.

Possible carbon electron configurations. N.d. n.p. Web. 28 Oct 2012. <http://www.grandinetti.org/resources/Teaching/Chem121/Lectures/MultiElectronAtoms/CarbonPossible.gif>.

Quantum wave function visualization. N.d. n.p. Web. 28 Oct 2012. <http://abyss.uoregon.edu/~js/images/wave_function.gif>.

Various emission spectra. N.d. n.p. Web. 28 Oct 2012. <http://wolfstone.halloweenhost.com/Lighting/colvis_VariousEmissionSpectra.png>

Werner Heisenberg. N.d. n.p. Web. 28 Oct 2012. <http://upload.wikimedia.org/wikipedia/commons/f/f8/Bundesarchiv_Bild183-R57262,_Werner_Heisenberg.jpg>.

Wolfgang Pauli. N.d. n.p. Web. 28 Oct 2012. <http://upload.wikimedia.org/wikipedia/commons/f/fb/Wolfgang_Pauli_ETH-Bib_Portr_01042.jpg>.B Caltech: The mechanical universe - 50 – particles and waves. 1985. n.p. VHS (original form). < http://video.google.com/videoplay?docid=5063999801799851614>
Chang, Raymond. Chemistry. 11. McGraw-Hill, 2011. Print.
Faizi, Sarah. "Electronic configurations." Chem Wiki. N.p.. Web. 28 Oct 2012. <http://chemwiki.ucdavis.edu/Inorganic_Chemistry/Electronic_Configurations>.
"Introduction to Quantum Theory." Quantwiki. N.p.. Web. 28 Oct 2012. <http://www.quantiki.org/wiki/Introduction_to_Quantum_Theory>.
Markus, Ehrenfried. "What is spin?." . N.p.. Web. 28 Oct 2012. <http://www.markusehrenfried.de/science/physics/hermes/whatisspin.html>. The electron configuration specifies all the orbitals of a atom
Each orbital is written in this notation: nl
Where n is the principal quantum number, l is the magnetic quantum number, and x is the number of electrons in the orbital
The orbitals are written with the principal quantum number in ascending order (with exceptions)
For a given n, all subshells are written before moving onto the next n (with exceptions). Example: 1s 2s 2p 2p 2p 3s...
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