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**George Mason University General Chemistry 211 Chapter 7 **

Quantum Theory and Atomic Structure Acknowledgements Course Text: Chemistry: the Molecular Nature of Matter and Change, 7th edition, 2011, McGraw-Hill Martin S. Silberberg & Patricia Amateis The Chemistry 211/212 General Chemistry courses taught at George Mason are intended for those students enrolled in a science /engineering oriented curricula, with particular emphasis on chemistry, biochemistry, and biology The material on these slides is taken primarily from the course text but the instructor has modified, condensed, or otherwise reorganized selected material. Additional material from other sources may also be included. Interpretation of course material to clarify concepts and solutions to problems is the sole responsibility of this instructor. 4/14/2017

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**Quantum Theory of The Atom**

How are electrons distributed in space? What are electrons doing in the atom? The nature of the chemical bond must first be approached by a closer examination of the electrons Electrons are involved in the formation of chemical bonds between atoms Quantum theory explains more about the electronic structure of atoms 4/14/2017 2

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**Quantum Theory of The Atom**

Origin of Atomic Theory When burned in a flame metals produce colors characteristic of the metal This process can be traced to the behavior of electrons in the atom 4/14/2017

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**Quantum Theory of The Atom**

Emission (line) Spectra of Some Elements When elements are heated in a flame and their emissions are passed through a prism, only a few color lines exist and are characteristic for each element. Atoms emit light of characteristic wavelengths when excited (heated) 4/14/2017

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**Quantum Theory of The Atom**

Electrons and Light: Wave Nature of Light Light moves (propagates) along as a wave (similar to ripples from a stone thrown in water) Light consists of oscillations of electric and magnetic fields that travel through space All electromagnetic radiation Visible light, Microwaves, Radio Waves, Ultraviolet Light, X-rays, Infrared Light consists of energy propagated by means of electric and magnetic fields that alternately increase and decrease in intensity as they move through space 4/14/2017 2

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**Quantum Theory of The Atom**

A Light Wave is Propagated as an Oscillating Electric Field (Energy) The Wave properties of electromagnetic radiation are described by two independent variables: wavelength and frequency Crest Crest = wavelength, Distance from crest to crest = c/ = frequency = Speed light / wavelength c = speed of electromagnetic radiation (3 x 108 m/s) 4/14/2017

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**Quantum Theory of The Atom**

Wave Nature of Light Wavelength – the distance between any two adjacent identical points (crests) in a wave (given the notation (lamba) Frequency – number of wavelengths that pass a fixed point in one unit of time (usually per second, given the notation (). The common unit of frequency is the hertz (Hz) = 1 cycle per second (1/sec) Propagation (Velocity) of an Electromagnetic wave is given as c = (1/sec * m) = m/sec c = velocity of light = 3.0 x 108 m/s in a vacuum c is independent of or in a vacuum 4/14/2017

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**Quantum Theory of The Atom**

Relationship Between Wavelength and Frequency Wavelength () and frequency () are inversely proportional = c/ (frequency = Speed light / wavelength) 4/14/2017

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**The electromagnetic spectrum**

High Frequency () Low High Energy (E) Low Short Wavelength () Long 4/14/2017

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**Distinction between Energy & Matter**

At the macro scale level – everyday life – energy (waves) & matter (particles) behave differently Waves (Energy) Particles (Matter) Wave passing from air to water is refracted (bends at an angle and slows down). The angle of refraction is a function of the density. White light entering a prism is dispersed into its component colors – each wavelength is refracted slightly differently A particle entering a pond moves in a curved path downward due to gravity and slows down dramatically because of the greater resistance (drag) of the water A wave is diffracted through a small opening giving rise to a circular wave on the other side of the opening. When a collection of particles encounter a small opening, they continue through the opening on a straight line along their original path (until gravity pulls them down If light waves pass through two adjacent slits, the emerging waves interact (interference). Constructive Interference – Crests coincide in phase Destructive Interference – Crests coincide with troughs, cancelling out. Particles passing through adjacent openings continue on straight paths, some colliding with each other moving at different angles 4/14/2017

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**Distinction between Energy & Matter**

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**Distinction between Energy & Matter**

The diffraction pattern caused by waves passing through two adjacent slits Constructive and destructive interference occurs as water waves viewed from above pass through two adjacent slits in a ripple tank As light waves pass through two closely spaced slits, they also emerge as circular waves and interfere with each other They create a diffraction (interference) pattern of bright regions where crests coincide in phase and dark regions where crests meet troughs (out of phase) cancelling each other out 4/14/2017

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**Quantum Theory of The Atom**

Quantum Effects – Wave-Particle Duality Wave-Particle Duality is a central concept in Chemistry & Physics All matter and energy exhibit both wave-like and particle-like properties Duality applies to: macroscopic (large scale) objects microscopic objects (atoms and molecules) quantum objects (elementary particles – protons, neutrons, quarks, mesons) As atomic theory evolved, matter was generally thought to consist of particles At the same time, light was thought to be a wave 4/14/2017

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**Quantum Theory of The Atom**

Quantum Effects - Wave-Particle Duality Christiaan Huygens proposed the wave theory of light Huygen’s wave theory was displaced by Isaac Newton’s view that light consisted of a beam of particles In the early 1800s Young and Fresnel showed that light, like waves, could be diffracted and produce interference patterns, confirming Huygen’s view In the late 1800s James Maxwell developed equations, later verified by experiment, that explained light as a propagation of electromagnetic waves At the turn of the 20th century, physicists began to focus on 3 confounding phenomena to explain Wave-Particle Duality Black Body radiation The Photoelectric Effect Atomic Spectra 4/14/2017 2

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**Quantum Theory of The Atom**

Quantum Effects – Wave-Particle Duality Black Body Radiation - As the temperature of an object changes, the intensity and wavelength of the emitted light from the object changes in a manner characteristic of the idealized “Blackbody” in which the temperature of the body is directly related to the wavelengths of the light that it emits In 1901, Max Planck developed a mathematical model that reproduced the spectrum of light emitted by glowing objects His model had to make a radical assumption (at that time): A given vibrating (oscillating) atom can have only certain quantities of energy and in turn can only emit or absorb only certain quantities of energy 4/14/2017 2

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**Quantum Theory of The Atom**

Quantum Effects – Wave-Particle Duality Planck’s Model: E (Energy of Radiation) v (Frequency) n (Quantum Number) = 1,2,3… h (Planck’s Constant, a Proportionality Constant) 6.626 x J s) 6.626 x kg m2/s Atoms, therefore, emit only certain quantities of energy and the energy of an atom is described as being “quantized” Thus, an atom changes its energy state by emitting (or absorbing) one or more quanta 4/14/2017

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**Quantum Theory of The Atom**

Wave-Particle Duality – The Photoelectric Effect The Planck model views emitted energy as waves Wave theory associates the energy of the light with the amplitude (intensity) of the wave, not the frequency (color) Wave theory predicts that an electron would break free of the metal when it absorbed enough energy from light of any color (frequency) Wave theory would also imply a time lag in the flow of electric current after absorption of the radiation Both of these observations are at odds with the Photoelectric Effect 4/14/2017 2

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**Quantum Theory of The Atom**

Wave-Particle Duality – The Photoelectric Effect Photoelectric Effect Flow of electric current when monochromatic light of sufficient frequency shines on a metal plate Electrons are ejected from the metal surface, only when the frequency exceeds a certain threshold characteristic of the metal Radiation of lower frequency would not produce any current flow no matter how intense Violet light will cause potassium to eject electrons, but no amount of red light (lower frequency) has any effect Current flows immediately upon absorption of radiation 4/14/2017 2

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**Quantum Theory of The Atom**

Wave-Particle Duality – The Photoelectric Effect Einstein resolved these discrepancies He reasoned that if a vibrating atom changed energy from nhv to (n-1)hv, this energy would be emitted as a quantum (hv) of light energy he called a photon He defined the photon as a Particle of Electromagnetic energy, with energy E, proportional to the observed frequency of the light. The energy (hv) of an impacting photon is taken up (absorbed) by the electron and ceases to exist The Wave-Particle Duality of light is regarded as complimentary views of wave and particle pictures of light 4/14/2017

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**Quantum Theory of The Atom**

In 1921 Albert Einstein received the Nobel Prize in Physics for discovering the photoelectric effect Electrons in metals exist in different and specific energy states Photons whose frequency matches or exceeds the energy state of the electron will be absorbed If the photon energy (frequency) is less than the electron energy level, the photon is not absorbed The electron moves to a higher energy state and is ejected from the surface of the metal The electrons are attracted to the positive anode of a battery, causing a flow of current 4/14/2017

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Practice Problem Light with a wavelength of 478 nm lies in the blue region of the visible spectrum. Calculate the frequency of this light Speed of Light = 3 x 108 m/s Ans: 4/14/2017

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Practice Problem The green line in the atomic spectrum of Thallium (Tl) has a wavelength of 535 nm. Calculate the energy of a photon of this light? 4/14/2017

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Practice Problem At its closest approach, Mars is 56 million km from earth. How many minutes would it take to send a radio message from a space probe of Mars to Earth when the planets are at this closest distance? 4/14/2017

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**Quantum Theory of The Atom**

Atomic Line Spectra When light from “excited” (heated) Hydrogen atoms or other atoms passes through a prism, it does not form a continuous spectrum, but rather a series of colored lines (Line Spectra) separated by black spaces The wavelengths of these lines are characteristic of the elements producing them The spectra lines of Hydrogen occur in several series, each series represented by a positive integer, n 4/14/2017

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**Quantum Theory of The Atom**

4/14/2017

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**Quantum Theory of The Atom**

Atomic Line Spectra In 1885, J. J. Balmer showed that the wavelengths, l, in the visible spectrum of Hydrogen could be reproduced by a Rydberg Equation where: R = The Rydberg Constant = wavelength of the spectral line n1 & n2 are positive integers and n2 > n1 4/14/2017

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**Quantum Theory of The Atom**

Atomic Line Spectra For the visible series of lines the value of n1 = 2 The known wavelengths of the four visible lines for hydrogen correspond to values of n2 = 3, n = 4, n = 5, and n = 6 The Rydberg equation becomes The above equation and the value of ”R” are based on “data” rather than theory The following work of Niels Bohr makes the connection between the “data” model and Theory with n2 = 3, 4, 5, 6…. 4/14/2017

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**Quantum Theory of The Atom**

Bohr Theory of the Hydrogen Atom Prior to the work of Niels Bohr, the stability of the atom could not be explained using the then-current theories, i.e., How can electrons (e-) lose energy and remain in orbit? Bohr in 1913 set down postulates to account for (1) the stability of the hydrogen atom and (2) the line spectrum of the atom Energy level postulate: An electron can have only specific energy levels in an atom Transitions between energy levels: An electron in an atom can change energy levels by undergoing a “transition” from one energy level to another 4/14/2017 2

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**Quantum Theory of The Atom**

Energy x (J/atom) Bohr Theory Transitions of the Electron in the Hydrogen Atom 4/14/2017

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**Quantum Theory of The Atom**

Bohr Theory of the Hydrogen Atom Bohr’s Postulates Bohr derived the following formula for the energy levels of the electron in the hydrogen atom 4/14/2017 2

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**Quantum Theory of The At**

Bohr Theory of the Hydrogen Atom Bohr’s Postulates For the Hydrogen atom, Z = 1 For the energy of the ground state (n =1) 4/14/2017

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**Quantum Theory of The Atom**

Bohr Theory of the Hydrogen Atom Bohr’s Postulates When an electron undergoes a transition from a higher energy level (ni) to a lower one (nf), the energy is emitted as a photon 4/14/2017 2

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**Quantum Theory of The Atom**

Bohr Theory of the Hydrogen Atom Bohr’s Theory vs. Rydberg Data model If we make a substitution into the previous equation that states the energy of the emitted photon, h, equals hc/ Thus, from the classical relationships of charge and motion combined with the concept of discreet energy levels – theory matches data versus Bohr (theory) Rydberg (data) 4/14/2017 2

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Practice Problem From the Bohr model of the Hydrogen atom we can conclude that the energy required to excite an electron from n = 2 to n = 3 is ___________ the energy to excite an electron from n = 3 to n = 4 a. less than b. greater than c. equal to d. either equal to or less than e. either equal to or greater than Greater Than 4/14/2017

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Practice Problem An electron in a Hydrogen atom in the level n = 5 undergoes a transition to level n = 3. What is the wavelength of the emitted radiation? (R = x J) Note: For computation of frequency and wavelength the negative sign of the energy value can be ignored 4/14/2017

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**Quantum Theory of The Atom**

Bohr Theory of the Hydrogen Atom Bohr’s Postulates Bohr’s theory explains not only the emission of light, but also the absorption of light When an electron falls from n = 3 to n = 2 energy level, a photon of red light (wavelength, 685 nm) is emitted When red light of this same wavelength shines on a hydrogen atom in the n = 2 level, the energy is gained by the electron that undergoes a transition to n = 3 4/14/2017 2

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**Quantum Theory of The Atom**

Quantum Mechanics Bohr’s theory established the concept of atomic energy levels but did not thoroughly explain the “wave-like” behavior of the electron Current ideas about atomic structure depend on the principles of quantum mechanics A theory that applies to subatomic particles such as electrons Electrons show properties of both waves and particles 4/14/2017 2

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**Quantum Theory of The Atom**

Quantum Mechanics The first clue in the development of quantum theory came with the discovery in 1923 by Louis de Broglie who reasoned that if light exhibits particle aspects, perhaps particles of matter show characteristics of waves He postulated that a particle with mass, m and a velocity, v has an associated wavelength, The equation = h/mv is called the de Broglie relation 4/14/2017 2

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**Quantum Theory of The Atom**

Quantum Mechanics If matter has wave properties, why are they not commonly observed? The de Broglie relation shows that a baseball (0.145 kg) moving at a velocity of about 60 mph (27 m/s) has a wavelength of about 1.7 x m. This value is so incredibly small that such waves cannot be detected Electrons have wavelengths on the order of a few picometers (1 pm = m) 4/14/2017 2

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Practice Problem At what speed (v) must an neutron (1.67 x kg) travel to have a wavelength of 10.0 pm? 1 pm (picometer) = m (meter) 4/14/2017

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**Quantum Theory of The Atom**

Quantum mechanics is the branch of physics that mathematically describes the wave properties of submicroscopic particles We can no longer think of an electron as having a precise orbit in an atom To describe such an orbit would require knowing its exact position and velocity, i.e., its motion (mv) In 1927, Werner Heisenberg showed (from quantum mechanics) that: It is impossible to simultaneously measure the present position while also determining the future motion of a particle, or of any system small enough to require quantum mechanical treatment 4/14/2017 2

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**Quantum Theory of The Atom**

Max Born stated in his Nobel Laureate speech: To measure space coordinates and instants of time, rigid measuring rods and clocks are required On the other hand, to measure momenta and energies, devices are necessary with movable parts to absorb the impact of the test object and to indicate the size of its momentum (mass x velocity) Paying regard to the fact that quantum mechanics is competent for dealing with the interaction of object and apparatus, it is seen that no arrangement is possible that will fulfill both requirements simultaneously 4/14/2017

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**Quantum Theory of The Atom**

Mathematically, the uncertainty relation between position and momentum, i.e., the variables, arises due to the fact that the expressions of the wavefunction in the two corresponding bases (variables) are Fourier Transforms of one another According to the Uncertainty Principle of Heisenberg, if the two operators representing a pair of variables do not commute, then that pair of variables are mutually complementary, which means they cannot be simultaneously measured or known precisely In the mathematical formulation of quantum mechanics, changing the order of the operators changes the end result, i.e., the operators are non- commuting, and are subject to similar uncertainty limits 4/14/2017

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**Quantum Theory of The Atom**

Quantum Mechanics Heisenberg’s uncertainty principle is a relation that states that the product of the uncertainty in position (Dx) and the uncertainty in momentum (mDvx) of a particle can be no smaller than: When m is large (for example, a baseball) the uncertainties are very small, but for electrons, high uncertainties disallow defining an exact orbit 4/14/2017 2

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Practice Problem Heisenberg's Uncertainty Principle can be expressed mathematically as: Where x is the uncertainty in Position p (= mv) is the uncertainty in Momentum h is Planck's constant (6.626 x kg m2/s) What would be the uncertainty in the position (∆x) of a fly (mass = g) that was traveling at a velocity of m/s if the velocity has an uncertainty of 2.72%? Planck’s Constant h = x J s 1 J = 1 kg m2/s2 h = x kg m2/s 4/14/2017

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**Quantum Theory of The Atom**

Quantum Mechanics Acceptance of the dual nature of matter and energy (E = mc2) and the “Uncertainty” Principle culminated in the field of Quantum Mechanics: Wave Nature of objects on the Atomic Scale Erwin Schrodinger developed quantum mechanical model of the Hydrogen atom, where An Atom has certain allowed quantities of energy An Electron’s behavior is wavelike, but its exact location is impossible to know The Electron’s Matter-Wave occupies 3-dimentional space near nucleus The Matter-Wave experiences continuous, but varying influence from the nuclear charge 4/14/2017

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**Quantum Theory of The Atom**

Quantum Mechanics Schrodinger Equation: H = = Energy of the atom = Wave Function H = Hamiltonian Operator – Mathematical operations that when carried out on a particular wave yields the allowed energy value Each solution of the wave equation is associated with a given “atomic orbital”, which bears no resemblance to an orbit in the Bohr model An “Orbital” is a mathematical function, which like a Bohr Orbit, represents a particular energy level of the orbiting electron, but it has no direct physical meaning 4/14/2017

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**Quantum Theory of The Atom**

Quantum Mechanics Heisenberg's uncertainty principle says we cannot precisely define an electron’s orbit The wave function (atomic orbital) has no direct physical meaning The square of the wave function, 2, however, is defined as the probability density, a measure of the probability that the electron can be found within a particular tiny volume of the atom 4/14/2017 2

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**Quantum Theory of The Atom**

Probability of Finding an Electron in a Spherical Shell About the Nucleus 4/14/2017

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**Quantum Theory of The Atom**

Quantum Numbers and Atomic Orbitals According to quantum mechanics each electron is described by 4 quantum numbers Principal Quantum Number (n) Angular Momentum Quantum Number (l) Magnetic Quantum Number (ml) Spin Quantum Number (ms) The first three quantum numbers define the wave function of the electron’s atomic orbital The fourth quantum number refers to the spin orientation of the 2 electrons that occupy an atomic orbital 4/14/2017 2

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**Quantum Theory of The Atom**

Quantum Numbers and Atomic Orbitals The Principal Quantum Number (n) represents the “Shell Number” in which an electron “resides” It represents the relative size of the orbital Equivalent to periodic chart Period Number Defines the principal energy of the electron The smaller “n” is, the smaller the orbital The smaller “n” is, the lower the energy of the electron n can have any positive value from 1, 2, 3, 4 … (Currently, n = 7 is the maximum known) 4/14/2017 2

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**Quantum Theory of The Atom**

Quantum Numbers and Atomic Orbitals (Con’t) The Angular Momentum Quantum Number (l) distinguishes “sub shells” within a given shell Each main “shell,” designated by quantum number “n,” is subdivided into: l = n - 1 “sub shells” (l) can have any integer value from 0 to n - 1 The different “l” values correspond to the s, p, d, f designations used in the electronic configuration of the elements Letter s p d f l value 4/14/2017 2

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**Quantum Theory of The Atom**

Quantum Numbers and Atomic Orbitals (Con’t) The Magnetic Quantum Number (ml) defines atomic orbitals within a given sub-shell Each value of the angular momentum number (l) determines the number of atomic orbitals For a given value of “l,” ml can have any integer value from -l to +l ml = -l to +l ( ) Each orbital has a different shape and orientation (x, y, z) in space Each orbital within a given angular momentum number sub shell (l) has the same energy 4/14/2017 2

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**Quantum Theory of The Atom**

Quantum Numbers and Atomic Orbitals (Con’t) The Spin Quantum Number (ms) refers to the two possible spin orientations of the electrons residing within a given atomic orbital Each atomic orbital can hold only two (2) electrons Each electron has a “spin” orientation value The spin values must oppose one another The possible values of ms spin values are: +1/2 and –1/2 4/14/2017 2

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**Summary of Quantum Numbers**

Name Symbol Permitted Values Property principal n positive integers (1, 2, 3, …) orbital energy (size) angular momentum l integers from to n -1 orbital shape The l values , 1, 2, and 3 correspond to s, p, d, and f orbitals, respectively magnetic ml integers from -l to 0 to +l orbital (x,y,z) orientation spin ms +1/2 or -1/2 e- spin orientation 4/14/2017

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**Quantum Numbers and Atomic Orbitals**

ml = 0 l=0 (1s) n=1 n=2 ml =0 l=0 (2s) l=1 (2p) ml = n=3 l=0 (3s) l=2 (3d) l=1 (3p) The Hierarchy of Quantum Numbers for Atomic Orbitals n=4 ml = 0 l=0 (4s) l=1 (4p) l=2 (4d) l=3 (4f) n=5 ml = 0 l=0 (5s) l=1 (5p) l=2 (5d) l=3 (5f) Note: n > 7 & l > 3 not defined for the current list of elements in the Periodic Table n=6,7 ml = 0 l=0 (6s,7s) l=1 (6p,7p) l=2 (6d) l=3 (6f) 4/14/2017

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**Quantum Numbers and Atomic Orbitals**

Using calculated probabilities of electron “position,” the shapes of the orbitals can be described The s (n = 1) sub shell orbital (there is only one) is spherical The p (n = 2) sub shell orbitals (there are three) are dumbbell shape The d (n = 3) sub shell orbitals (there are five) are a mix of cloverleaf (pear-shaped lobes) and dumbbell shapes 4/14/2017 2

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**Quantum Numbers and Atomic Orbitals**

Cross-sectional Representations of the Probability Distributions of “s” Orbitals (spherical) 4/14/2017

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**Quantum Numbers and Atomic Orbitals**

Cutaway Diagrams Showing the Spherical Shape of “s” Orbitals 4/14/2017

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**Quantum Numbers and Atomic Orbitals**

Radial Probability Distribution of the Three 2p Orbitals (dumbell shapes) n = l = 2 – 1= 1 (p) ml = 4/14/2017

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**Quantum Numbers and Atomic Orbitals**

Radial Probability Distribution of the Five 3d Orbitals (Cloverleaf & Dumbells) n = l = n -1 = 3 – 1= 2 (d) ml = 4/14/2017

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**Quantum Numbers and Atomic Orbitals**

Radial Probability Distribution of the Seven 4f Orbitals n = l = 4 – 1= 3 (f) ml = 4/14/2017

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**Quantum Numbers and Atomic Orbitals**

Orbital Energies of the Hydrogen Atom 4/14/2017

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Practice Problems If the n quantum number of an atomic orbital is 4, what are the possible values of the l quantum number? Ans: (l) can have any integer value from 0 to n –1 l = n - 1 = 4 – 1 = 3 Values of l = 4/14/2017

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Practice Problem If the l quantum number is 3, what are the possible values of ml? Ans: ml can have any integer value from -l to +l Since l = 3 ml = 4/14/2017

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Practice Problem State which of the following sets of quantum numbers would be possible and which impossible for an electron in an atom? a. n = 0, l=0, ml = 0, ms = +1/2 b. n = 1, l=0, ml = 0, ms = +1/2 c. n = 1, l=0, ml = 0, ms = -1/2 d. n = 2, l=1, ml = -2, ms = +1/2 e. n = 2, l=1, ml = -1, ms = +1/2 Ans: Possible Impossible b c e a “n ” must be positive 1, 2, 3... d ml can only be 4/14/2017

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Summary Equations Light c = c = 3 x 108 m/s Planck’s Model Photoelectric Balmer Rydberg Bohr Model Bohr Postulate De Broglie Heisenberg 4/14/2017

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