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The Wave Nature of Matter de Broglie, the Uncertainty Principle, and Quantum Mechanics Dual Nature of Matter?! As a result of Planck’s and Einstein’s work, light was found to have certain characteristics of particulate matter No longer purely wavelike But is the opposite also true? In 1923, Prince Louis de Broglie of France had an idea He proposed that everything propagates like a wave, and that everything interacts like a particle de Broglie and Wave Nature of Matter de Broglie rewrote Einstein’s formula for the momentum of a photon and applied it to any object NOT traveling at the speed of light: h λ= mν λ h m v = = = = wavelength, meters Planck’s constant mass, kg velocity, m/s De Broglie Waves h λ= mν Using de Broglie’s equation and the fact that Bohr had already reported the speed of an electron as 2.2 x 106 m s-1, we can calculate the wavelength of an electron: λ= This equation shows that the more massive the object, the smaller its associated wavelength and vice versa! 6.6 x 10-34 kg m2 s-1 (9.1 x 10-31 kg)(2.2 x 106 m s-1) For example, a thrown baseball only has a wavelength of about 10-35 meters! Hence the reason why on the macroscopic level, objects do not seem to act as waves! Wave-Particle Duality of Nature • It is important to remember that Planck’s constant is very tiny • Roughly speaking, this means that in our everyday world, quantum effects like the wave-particle duality make a difference only in the 34th decimal place when predicting the behavior of a moving baseball • Bottom line is large objects obey Newton’s laws and subatomic particles defy classical physics and obey quantum mechanics Interpretation of de Broglie’s Work Electrons bound to the nucleus are similar to standing waves Standing waves do not propagate through space Standing waves are fixed at both ends Think of a guitar or violin A string is attached to both ends and vibrates to produce a musical tone Waves are “standing” because they are stationary – the wave does not travel along the length of the string The Dual Nature of Electrons and Its Limitations The velocity of an electron is related to its wave nature The position of an electron is related to its particle nature Particles have well-defined positions; waves do not We are unable to observe an electron simultaneously as both a particle and a wave This is because in order to observe an electron, one would need to hit it with photons having a very short wavelength and high frequency If one were to hit an electron, it would cause the motion and the speed of the electron to change Lower energy photons would have a smaller effect but would not give precise information The Electron and Complementary Properties As a result of this, we cannot simultaneously measure its position AND velocity Position and velocity are complementary properties Complementary properties mean that the more you know about one, the less you know about the other Heisenberg Uncertainty Principle and Complementary Properties A physicist named Werner Heisenberg stated that “[t]here is a fundamental limitation on how precisely we can know both the position and momentum of a particle at a given time” In other words, it is impossible to know both the velocity and location of an electron at the same time He developed the following relationship: h ∆x ∆(mv) > 4πm As the mass of an object gets smaller, the product of the uncertainty of its position (∆x) and speed (∆v) increase Again, How Can Something be Both a Particle and a Wave? Saying that an object is both a particle and a wave is saying that an object is both a circle and a square – a contradiction Complementary solves this problem An electron is observed as either a particle or a wave, but never both at once! Energies and Electrons – Introducing the Quantum Mechanical Model Many properties of an element depend on the energies of its electrons Remember, position and velocity of the electron are complementary properties Since velocity is directly related to energy via ½ mv2, position and energy are also complementary properties Therefore, we can specify the energy of the electron precisely, but not its location at a given instant Instead, the electron’s position is described as a probable location where the electron is likely to be found called an orbital Enter Schrӧdinger… Erwin Schrödinger and the Quantum Model of the Atom Schrödinger developed an equation to describe the behavior and energies of electrons in atoms ℎ2 𝑑2Ψ − + 𝑉Ψ = 𝐸Ψ 2 2 8𝜋 𝑚 𝑑𝑥 General equation: ĤΨ = 𝐸Ψ Ĥ = set of mathematical instructions called an “operator” that represent the total energy (kinetic and potential) of the electron within the atom Ψ = Wave function that describes the wavelike nature of the electron Orbitals Orbitals…What?! Orbitals are NOT circular orbits for electrons Orbitals ARE areas of probability for locating electrons Square of absolute value of the wave function gives a probability distribution Ψ2 Electron density maps (probability distribution) indicates the most probable distance from the nucleus Limitations of Orbitals Wave functions and probability maps DO NOT describe: How an electron arrived at its location Where the electron will go next When the electron will be in a particular location More on the Schrödinger Equation While the equation is too complicated to actually use in this class, we can still use the results! His equation is used to plot the position of the electron relative to the nucleus as a function of time The solution of the equation has demonstrated that E (energy) must occur in integer multiples (quanta) and that each electron can be described in terms of its quantum numbers What is a Quantum Number? Think that each electron has a specific “address” in the space around a nucleus An electrons “address” is given as a set of four quantum numbers Each quantum number provides specific information on the electrons location Electron Configuration Analogy state town house number street What are The 4 Quantum Numbers? State (energy level) - quantum number n Town (shape of orbital) - quantum number l Street (orbital room) - quantum number ml House number (electron spin) - quantum number ms Principal Quantum Number (n) Same as Bohr’s n Indicates probable distance from the nucleus Higher numbers = greater distance from nucleus Greater distance = less tightly bound = higher energy Ranges are integral values: 1, 2, 3, …. Angular Momentum Quantum Number (l) Indicates the number of subshells that a principal level contains It also tells the shape of the atomic orbitals Ranges with integral values from 0 to n - 1 for each principal quantum number n Value of l 0 1 2 3 4 Letter used s p d f g More on Angular Quantum Number, l Size of orbitals is defined as the surface that contains 90% of the total electron probability Orbitals of the same shape (s, for instance) grow larger as principal quantum number (n) increases # of nodes (areas in which there is zero electron probability) increase as well Why Do We Care About the Shape of the Orbitals? Covalent chemical bonds depend on the sharing of the electrons that occupy these orbitals Shape of overlapping orbitals determine the shape of the molecule! The s Orbital 1s The1s 2sorbital orbitalhas hasaamaximum maximumprobability probability The regionat ataaradius radiusof of1~Å3 from Å from nucleus. region thethe nucleus. 2s 1s 2s 0 1 2 3 4 5 Distance from nucleus, r (Å) The s orbital is a sphere Every level has one s orbital S-Electron Probability Distributions for First Three Energy Levels The lower the energy level an electron is located at, the higher chance it has of being found near the nucleus p sub-level (l = 1) p (x) y-axis z-axis p (y) x-axis There are three p orbitals: px, py and pz p (z) Comparison of p- and sElectron Probability Distributions Electrons of the s-orbital are found closer to the nucleus than p-orbital electrons d sub-level (l = 2) five clover-shaped orbitals seen in all energy levels n=3 and above Combined Orbitals - n = 1, 2 & 3 1s, 2s, 2p, 3s, 3p and 3d sublevels Comparison of d-, p- and sElectron Probability Distributions As electrons get to higher and higher energy levels, the harder it is to locate it because the radius of the orbital is greater f sub-level (l = 3) seven equal energy orbitals shape is not well-defined seen in all energy levels n=4 and above Magnetic Quantum Number (ml) Relates to the orientation of the orbital in space relative to an x, y, z plot 3-D orientation of each orbital Integral values from l to -l, including zero For example, if l = 1, then ml would have values of -1, 0, +1 Knowing all three quantum numbers provides us with a picture of all of the orbitals Magnetic Quantum Number Electron Spin Quantum Number (ms) Wolfgang Pauli added one additional quantum number that would allow only two electrons to be in an orbital This means that an orbital can hold only two electrons, and they must have opposite spins He also developed the Pauli Exclusion Principle: Spin can have two values, +1/2 and -1/2 "In a given atom no two electrons can have the same set of four quantum numbers" Therefore, each electron has its own specific location around the nucleus of the atom Orbitals and Their Energies For the hydrogen atom, the principal quantum number determines the energy of the orbital Electron closest to the nucleus (n=1) was lowest in energy – the ground state When an atom absorbs energy, electrons may move to higher energy levels –the excited state Things get a bit more complex where more than one electron is involved… Orbital Energies in Polyelectronic Atoms Must make approximations with quantum mechanical model to compensate for repulsions between electrons Variations in energy within the same quantum level En-s < En-p < En-d < En-f Orbital Energies Work on Quantum Numbers WS Using the Periodic Table to Predict Electron Locations Electron Configurations Using the Periodic Table to Predict Electron Locations Use the Aufbau principle “aufbau” This German for ‘build up or construct’ principle states that electrons are added one at a time to the lowest energy orbitals available until all the electrons of the atom have been accounted for If two or more orbitals exist at the same energy orbital, do not pair the electrons until you first have one electron per orbital The lone electrons will have the same direction of spin Called Hund’s Rule Hund’s Rule and Magnetism The existence of unpaired electrons can be tested for since each acts like a tiny electromagnet An atom that is paramagnetic is attracted to magnetic field Indicates the presence of unpaired electrons An atom that is diamagnetic is unaffected by a magnetic field Indicates that all electrons are paired The Aufbau Principle 1s2 2s2 2p6 3s2 3p6 3d10 Blue sublevels not occupied… Not enough e- 4s2 4p6 4d10 4f14 5s2 5p6 5d10 5f14 5g18 6s2 6p6 6d10 6f14 6g18 6h22 7s2 7p6 7d10 7f14 7g18 7h22 7i26 s The Periodic Table and Its Classification by Sublevels p H He d Li Be Na Mg K Ca Sc Rb Sr Y Ti V B C N O F Al Si P S Cl Ar Cr Mn Fe Co Ni Cu Zn Ga Ge As Se Br Zr Nb Mo Tc Ru Rh Pd Ag Cd In Cs Ba Lu Hf Fr Ra Lr Ne Ta W Re Os Ir Pt Au Hg Tl Sn Sb Te I Xe Pb Bi Po At Rn La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb f Kr Ac Th Pa U Np Pu AmCm Bk Cf Es Fm Md No Orbital Diagrams and Electron Configurations Electrons fill in order from lowest to highest energy The Pauli exclusion principle holds. An orbital can hold only two electrons Two electrons in the same orbital must have opposite signs You must know how many electrons can be held by each orbital 2 for s 6 for p 10 for d 14 for f Hund’s rule applies. The lowest energy configuration for an atom is the one having the maximum number of unpaired electrons for a set of orbitals By convention, all unpaired electrons are represented as having parallel spins with the spin “up” Writing Electron Configurations Electron configurations can be written for atoms or ions Start with the ground-state configuration for the atom For positively-charged cations, remove a number of the outermost electrons equal to the charge For negatively-charged anions, add a number of outermost electrons equal to the charge Practice! Full configurations O 1s 2 2s 2 2p 4 Ti 1s 2 2s 2 2p 6 3s 2 3p 6 4s 2 3d 2 Br 1s 2 2s 2 2p 6 3s 2 3p 6 4s 2 3d 10 4p 5 Noble gas configurations O [He] 2s 2 2p 4 Ti [Ar] 4s 2 3d 2 Br [Ar] 4s 2 3d 10 4p 5 More Practice! Example - Cl- First, write the electron configuration for chlorine: Cl 17 e1s 2 2s 2 2p 6 3s 2 3p 5 or [Ne] 3s 2 3p 5 Because the charge is 1-, add one electron Cl- 1s 2 2s 2 2p 6 3s 2 3p 6 or [Ne] 3s 2 3p 6 or [Ar] Orbital Notation • A simple method used to show only the electrons in the highest main energy level • • The d and f sublevels are not shown unless they are partially filled Indicates which orbitals are occupied and whether or not electrons are paired with other electrons in the orbitals Practice! Time for Electron Configuration Battleship!