Chem 103, Section F0FUnit II - Quantum Theory and Atomic StructureLecture 7•The Quantum mechanical model of the atomLecture 7 - Atomic Structure•Reading in Silberberg-Chapter 7, Section 4 The Qunatum-Mechanical Model of the Atom2Lecture 7 - IntroductionThe discoveries of Planck, Einstein, de Broglie and Heisenberg, showed that matter and energy share common wave-like and particle-like properties•These discoveries lead to the development of a new field of physics called quantum mechanics3Wave-likePropertiesParticle-likePropertiesRefractionDiscrete energy valuesDiffractionMomentumInterferenceLecture 7 - IntroductionNiels Bohr’s model of the hydrogen was able to successfully explain the particle-like behavior of light that was emitted from excited hydrogen atoms.•Unfortunately, his model failed to work with atoms that contained more than one electron.4Lecture 7 - The Quantum Mechanical ModelQuantum mechanics provides a solution to this problem by considering the wave-like properties of the electron in an atom.•The quantum mechanical model of the atom was first proposed in 1926 Edwin Schrödinger.-E represents the energy of the electron-!, represents the wave function for the electron with energy E and describes its wave-like properties.-H is the Hamiltonian operator, which is applied to ! and gives the allowed energy states for the electron.5 H!!!!= E!Lecture 7 - The Quantum Mechanical ModelThe Hamiltonian operator, H , is a complex function of the electron’s potential and kinetic energies:•The good news is that you will never in this course be asked to solve Schrödinger’s equation.6 H = !h28"2me#2#x2+#2#y2+#2#z2$%&'()+ V x, y, z( )*+,-./K.E. P.E.Lecture 7 - The Quantum Mechanical ModelThe solutions to Schrödinger’s equation, however, do provide us with very valuable information about the physical and chemical properties of elements and compounds.•Later we will see how the quantum mechanical model of the atom helps us to understand why the elements are arranged the way they are on the periodic table.Edwin Schrödinger shared the 1933 Nobel Prize in Physics for his quantum mechancial model of the atom7Lecture 7 - The Quantum Mechanical ModelThe wave function, !, is also called an atomic orbital.•There is a different wave function for each of the different energy states that an electron can have in an atomWhile the wave function, !, has no physical meaning, the square of the wave function, !2, is does.•!2 is called the probability density and gives the probability that the electron will be found at a particular location in an atom.•As shown by Heisenberg’s uncertainty principle, we cannot know the exact location in an atom of the electron at any given time-the best we can do and calculate a probability of finding it at a particular location.8Lecture 7 - The Quantum Mechanical ModelThe probability density functions, !2, which correspond to the allowed energy values for the electron, provides us with valuable 3-dimensional pictures of where we can expect to find the electron in an atom9The probability density, !2, as a function of distance from the nucleus.Lecture 7 - The Quantum Mechanical ModelThe radial probability distribution is calculated from the probability density and tells us the probability of finding the electron at a particular distance from the nucleus.10The radial probability distribution is calculated by summing up the volume times !2, as you move out from the nucleus.Lecture 7 - The Quantum Mechanical ModelCalculating the radial probability distribution is analogous to collecting apples from the ground under an apple tree.11The radial probability distribution for the apples is obtained by collecting the apples in each ring into a separate box, and then counting the apples in each boxLecture 7 - The Quantum Mechanical ModelThe surface of an atomic orbital is often represented by the surface within which there is a 90% chance of finding the electron.•This is the way we will be picturing at the atomic orbitals.12Lecture 7 - The Quantum Mechanical ModelThe spherical atomic orbital described so far is the ground state orbital in the hydrogen atom.•When the hydrogen atom is excited, the electron can transfer to a higher energy state, which will have a different wave function (atomic orbital).Each different atomic orbital is characterized by a set of numbers called quantum numbers.•These numbers determine the size, shape and orientation of the orbitals.•We will see that not all of the possible orbitals are spherical13Lecture 7 - The Quantum Mechanical ModelThe spherical atomic orbital described so far is the ground state orbital in a hydrogen atom.•When the hydrogen atom is excited, the electron can transfer to higher energy state with a different wave function (atomic orbital).Each different atomic orbital is characterized by a set of numbers called quantum numbers.•These numbers determine the size, shape and orientation of the orbitals.•We will see that not all of the orbitals are spherical14Lecture 7 - The Quantum Mechanical ModelThere are four types of quantum numbers, we will focus first on three of these:• n, is the principal quantum number and is related to the size of the orbital.-It is a positive integer, 1, 2, 3, …• l, is the angular momentum quantum number and is related to the shape of the orbital.-It has an integer that runs from 0 to (n-1), where n is the corresponding principal quantum number.• ml, is the magnetic quantum number and is related to the orientation of the orbital.-It is an integer that runs from -l to +l, where l is the corresponding angular momentum quantum number.15Lecture 7 - The Quantum Mechanical ModelFor the first three principal quantum numbers, n, Table 7.2 shows the possible values for the l and ml quantum numbers.16Lecture 7 - Question 117Which of the following lists gives all possible values for the magnetic quantum number, ml, when l = 2?A) 0B) 0, 1C) -1, 0, 1D) -2, 1, 0, 1, 2Lecture 7 - The Quantum Mechanical ModelOther terminology related to the quantum numbers:•The principal quantum number, n, designates the energy level or shell that an electron is in.•The angular momentum quantum number, l, designates the sublevel or subshell that an electron is in.-Letters are used to represent these sublevels:•For example, the sublevel with the quantum numbers n = 2 and l = 1, is called the 2p sublevel.18l = 0, sl = 1, pl = 2, dl = 3, fl = 4, gLecture 7 - The Quantum Mechanical