Chapter 7: The Quantum-Mechanical Model of the Atom PDF

Fall 2022 CHAPTER 7 The Quantum-Mechanical Model of the Atom 1 The Wave Nature of Light Light is electromagnetic radiation – a type of energy comprised of an electric component and a magnetic one perpendicular to one another The Wave Nature of Light Waves can be descr...

Fall 2022 CHAPTER 7 The Quantum-Mechanical Model of the Atom 1 The Wave Nature of Light Light is electromagnetic radiation – a type of energy comprised of an electric component and a magnetic one perpendicular to one another The Wave Nature of Light Waves can be described by: Frequency (ν) in s-1 (or hertz, Hz) Wavelength (λ) in m The Wave Equation relates the two: c=νλ c = 3.00 x 108 m/s The Electromagnetic Spectrum Note: longer wavelengths to the left. 4 The Particle Nature of Light The photoelectric effect showed light has a particle nature as well as a wave nature One “particle” or “quantum” of light called a photon, and has energy E = hv h = Planck’s constant 5 A theory to explain these observations E = hν and because v = c/λ then E = hc/λ E: energy in J h: Planck's constant = 6.63 x 10-34 J s v: frequency in s-1 λ: wavelength in m c: speed of light in m/s = 3.00 x 108 m/s 6 Polling question Which of the following statements concerning light is not correct? A) Energy is directly proportional to frequency B) Energy is inversely proportional to wavelength C) Energy is directly proportional to wavelength D) Frequency is inversely proportional to wavelength 7 Let’s work a problem Atomic Spectroscopy and the Bohr Atom When an atom absorbs energy it often re-emits that energy as light The light emitted by a low-pressure gas only contains certain distinct wavelengths unique to a particular element H2 Na (street lamp) The light emitted by an element can be separated into its various wavelengths by passing it through a prism yielding a series of bright lines called an emission spectrum 9 Atomic Spectroscopy and the Bohr Atom 10 Atomic Spectroscopy and the Bohr Atom The classical models of the atom could not explain why emission spectra were observed instead of continuous spectra To explain experimental results, N. Bohr developed a model for the atom, worked well to explain H atom 11 The Bohr Atom 1. Electrons (e-) move in a circular orbits 2. The electron has only a fixed set of allowed orbits, of a certain potential energy 3. An electron can pass only from one allowed orbit to another 4. Passing from a higher to a lower orbit emits light The Bohr Atom The energy of an electron in a hydrogen-like atom in any one of these orbits is 2 𝑍 𝐸𝑛 = −2.18 × 10−18 𝐽 2 𝑛 where n = 1, 2, 3, … Z = atomic number The difference in energy levels, ∆E, nf and ni, (final and initial states) is 𝑍 2 𝑍 2 ∆𝐸𝑛 = −2.18 × 10−18 𝐽 2 − 2 𝑛𝑓 𝑛𝑖 Note: ∆E is negative if nf < ni ∆E is positive if nf > ni Let’s work a problem The Bohr Atom The emission spectrum of hydrogen contains some transitions in the visible region of the electromagnetic spectrum, but many are outside that range So what is wrong with the Bohr’s model? Cannot explain emission spectra from multi-electron atoms No reasoning for the fixed, quantized orbits The Bohr model served as an intermediate between the classical view of the electron and the Schrodinger model Experiments would later show wave/particle duality existed for matter as well as light, leading to the uncertainty principle 16 Double-Slit Experiment (Electrons) Dr. Quantum video https://www.youtube.com/watch?v=DfPeprQ7oGc& More advanced video https://www.youtube.com/watch?v=iVpXrbZ4bnU 17 The de Broglie Wavelength An electron (or any other particle) has a wave nature therefore can be described by a wavelength (De Broglie, 1924) ℎ 𝜆= 𝑚𝑣 18 Heisenberg’s Uncertainty Principle The wave nature and the particle nature of the electron are said to be complementary properties – the more you know about one, the less you know about the other This is illustrated by Heisenberg’s uncertainty principle ℎ (∆𝑥)(𝑚∆𝑣) ≥ 4𝜋 The position of an object is a particle property The velocity of an object is related to its wave nature For a typical electron in an atom, Δx is about the size of the atom itself 19 Heisenberg’s Uncertainty Principle Bohr atom: electrons occupy fixed orbits as known radii from the nucleus Updated quantum-mechanical atom: due to H.U.P. we can only determine the probability of finding the electron in a certain region around the nucleus Indeterminacy: Future electron positions can not be exactly calculated, but only described statistically 20 Quantum Mechanics and the Atom The Schrӧdinger equation allows us to solve for electron positions within an atom by giving a wavefunction (Ψ) ꞪΨ = EΨ Hamiltonian operator (Ɦ) involves solving a differential equation E is the total energy of the electron Solving for Ψ gives the wavefunction, called an orbital Squaring the wavefunction (Ψ2) gives position probability Particle in a 1-D box not required (p. 266-267) 21 Solutions to the Schrӧdinger Equation for the Hydrogen Atom Solving the S.E. for a hydrogen atom gives families of ψ solutions (many orbitals) with three variables (n, l and ml) that can take on multiple values Principal quantum number (n) − Mostly describes overall size of orbital Angular momentum quantum number (l) − Mostly describes shape of the orbital Magnetic quantum number (ml) − Mostly describes spatial orientation of the orbital 22 What do the wavefunctions look like mathematically? 23 Definitions of Quantum Numbers Principal quantum number: n n = 1, 2, 3,… (relates to energy and most probable distance of an electron from the nucleus) For the hydrogen atom, the energy of an electron in an orbital only depends on n Energy levels get closer together as n increases 24 Definitions of Quantum Numbers Angular momentum quantum number: l l = 0, 1, 2, … (n - 1) (relates to shape of the orbital) Value of l Letter Designation l=0 s l=1 p l=2 d l=3 f 25 Definitions of Quantum Numbers Magnetic quantum number: ml ml = - l, (-l + 1), (-l +2), …0…, (l – 2), (l – 1), l (relates to the orientation of the orbital) Value of l Values of ml l=0 0 l=1 -1, 0, 1 l=2 -2, -1, 0, 1, 2 l=3 -3, -2, -1, 0, 1, 2, 3 26 Shells and Sub-Shells 27 Shells and Sub-Shells The number of orbitals in any level is equal to n2 − When n = 1, 1 “s” only (# of orbitals = 12 = 1) − When n = 2, 1 “s” + 3 “p” (# of orbitals = 22 = 1+3 = 4) − When n = 3, 1 “s” + 3 “p” + 5 “d” (# of orbitals = 32 = 1+3+5 = 9) − When n = 4, 1 “s” + 3 “p” + 5 “d” + 7 “f” (# of orbitals = 42 = 1+3+5+7 = 16) 28 Polling question The sets of quantum numbers are each supposed to specify an orbital. One set, however, is incorrect and can not exist. Which one? A. n = 3; l = 0; ml = 0 B. n = 1; l = 0; ml = 0 C. n = 2; l = 1; ml = -1 D. n = 4; l = 1; ml = -2 29 The hydrogen atom Neutral H atom has 1 electron so what orbital would it be “found in”? For a H atom, electron potential energy is lowest at n = 1 Ground state configuration of the H atom: Excited states are also possible, but “relaxation” to ground state would occur quickly 30 What do the wavefunctions look like mathematically? 31 Shapes of Atomic Orbitals s orbitals (l = 0) The lowest energy orbital is the spherically symmetric 1s orbital 32 Shapes of Atomic Orbitals We typically draw orbitals to show >90% probability within a given space Remember: orbitals are not physical objects, they are mathematical functions that represent probability 33 Shapes of Atomic Orbitals Ψ2 alone is not overly useful since it only describes probability at one point at a specific radius Radial probability takes into account total probability of finding the electron at any point along the given radius (like an imaginary spherical surface) 34 Shapes of Atomic Orbitals The 2s orbital is also spherical but has a radial node where the probability is exactly 0. 35 Shapes of Atomic Orbitals The 3s orbital is spherical and has two radial nodes (notice a pattern?) 36 Shapes of Atomic Orbitals p orbitals (l = 1) Each principal level with n ≥ 2 contains three p orbitals (ml = -1, 0, +1) The p orbitals are not spherically symmetric like the s orbitals − There are two lobes of electron density, one on each side of the nucleus separated by angular node 37 Phases of Atomic Orbitals Why do the figures for the p orbitals have lobes with different colours? Phase of atomic orbitals is important in bonding (see Chapter 10) 38 Shapes of Atomic Orbitals 2p orbitals have no radial nodes 3p orbitals have 1 radial node (notice a pattern)? Number of radial nodes = n – l – 1 Don’t confuse radial and angular nodes Polling question Which of the following radial probability diagrams could show a 3d orbital? 40 Shapes of Atomic Orbitals d orbitals (l = 2) 𝑑𝑥 2 −𝑦2 𝑑𝑧 2 Each principal level with n ≥ 3 contains five d orbitals (ml = -2, -1, 0, +1, +2) 𝑑𝑥𝑦 𝑑𝑥𝑧 𝑑𝑦𝑧 41 Orbital Energy Diagram for one-electron atom 42 Electron Spin and the Pauli Exclusion Principle The electron configuration of hydrogen may be represented by an orbital diagram as follows: H ↿ 1s The direction of the arrow indicates electron spin Electron spin was first demonstrated by the Stern-Gerlach experiment 43 Electron Spin For atoms with an odd number of electrons, the atom experiences a force when placed in a magnetic field. This is best explained using the particle nature of the electron if the electron can spin. This spin induces a magnetic field. The spin is described by a fourth quantum number, the spin quantum number (ms) with values of +½ or -½ 44 Pauli Exclusion Principle No two electrons in an atom can have the same four quantum numbers. Recall that n, l, ml, specify a particular orbital. Therefore ms limits the number of electrons in a given orbital to two, one with ms = +½, the other ms = -½. Helium is the simplest multielectron element. Its ground state electron configuration is 1s2 for which the orbital diagram is: 45 Electron Configurations The solutions for the Schrӧdinger equation (i.e. the atomic orbitals and their energies) are for the hydrogen atom. Solutions for other atoms cannot be determined exactly but approximations indicate that they are very similar to those for hydrogen Orbital energies more complicated than hydrogen since more than 1 electron to consider 46 Orbital Energy Diagram for multi-electron atoms 47 Sublevel Energy Splitting in Multielectron Atoms In multielectron atoms, the energies depend on both n and l (the energies of sublevels are split). In general, the lower the value of l with a given principle level, the lower the energy of the corresponding orbital) In a multielectron atom, for a given n: E(s orbital) < E(p orbital) < E(d orbital) 48 Sublevel Energy Splitting in Multielectron Atoms 1. Coulomb’s Law − Electrons will repel one another − Electrons will be attracted to the positive charge of protons in the nucleus − the magnitude of the interaction between an electron and the nucleus will increase as the number of protons in the nucleus increases 49 Sublevel Energy Splitting in Multielectron Atoms 2. Shielding In multielectron atoms, any electron will experience repulsive forces with other electrons These electrons partially “block out” the attractive force to the nucleus, increasing the energy of that orbital (weaker effective nuclear charge) Electrons in inner orbitals better shield electrons from the nucleus than electrons in the same orbital 50 Sublevel Energy Splitting in Multielectron Atoms 3. Penetration Electrons in outer orbitals with radial probabilities that overlap an inner orbital will experience less shielding (greater effective nuclear charge) 2s orbital penetrates into the 1s orbital and are less shielded than 2p orbitals 51 Polling question Which statement is true? An orbital that penetrates into the region occupied by inner electrons… A) is more shielded from nuclear charge than a non- penetrating orbital and will have a higher energy B) is less shielded from nuclear charge than a non- penetrating orbital and will have a higher energy C) is less shielded from nuclear charge than a non- penetrating orbital and will have a lower energy D) is more shielded from nuclear charge than a non- penetrating orbital and will have a lower energy 52 Sublevel Energy Splitting in Multielectron Atoms Penetration continued… Same effect noted in n = 3 shell 3s orbital penetrates into inner electron orbitals more than 3p 3p orbital penetrates into inner electron orbitals more than 3d In general, penetrating ability s > p > d 53 Electron Configurations in Ground State Multielectron Atoms 1. Aufbau principle: electrons are “filled” into orbitals from lowest energy to highest energy (1s, then 2s, 2p, 3s, 3p, etc) The following “diagonal” representation illustrates this for most atoms: 54 Electron Configurations in Ground State Multielectron Atoms 2. Pauli exclusion principle: Orbitals hold no more than two electrons because no two electrons can have the same four quantum numbers (they must have opposing spins in the same orbital) 3. Hund’s rule: When orbitals of identical energy are available (ex: the three 2p orbitals), electrons first occupy these orbitals singly. Once the orbitals of equal energy are filled, the electrons spin-pair in the same orbitals 55 Electron Configurations in Ground State Multielectron Atoms Going back to this figure, we can see how electrons will fill the orbitals in a multi- electron atom: 56 Electron Configurations in Ground State Multielectron Atoms 57 Electron Configurations in Ground State Multielectron Atoms For convenience, condensed electron configurations are written using the noble gas configuration that precedes the atom of interest Na: 1s22s22p63s1 or P: 1s22s22p63s23p3 or Electrons in the “core” orbitals are usually not important for chemical reactions 58 Ambiguity: 3d and 4s orbitals 3d and 4s orbitals are very close in energy, the first orbital filled depends on the element For potassium and calcium, 4s orbital filled before 3d because 4s is slightly lower in energy K: [Ar] 4s1 Ca: [Ar] 4s2 59 Electron Configurations for Transition Metals First row transition metals start with scandium (Sc), and 4s is usually filled with 2 electrons along with 3d electrons for the leftover electrons Sc: [Ar] 4s23d1 V: [Ar] 4s23d3 Ni: [Ar] 4s23d8 Exceptions: Cr and Cu are 4s1 with remainder 3d Cu: [Ar] 4s 13d10 60 Electron Configurations of Ions Ions are atoms that have either gained or lost electrons For anions (negative charge) simply add the number of electrons indicated by the charge F- : O- : 61 Electron Configurations of Ions For cations, remove electrons from orbitals with the highest value of n If there is more than one orbital for the highest value of n, electrons are first removed from the one with the highest value of l V: V2+: V3+: As: As3+: As5+: 62 Polling question Which of the following is the condensed electron configuration for Fe3+? A) [Ar]4s23d9 B) [Ar]4s13d4 C) [Ar]3s14d10 D) [Ar]4s23d3 E) [Ar]3d5 63 Magnetic Properties An atom, or an ion, that contains unpaired electrons will be attracted to an external magnetic field and is said to be paramagnetic Examples: C, V3+, Cu2+ 64 Magnetic Properties An atom, or an ion, that contains no unpaired electrons will not be attracted to an external magnetic field and is said to be diamagnetic Examples: He, Be, Zn2+ 65 Polling question Is a sulfur atom diamagnetic or paramagnetic? A) diamagnetic B) paramagnetic 66