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Ch. 1: Atoms: The Quantum World CHEM 4A: General Chemistry with Quantitative Analysis Fall 2009 Instructor: Dr. Orlando E. Raola Santa Rosa Junior College Overview 1.1The nuclear atom 1.2 Characteristics of electromagnetic radiation 1.3 Atomic spectra 1.4 Radiation, quanta, photons 1.5 Wave-particle duality 1.6 Uncertainty principle Spherical polar coordinates colatitude azimuth radius General formula of wavefunctions for the hydrogen atom ψ (r,θ ,ϕ ) = R(r )Y(θ ,ϕ ) For n = 1 ψ (r,θ ,ϕ ) = 2e − r a0 3 2 0 a × 1 2π 1 2 = e − r a0 1 3 2 0 (π a ) a0 = 4πε 0 2 me e 2 General formula of wavefunctions for the hydrogen atom ψ (r,θ ,ϕ ) = R(r )Y(θ ,ϕ ) 1 For n = 2 and E 2 = − hℜ 4 ψ (r,θ ,ϕ ) = 1 1 2 6 5 2 0 a re − r 2a0 1 2 r − ⎛ ⎞ ⎛ 3 ⎞ 1 1 2a0 ×⎜ sin θ cos φ = r e sin θ cos φ ⎜ 5⎟ ⎟ 4 ⎝ 2π a0 ⎠ ⎝ 4π ⎠ Quantum numbers n: principal quantum number determines the energy indicates the size of the orbital : angular momentum quantum number, relates to the shape of the orbital m : magnetic quantum number, possible orientations of the angular momentum around an arbitrary axis. magnetic quantum number principal quantum number orbital angular momentum quantum number Electron probability in the ground-state H atom. Radial probability distribution Allowable Combinations of Quantum Numbers l = 0, 1, …, (n – 1) ml = l, (l – 1), ..., -l No two electrons in the same atom have the same four quantum numbers. Higher probability of finding an electron Lower probability of finding an electron most probable radii The most probable radius increases as n increases. boundary surface • 90% likelihood of finding electron within radial nodes Wavefunction (Ψ) is nonzero at the nucleus (r = 0). For an s-orbital, there is a nonzero probability density (Ψ2) at the nucleus. radial nodes n=1 l=0 no radial nodes n=2 l=0 1 radial node n=3 l=0 2 radial nodes 2p-orbital n=2 l = 1, 0, or -1 no radial nodes 1 nodal plane Plot of wavefunction is for yellow lobe along blue arrow axis. The three p-orbitals nodal planes The labels “x”, “y”, and “z” do not correspond directly to ml values (-1, 0, 1). The five d-orbitals n = 3, 4, … dark orange (+) l = 2, 1, 0, -1, -2 light orange (–) nodal planes The seven f-orbitals n = 4, 5, … dark purple (+) l = 3, 2, 1, 0, -1, -2, -3 light purple (–) Allowed orbitals Allowed subshells 2 electrons per orbital Maximum of 32 electrons for n = 4 shell Stern and Gerlach Experiment: Electron Spin Atoms with one type of electron spin Atoms with other type of electron spin Silver atoms (with one unpaired electron) Spin States of an Electron Spin magnetic quantum number (ms) has two possible values: Relative Energies of Orbitals in a Multi-electron Atom Z is the atomic number. After Z = 20, 4s orbitals have higher energies than 3d orbitals. Probability maxmima for orbitals within a given shell are close together. A 3s-electron has a greater probability of being found near the nucleus than 3p- and 3d-electrons due to contribution of peaks located closer to the nucleus. Paired spins Lower energy Parallel spins Higher energy Electron Configurations: H and He 1s electron (n, l, ml, ms) • 1, 0, 0, (+½ or –½) 1s electrons (n, l, ml, ms) • 1, 0, 0, +½ • 1, 0, 0, –½) Electron Configurations: Li and Be 1s electrons (n, l, ml, ms) • 1, 0, 0, +½ • 1, 0, 0, –½ 1s electrons (n, l, ml, ms) • 1, 0, 0, +½ • 1, 0, 0, –½ 2s electron* • 2, 0, 0, +½ 2s electrons • 2, 0, 0, +½ • 2, 0, 0, –½ * one possible assignment Electron Configurations: B and C 1s electrons (n, l, ml, ms) • 1, 0, 0, +½ • 1, 0, 0, –½ 1s electrons (n, l, ml, ms) • 1, 0, 0, +½ • 1, 0, 0, –½ 2s electrons • 2, 0, 0, +½ • 2, 0, 0, –½ 2s electrons • 2, 0, 0, +½ • 2, 0, 0, –½ 2p electron* • 2, 1, +1, +½ 2p electrons* • 2, 1, +1, +½ • 2, 1, 0, +½ * one possible assignment * one possible assignment Filling order for orbitals subshell being filled maximum number of electrons in subshell The Hydrogen atom: atomic orbitals The potential in a hydrogen atom can be expressed as 2 e V(x) = − 4πε 0r Schrödinger (1927) found that the exact solutions for his equation give expression for the energy as hℜ E=− 2 n ℜ= me e 4 8h ε 3 2 0 n = 1,2,3.... Quantum Numbers and Atomic Orbitals An atomic orbital is specified by three quantum numbers. n the principal quantum number - a positive integer ℓ the angular momentum quantum number - an integer from 0 to n-1 mℓ the magnetic moment quantum number - an integer from -ℓ to +ℓ Quantum Numbers 1.Principal (n = 1, 2, 3, . . .) - related to size and energy of the orbital. 2.Angular Momentum (ℓ = 0 to n 1) - relates to shape of the orbital. 3.Magnetic (mℓ = ℓ to ℓ) - relates to orientation of the orbital in space relative to other orbitals. 4.Electron Spin (ms = +1/2, 1/2) - relates to the spin states of the electrons. Table 7.2 The Hierarchy of Quantum Numbers for Atomic Orbitals Name, Symbol Allowed Values (Property) Principal, n Positive integer (size, energy) (1, 2, 3, ...) Quantum Numbers 1 2 3 Angular momentum, ℓ (shape) 0 to n-1 Magnetic, mℓ (orientation) -ℓ,…,0,…,+ℓ 0 0 0 0 1 0 1 2 0 -1 0 +1 -1 -2 0 +1 -1 0 +1 +2 Sample Problem 7.5 Determining Quantum Numbers for an Energy Level PROBLEM: What values of the angular momentum (ℓ) and magnetic (m ) ℓ quantum numbers are allowed for a principal quantum number (n) of 3? How many orbitals are allowed for n = 3? PLAN: Follow the rules for allowable quantum numbers found in the text. l values can be integers from 0 to n-1; mℓ can be integers from -ℓ through 0 to + ℓ. SOLUTION: For n = 3, ℓ = 0, 1, 2 For ℓ = 0 mℓ = 0 For ℓ = 1 mℓ = -1, 0, or +1 For ℓ= 2 mℓ = -2, -1, 0, +1, or +2 There are 9 mℓ values and therefore 9 orbitals with n = 3. Sample Problem 7.6 Determining Sublevel Names and Orbital Quantum Numbers PROBLEM: Give the name, magnetic quantum numbers, and number of orbitals for each sublevel with the following quantum numbers: (a) n = 3, ℓ = 2 (b) n = 2 ℓ= 0 (c) n = 5, ℓ = 1 (d) n = 4, ℓ = 3 PLAN: Combine the n value and ℓ designation to name the sublevel. Knowing ℓ, we can find mℓ and the number of orbitals. SOLUTION: n ℓ sublevel name possible mℓ values # of orbitals (a) 3 2 3d -2, -1, 0, 1, 2 5 (b) 2 0 2s 0 1 (c) 5 1 5p -1, 0, 1 3 (d) 4 3 4f -3, -2, -1, 0, 1, 2, 3 7 1s 2s 3s The 2p orbitals. Representation of the 1s, 2s and 3s orbitals in the hydrogen atom Representation of the 2p orbitals of the hydrogen atom Representation of the 3d orbitals Representation of the 4f orbitals Pauli Exclusion Principle In a given atom, no two electrons can have the same set of four quantum numbers (n, ℓ, mℓ, ms). Therefore, an orbital can hold only two electrons, and they must have opposite spins. Types of Atomic Orbitals Levels and sublevels s orbital are spherical Dot picture of electron cloud in 1s orbital. Surface density 4πr2y versus distance Surface of 90% probability sphere 1s orbital 2s orbitals 3s orbital p orbitals When n = 2, then ℓ = 0 and 1 Therefore, in n = 2 levell there are 2 types of orbitals — 2 sublevels For ℓ = 0 mℓ = 0 this is a s sublevel For ℓ = 1 mℓ = -1, 0, +1 this is a p sublevel with 3 orbitals p Orbitals The three p orbitals lie 90o apart in space 2px Orbital 3px Orbital d Orbitals When n = 3, what are the values of ℓ? ℓ = 0, 1, 2 and so there are 3 sublevels in level n=3. For ℓ = 0, mℓ = 0 s sublevel with single orbital For ℓ = 1, mℓ = -1, 0, +1 p sublevel with 3 orbitals For ℓ = 2, mℓ = -2, -1, 0, +1, +2 d sublevel with 5 orbitals s orbitals have no planar node (ℓ = 0) and so are spherical. p orbitals have ℓ = 1, and have 1 planar node, and so are “dumbbell” shaped. This means d orbitals (with ℓ = 2) have 2 planar nodes One of 7 possible f orbitals. All have 3 planar surfaces. Can you find the 3 surfaces here? 2 s orbital Summary of Quantum Numbers of Electrons in Atoms Name Symbol Permitted Values Property principal n positive integers(1,2,3,…) orbital energy (size) angular momentum ℓ integers from 0 to n-1 magnetic mℓ integers from -ℓ to 0 to +ℓ orbital shape (The ℓ values 0, 1, 2, and 3 correspond to s, p, d, and f orbitals, respectively.) orbital orientation spin ms +1/2 or -1/2 direction of e- spin Experimental observation of the spin of the electron (Stern and Gerlach, 1920) A comparison of the radial probability distributions of the 2s and 2p orbitals The radial probability distribution for an electron in a 3s orbital. The radial probability distribution for the 3s, 3p, and 3d orbitals. The 3d orbitals One of the seven possible 4f orbitals. Schematic representation of the energy levels of the hydrogen atom CHEM 4A: General Chemistry with Quantitative Analysis Dr. Orlando E. Raola FALL 2008 1 / 25 Ch.1: Atoms: The Quam World 2 / 25 Expression forms for the wavefunction ψ 1. The wavefunction ψ describes the movement of the particle. 2. The probability density ψ 2 describes the probability of finding the electron in a region of space . 3. The radial distribution function P(r ) = r 2 R 2 (r ) is the probability that the electron would be found at a certain distance r from the nucleus. 3 / 25 Allowable Combinations of Quantum Numbers ` = 0, 1, 2, ...,n-1; m` = -`,...,0,...` 4 / 25 Stern and Gerlach Experiment (1925) 5 / 25 Stern and Gerlach Experiment (1925) 6 / 25 Stern and Gerlach Experiment (1925) 7 / 25 Electron spin Electrons (and many other subatomic particles) have intrinsic magnetic moment. This is a purely quantum mechanical effect without parallel in classical physics. Spin is characterized by the spin magnetic quantum number ms , that can have values + 21 and − 21 To help our understanding of this phenomenon, we resort to a rather crude physical model: 8 / 25 EPR: electron paramagnetic resonance 9 / 25 Energy of hydrogen electronic orbitals 10 / 25 Many-electron atoms: The form of the potential for the helium atom V =− 2e2 2e2 e2 − + 2 4π0 r12 4π0 r22 4π0 r12 11 / 25 Orbital energies in many-electron atoms 12 / 25 Shieding: Effective nuclear charge As a result of electron-electron repulsion, in many-electron atoms, the pull of the nucleus in an electron is decreased, as if the charge of the nucleus were smaller than it actually is. En = − 2 h < Zeff n2 13 / 25 Penetration: how close to the nucleus Because s electrons are in average closer to the nucleus, they experience less shielding and “see” a stronger effective nuclear charge than p electrons. Penetration effects can explain why the 4s orbital has such a low energy than even the 3d. 14 / 25 Pauli exclusion principle, 1925 In an atomic system, no more than two electrons can occupy any given orbital. When two electrons occupy one orbital, their spins must be antiparallel. No two electrons in an atom can have the same set of four quantum numbers. 15 / 25 Aufbau principle To predict the electronic configuration of a neutral atom of an element: 1. Electrons are added, one by one, to the orbitals in the order of their increasing energies. No more than two electrons per orbital (Pauli exclusion priniciple). 2. If more than one orbital in a subshell is available, electrons are added to the orbitals in that subshell with parallel spins until the subshell is half-full. Then electrons are start to be paired with antiparallel spins (Hund’s rule of maximum multiplicity). 16 / 25 Valence shell and the Periodic Table Interactive periodic table 17 / 25 Periodic properties: Effective nuclear charge 18 / 25 Atomic radius Three types of atomic radii: 1. metallic radius 2. covalent radius 3. van der Waals radius 19 / 25 Periodic trends: atomic radius 20 / 25 Ionic radius Ionic radii are determined on the distance between neighboring ions in an ionic solid, based on the asumption that the radius of the oxide ion (O 2− ) is 140. pm 21 / 25 Periodic trends: ionic radius 22 / 25 Ionization energy Ionization energy is the enery required to remove one electron from an atom in the gas phase. Mg(g) → Mg + (g) + e− (g) I = E(X + ) − E(x) 23 / 25 Succesive ionization energies 24 / 25 Electron affinity 25 / 25