Lecture #2: August 25, 2020 Goal is to define in

• Bohr and Principal Energy Levels from “orbits”; Balance of electrostatic attraction and centripetal force: classical • Inability to account for emission lines => particle/wave description of atom and application of wave mechanics • Solutions of Schrodinger’s equation, Hψ = Eψ Required boundaries => numbers (and the Pauli Exclusion Principle)

• configurations. C: 1s2 2s2 2p2 or [He]2s2 2p2

Na: 1s2 2s2 2p6 3s1 or [Ne] 3s1 => Na+: [Ne] Cl: 1s2 2s2 2p6 3s23p5 or [Ne]3s23p5 => Cl-: [Ne]3s23p6 or [Ar]

What you already know: Quantum Numbers: n, l, ml , ms n is the principal , indicates the size of the orbital, has all positive integer values of 1 to ∞(infinity) (Bohr’s discrete orbits) l (angular ) orbital 0s l is the quantum number, 1p represents the shape of the orbital, has integer values of (n – 1) to 0 2d 3f ml is the , represents the spatial direction of the orbital, can have integer values of -l to 0 to l Other terms: , noble gas configuration, valence shell ms is the quantum number, has little physical meaning, can have values of either +1/2 or -1/2 Pauli Exclusion principle: no two electrons can have all four of the same quantum numbers in the same atom (Every electron has a unique set.) Hund’s Rule: when electrons are placed in a set of degenerate orbitals, the ground state has as many electrons as possible in different orbitals, and with parallel spin. Aufbau (Building Up) Principle: the ground state electron configuration of an atom can be found by putting electrons in orbitals, starting with the lowest energy and moving progressively to higher energy. Finding these familiar descriptors within the models and Emission Spectra

Sun Albert Einstein, 1925 Helium Mercury

Uranium Taken from John L. Heilbron’s “History: The Path to the 700nm600nm 500nm 400nm quantum Atom”, Nature 498, 27-30, (06 June 2013) What do these Two objects have In common?

The plane is a Model of the bird; better and better. What do these Two objects have In common?

Early models of the bird quite different, Still very useful for future development. The Bohr Atom: electrons in concentric rings

The Balmer formula expresses the frequencies of some lines in the spectrum of hydrogen in simple algebra:

2 2 νn = R(1/2 –1/n )

where νn is the (1/λ) of the nth Balmer line and R is the universal for frequency, named in honor of the Swedish spectroscopist Johannes Rydberg, who generalized Balmer’s formula to apply to elements beyond hydrogen.

Each level can accommodate Taken from John L. Heilbron’s “History: The Path to the 2 n2 electron: quantum Atom”, Nature 498, 27-30, (06 June 2013) Rows Quantum Energy Number n The Hydrogen 0 ∞ 5 -1/16 RH 4 Atom Spectrum Paschen Series (IR) -1/9 RH 3 and Energy Levels (visible transitions shown) -1/4 RH 2

2 2 νn = R(1/nf –1/ni ) Lyman + Series (UV)

Niels Bohr’s view of the atom

-RH 1 Hydrogen 656.3 486.1 434 410.1

700 600 500 400 Linking the emitted to the Bohr atom model:

The Rydberg constant represents the limiting value of the highest wavenumber (1/λ) of any that can be emitted from the , or, alternatively, the wavenumber of the lowest-energy photon capable of ionizing the hydrogen atom from its ground state. E = hc/λ Quantum Energy Number n 0 ∞ For Hydrogen: 5 -1/16 RH 4 E = -RH Paschen Series (IR) 2 -1/9 RH 3 n Balmer Series Rydberg constant for hydrogen,RH (visible transitions shown) 4 -18 RH = mee = 2.179x10 J -1/4 R 2 2 H 2 8 εo h = 13.6 eV

General equation for Rydberg constant for any element R= -Z2 e4 (UV) 2 2 8 εo h

Note: For Hydrogen atom, RH is simply the ionization energy! +13.6 eV Required to remove electron from -R H 1 lowest ; -13.6 eV emitted as ionized electron falls into the lowest Energy level, n = 1. Quantum Energy Number n 0 ∞ For Hydrogen: 5 -1/16 RH 4 E = -RH Paschen Series (IR) 2 -1/9 RH 3 n Balmer Series Rydberg constant for hydrogen,RH (visible transitions shown) 4 -18 RH = mee = 2.179x10 J -1/4 R 2 2 H 2 8 εo h = 13.6 eV

General equation for Rydberg constant for any element Lyman Series R= -Z2 e4 (UV) 2 2 8 εo h

Note: The predicted emission spectra using the Rydberg constant was only successful for simple elements such as -R H 1 H and failed for heavier atoms due to the limitations of the Bohr view of the atom. This led to the foundations of . Inorganic Chapter 1: Figure 1.5 Properties of waves: Addition for reinforcement or cancellation

© 2009 W.H. Freeman Properties of waves: Squared = amplitude; can go through zero; Boundaries => Restrictions on values => quantized

a “node”

Copyright © 2014 Pearson Education, Inc. Inorganic Chemistry Chapter 1: Figure 1.4 Time-independent Schrödinger equation (general—one dimension)

E ψ = H ψ

a “node”

© 2009 W.H. Freeman Need for Spherical Coordinates and Volume Elements

- -

+ +

Need both radial and angular functions Copyright © 2014 Pearson Education, Inc.

Summarizing: Solutions Required Quantum Numbers Quantum Numbers: n, l, ml , ms n is the , indicates the size of the orbital, has all positive integer values of 1 to ∞(infinity) l (angular momentum) orbital 0s l is the angular momentum quantum number, 1p represents the shape of the orbital, has integer values of (n – 1) to 0 2d 3f ml is the magnetic quantum number, represents the spatial direction of the orbital, can have integer values of -l to 0 to l Other terms: electron configuration, noble gas configuration, valence shell ms is the , has little physical meaning, can have values of either +1/2 or -1/2 Pauli Exclusion principle: no two electrons can have all four of the same quantum numbers in the same atom (Every electron has a unique set.) Hund’s Rule: when electrons are placed in a set of degenerate orbitals, the ground state has as many electrons as possible in different orbitals, and with parallel spin. Aufbau (Building Up) Principle: the ground state electron configuration of an atom can be found by putting electrons in orbitals, starting with the lowest energy and moving progressively to higher energy. While n is the principal energy level, the l value also has an effect; l = 3 Related to no. of nodes in Radial function!

l = 2

l = 1 l = 0 l = 1

l = 0

l = 0 Radial Wave Functions and Nodes

3s 3p 3d

2s 2p # Radial Nodes = n - l - 1

Summary: 1s # Radial Nodes = n - l -1 # Angular Nodes = l

Total # Nodes = n - 1

Copyright © 2014 Pearson Education, Inc. Radial Probability Functions and Nodes

3s 3p 3d

2s 2p

1s # Radial Nodes = n - l - 1

Copyright © 2014 Pearson Education, Inc.

Where are the nodes in orbitals??? (MFT, Figure 2.8 shows both radial and angular)

Cl, 3pz C, 2pz Cl, 3s

3+ Ti , 3dz2 3+ 3+ Ti , 3dx2-y2 Ti , 3dx2-y2

Copyright © 2014 Pearson Education, Inc. Note the orientation of the viewer: down z or x or y axes Orbitals and Shapes/Electron Distribution Each p-orbital has two lobes with positive and negative values The p-orbitals (phases) of the wavefunction either side of the nucleus separated by a nodal plane where the wavefunction is zero.

The s-orbital

Two angular nodes Quantum Numbers n is the principal quantum number, indicates the size of the orbital, has all positive integer values of 1 to ∞(infinity) l (angular momentum) orbital 0s l is the angular momentum quantum number, represents the shape of the orbital, has integer 1p values of n-1 to 0 2d 3f ml is the magnetic quantum number, represents the spatial direction of the orbital, can have integer values of -l to 0 to l Other terms: electron configuration, noble gas configuration, valence shell ms is the spin quantum number, has little physical meaning, can have values of either +1/2 or -1/2 Pauli Exclusion principle: no two electrons can have all four of the same quantum numbers in the same atom Hund’s Rule: when electrons are placed in a set of degenerate orbitals, the ground state has as many electrons as possible in different orbitals, and with parallel spin. Aufbau (Building Up) Principle: the ground state electron configuration of an atom can be found by putting electrons in orbitals, starting with the lowest energy and moving progressively to higher energy. The f-orbitals Energy Levels for Electron Configurations

The The Pauli Exclusion Principle Hund’s Rules While n is the principal energy level, the l value also has an effect Screening:

The 4s electron “penetrates” Inner shell electrons more efficiently than does 3d in neutral atoms. Reverses in positive ions. How to handle atoms larger than H? Effective Nuclear Charge or Z eff 2 4 EH = -Z e 2 2 8 εo h