Part 2: Many-Electron Atoms and the Periodic Table

Wavefunctions for Many Electron Atoms For helium (He): function of six position variables, x 1, y 1, and z 1 for electron 1 and x 2, y 2, and z 2 for electron 2. For an atom with N electrons

Schrodinger equation for helium (He) potential energy term kinetic energy term Potential energy term: electron-electron repulsion electron-nuclear attraction Electron-electron repulsions not present in hydrogen.

Without electron-electron repulsions where  denotes an orbital for an individual electron leads to unsatisfactory results. Solution: self-consistent field (Hartee/SCF) method

Self-consistent field (Hartee/SCF) method For an atom with N electrons

Schematic representation of SCF method SCF orbitals can be described using the same set of quantum numbers (n, l, m l ). The four quantum number (n, l, m l, m s ) completely label an electron in any orbital in any atom. Computationally intensive accomplished by sophisticated computer programs.

electron configuration – how electrons are distributed among the various atomic orbitals orbital diagram – pictorial representation of the electron configuration which shows the spin of the electron

Pauli Exclusion Principle No two electrons in an atom can have the same four quantum numbers (n, l, m l, m s ). Same values of m s Effect of radial probability function for Ar 3 distinct shells Correct representation

Paramagnetic unpaired electrons 2p Diamagnetic all electrons paired 2p Diamagnetism and Paramagetism

Paramagnetic - attracted by a magnet Diamagnetic – slightly repelled by a magnet Gouy balance - provides direct evidence of electron configurations

Aufbau (buiding-up) Prinicple orbital energy – negative amount of energy required to remove an electron from a given orbital An electron configuration is constructed according to the Paul exclusion principle, so that the total energy of the configuration is a minimum.

Energy of orbitals in a single electron atom Energy only depends on principal quantum number n E n = -R H ( ) 1 n2n2 n=1 n=2 n=3

Factors Affecting Atomic Orbital Energies Additional electron in the same orbital An additional electron raises the orbital energy through electron-electron repulsions. Additional electrons in inner orbitals Inner electrons shield outer electrons more effectively than do electrons in the same sublevel. Higher nuclear charge lowers orbital energy (stabilizes the system) by increasing nucleus-electron attractions. The Effect of Nuclear Charge (Z effective ) The Effect of Electron Repulsions (Shielding)

The effect of nuclear charge on orbital energy.

Shielding

Electron density varies with distance from nucleus as shown by radial probability plots. For same n: s<p<d<f

The effect of orbital shape

Energy of orbitals in a multi-electron atom Energy depends on n and l n=1 l = 0 n=2 l = 0 n=2 l = 1 n=3 l = 0 n=3 l = 1 n=3 l = 2

Order of orbitals (filling) in multi-electron atom 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s Graphical representation of Madelung’s rule

C 6 electrons The most stable arrangement of electrons in subshells is the one with the greatest number of parallel spins C 1s 2 2s 2 2p 2 N 7 electrons N 1s 2 2s 2 2p 3 O 8 electrons O 1s 2 2s 2 2p 4 F 9 electrons F 1s 2 2s 2 2p 5 Ne 10 electrons Ne 1s 2 2s 2 2p 6 Hund’s Rule

Aufbau (buiding-up) Prinicple - revisited As protons are added one by one to the nucleus to build up the elements, electrons are similarly added to the atomic orbitals in accordance with the Pauli exclusion principle. noble gas core

Noble gases – completely filled s and p subshells except He Transition metals – incompletely filled d subshells or give rise to cations with incompletely filled d subshells

Lanthanide series (rare earth elements) – incompletely filled 4f subshell and or give rise to cations with incompletely filled 4f subshells Actinide series (rare earth elements) –filling 5f subshell and are not found free in nature but have been synthesized.

Outermost subshell being filled with electrons

Electron Configuration of Anions and Cations Cations – lose electrons to achieve a more stable configuration nontransition metals transition metals – through loss of s and then n-1 d electrons form more than one cation heavy nontransition metals – inert pair effect

Electron Configurations of Cations of Transition Metals Fe: [Ar]4s 2 3d 6 Fe 2+ : [Ar]4s 0 3d 6 or [Ar]3d 6 Fe 3+ : [Ar]4s 0 3d 5 or [Ar]3d 5 Mn: [Ar]4s 2 3d 5 Mn 2+ : [Ar]4s 0 3d 5 or [Ar]3d 5 When a cation is formed from an atom of a transition metal, electrons are always removed first from the ns orbital and then from the (n – 1)d orbitals.

Anions – addition of one or more electrons to partially filled outermost n shell

When the Elements Were Discovered

ns 1 ns 2 ns 2 np 1 ns 2 np 2 ns 2 np 3 ns 2 np 4 ns 2 np 5 ns 2 np 6 d1d1 d5d5 d 10 4f 5f Ground State Electron Configurations of the Elements

Classification of the Elements

Effective nuclear charge (Z eff ) is the “positive charge” felt by an electron Na Mg Al Si Z eff Core Z Radius (pm) Z eff = Z -  0 <  < Z (  = shielding constant) Z eff  Z – number of inner or core electrons

Metals and atoms in three-dimensional arrays Diatomic molecules Atomic Radii One-half the distance between neighboring nuclei One-half the distance between the atomic nuclei

General Trend in Atomic Radius Increasing Atomic Radius

Atomic Radii

Comparison of Atomic Radii with Ionic Radii CationsAnions

Cation is always smaller than atom from which it is formed. Anion is always larger than atom from which it is formed.

The Radii (in pm) of Ions of Familiar Elements

The 3 rd Liquid Element? Liquid? 113 elements, 2 are liquids at 25 0 C, Br 2 and Hg 223 Fr, t 1/2 = 21 minutes

Ionization energy is the minimum energy (kJ/mol) required to remove an electron from a gaseous atom in its ground state. I 1 + X (g) X + (g) + e - I 2 + X + (g) X 2 + (g) + e - I 3 + X 2+ (g) X 3 + (g) + e - I 1 first ionization energy I 2 second ionization energy I 3 third ionization energy I 1 < I 2 < I 3

Filled n=1 shell Filled n=2 shell Filled n=3 shell Filled n=4 shell Filled n=5 shell Variation of the First Ionization Energy with Atomic Number

General Trend in First Ionization Energies Increasing First Ionization Energy

Electron affinity is the negative of the energy change that occurs when an electron is accepted by an atom in the gaseous state to form an anion. X (g) + e - X - (g) F (g) + e - X - (g) O (g) + e - O - (g) E ea = +328 kJ/mol E ea = +141 kJ/mol

Variation of Electron Affinity With Atomic Number (H – Ba)