Electrons in Atoms

• $e^{-}$
• $9.11\cdot 10^{-31}\text{ kg}$
• forms cloud around nucleus
• live in orbitals around nucleus
  • orbitals have size, shape, energy
  • orbitals are associated with quantum numbers

Quantum Numbers §

• solution of Schrodinger’s Equation for one-electron atom when replacing potential energy with Coulomb potential quantizes 3 quantum numbers : $n,l,m$
• electrons with quantum numbers $n,l,m$ (aka every Hartree orbital) are associated with a wave function ${\Psi }_{nlm}$, where $\Psi (p{)}^2$ is the probability of finding the electron at position $p$.

$${\Psi }_{nlm}(r,\theta ,\varphi )=R_{nl}(r)Y_{lm}(\theta ,\varphi ).$$

4 Quantum Numbers §

Principal Quantum Number ($n$) §

• indexes energy level
• comes from quantization of energy

$$E_n=-\frac{Z^2{e}^4{m}_e}{8{ϵ}_0^2{n}^2{h}^2}n=1,2,3,\dots$$

• $n\propto$ energy of electron
• $n\propto r$ (distance from nucleus)

Angular Momentum Quantum Number ($l$) §

• quantization of $L^2$ (angular momentum squared)

$$L^2=l(l+1)\frac{h^2}{4{\pi }^2}l=0,1,\dots ,n-1$$

• related to shape
• $s=0,p=1,d=2,f=3$

Magnetic Quantum Number ($m_l$) §

• quantization of angular momentum

$$L_z=m\frac{h}{2\pi }m=-l,-1+1,\dots ,0,\dots l-1,l$$

• describes how energy of atom shifts in external magnetic field

Electron Spin Quantum Number ($m_s$) §

• orientation of the magnetic moment of the electron is quantized according to Stern-Gerlach experiment
• $+\frac{1}{2},-\frac{1}{2}$
• 2 electrons live in orbital described by first 3 quantum numbers

Rules §

Basically, electrons ALWAYS go for LOWER ENERGY. This summarizes all of chemistry.

Aufbau Principle §

• describes ground state electron configuration of atom
  • ground state: lowest energy state of atom
• built up by arranging the Hartree atomic orbitals in order of increasing energy
• electrons always fill lower energy orbitals before higher energy orbitals
  • Periodic Table spdf categorization is based on this
    • start with $s$ on left, end with $p$ on right

• exceptions starting in period 4
  • transition metals
  • energy of $4s$ and $3d$ subshells are very similar
  • ${ϵ}_{3d}<{ϵ}_{4s}$
    • $K:[Ar]3d^{0}4s_1$
    • $Ca:[Ar]3d^{0}4s^2$
  • ${ϵ}_{4s}<{ϵ}_{3d}$
    • when putting >3 electrons into 3d orbitals, there is electrostatic repulsion energy (${ϵ}_{\text{er}}$)
    • ${ϵ}_{\text{er d-orbital}}\gg {ϵ}_{\text{er s-orbital}}$ because localized vs diffused
    • ${ϵ}_{\text{er d}}>|{ϵ}_{3d}-{ϵ}_{4s}|$
  • half filled + fully filled shells -> lower energy -> $4s$ electrons sometimes go to $3d$ to fill last spot for fully filled/half filled subshell
    • $Cu:[Ar]3d^{10}4s^1$
    • $Cr:[Ar]3d^{5}4s^1$

Pauli’s Exclusion Principle §

• no 2 atoms can have same 4 qn

Hund’s Rules §

• when electrons are added to Hartree orbitals of equal energy, all orbitals will be singly occupied before any is doubly occupied
  • lowest electron config has parallel spins

Electron Configurations §

• Example configurations:
  • H: $1s^1$
  • He: $1s^2$
  • Li: $1s^2,2s^1$
  • B: $1s^2,2s^2,2p^1$
  • C: $1s^2,2s^2,2p^2$

Ionization Energy §

$$\text{X}+\text{IE}\to {\text{X}}^{+}+{\text{e}}^{-}$$

• energy required to remove electron from orbitals to vacuum (0 energy)
• farther away from nucleus -> shielding lower effective nuclear charge -> higher energy -> lower ionization energy
• higher ionization energy -> more stable
• measured by photoelectron spectroscopy
  • shots photon at electron with certain energy
  • produces graph with peaks in energy that represents atomic subshells

• energy level differences $\to$ valence vs core electrons

• valence electrons
  • outermost shells
  • participate in bonding
• core electrons
  • all the others
  • cause shielding
  • (Ry) rydberg energy: $2.18\cdot {10}^{-18}$ J
    • boundary between valence and core electrons
    • $|e_a-e_b|<\text{Ry}⟹$ $e_a$ is valence electron if $e_b$ is valence electorn and vice versa

Electron Affinity §

$${\text{X}}^{-}+\text{EA}\to \text{X}+{\text{e}}^{-}$$

• energy required to detach electron from anion to yield neutral atom
  • energy released when atom gains an electron

Electronegativity §

• tendency of an atom to attract share $e^{-}$
  • bonds
  • relative
• atomic radius $\propto$ 1/EN
• nuclear charge $\propto$ EN
• shielding $\propto$ 1/EN
• example: Cl and Na
  • electron from Na will go to Cl because Cl more stable
  • $N{a}^{+}C{l}^{-}$
• causes dipoles in ionic bonds

There are two scales:

Mulliken §

$$EN=\frac{1}{2}C(1E_1+EA),$$

$C$ is a proportionality constant.

Pauling §

$$EN=\pi$$

• $\pi$ related to bond dissocation
• A-A
  • perfectly covalent
  • $\Delta E_{AA}$ = bond dissociation E
• B-B
  • $\Delta E_{BB}$ = bond dissociation E
• consider A-B
  • covalent character of A-B = $\sqrt{\Delta E_{AA}\Delta E_{BB}}$
  • ionic character = $\Delta$ (excess bond energy)
    • $\Delta =\Delta E_{AB}-\sqrt{\Delta E_{AA}\Delta E_{BB}}$
  • define $\pi$ in terms of $\Delta$
    • ${\pi }_A-{\pi }_B=0.102{\Delta }^{1/2}$
  • EN diff > 2 = ionic bond
  • EN diff = 0 = covalent bond