CHAPTER 8A-8D: Hydrogenic and Many-Electron Atoms Overview

Quantum number for energy level

n

Shape of an s-orbital

Spherical

Angular nodes in a 3p orbital

1

Total nodes in a 4d orbital

3

Radial distribution function

The likely distance from the nucleus to find an electron

Degenerate orbitals in hydrogen

Orbitals with the same n

Valid magnetic quantum number (mₗ) for l = 2

-2

Orbital with torus shape

d𝓏²

Energy formula for hydrogenic atom

E = -\frac{Z^2 R_H}{n^2}

Node representation in an orbital

Zero electron probability

Radial nodes in a 3s orbital

2

Quantum number for orbital shape

l

Orbitals appearing in third shell (n = 3)

d

Transition emitting most energy

2 → 1

Bohr radius definition

The most probable distance of 1s electron from nucleus

Orbitals with 2 angular nodes

4d

Allowed values of l for n = 3

0, 1, 2

Non-existent orbital in hydrogen atom

1p

Electron spin quantum number (mₛ) values

±½

Selection rules for hydrogenic transitions

Δl = ±1

Definition of a hydrogenic atom

An atom or ion with only one electron (e.g., H, He⁺, Li²⁺).

Importance of hydrogenic atoms in quantum mechanics

They are the only atoms for which Schrödinger's equation has an exact analytical solution.

Principal quantum number (n) representation

The main energy level of the electron and its average distance from the nucleus.

Angular momentum quantum number (l) determination

The shape of the orbital (s, p, d, etc.).

Reason for larger orbitals with higher n values

Because the electron has more energy and is on average further from the nucleus.

Cause of degeneracy in hydrogenic atoms

All orbitals with the same n have the same energy because there are no electron-electron repulsions.

Definition of a radial node

A spherical region where the probability of finding the electron is zero due to changes in sign of the radial wavefunction.

Difference between angular and radial nodes

Angular nodes are planes or cones where the angular part of the wavefunction equals zero.

Radial node

2s has one radial node due to the higher principal quantum number (n = 2).

Radial distribution function

The probability of finding the electron at a certain distance from the nucleus.

Total nodes in an orbital

Total nodes = n − 1.

Orbitals in the n = 3 shell

9 orbitals: 1 (3s) + 3 (3p) + 5 (3d).

1p orbital

Because for n = 1, l must be 0. A p orbital requires l = 1.

Bohr radius

The most probable distance between the electron and nucleus in a hydrogen 1s orbital (~0.529 Å).

Degenerate orbitals

They have the same energy despite having different shapes or orientations.

Schrödinger model vs Bohr's model

Schrödinger's model describes electrons as probability clouds, not fixed orbits.

Spectral lines in hydrogen atoms

Electrons transition between energy levels, emitting or absorbing photons of specific energy.

Purpose of selection rules

They determine which transitions between orbitals are allowed based on quantum mechanical constraints.

Shapes of dₓ²-ᵧ² and d𝓏² orbitals

Because they are solutions to different angular parts of the Schrödinger equation, describing different orientations of electron clouds.

|ψ|² vs ψ

|ψ|² gives the probability density—i.e., where the electron is likely to be found—while ψ can be positive or negative and has no direct physical meaning.

Electron repulsion in the same atom

Their negative charges naturally repel due to electrostatic (Coulombic) forces.

Shielding in an atom

When inner electrons block the attraction between the nucleus and outer electrons.

Effect of shielding on Z_eff

Shielding reduces Z_eff because outer electrons feel less pull from the nucleus.

Energy of s-orbitals vs p-orbitals

Because s-electrons penetrate closer to the nucleus, feeling more attraction.

Penetration of 2s vs 2p

2s has better penetration.

Effective nuclear charge formula

Zeff=Z−σ.

Z in the Z_eff formula

The atomic number (number of protons).

Z_eff across a period

Nuclear charge increases while shielding stays relatively constant.

Energy of 2p vs 2s

It is more shielded and less penetrating.

Energy of d and f orbitals

They are more shielded and penetrate less.

Degeneracy of 3d and 4s orbitals

No, 4s is lower in energy until 3d fills up.

Electron configuration of Cr

A half-filled d subshell is more stable than the expected configuration.

Electrons lost first in cation formation

4s electrons are lost first.

Hund's Rule

Electrons occupy degenerate orbitals singly before pairing.

Higher angular momentum and energy

They are less penetrating and more shielded.

Pauli Exclusion Principle

Two electrons in the same orbital must have opposite spins.

Aufbau principle

Electrons fill lower-energy orbitals before higher ones.

Z_eff

Net nuclear charge felt by an electron

Atomic radius

Distance from the nucleus to the outermost electrons

Degenerate orbitals

Orbitals that have the same energy

Penetration

Ability of an orbital to get close to the nucleus

Shielding

Reduction of effective nuclear charge on an electron due to other electrons

Hund's Rule

Electrons fill all orbitals of a sublevel singly before pairing

Aufbau principle

Electrons occupy the lowest energy orbitals first

Pauli Exclusion Principle

No two electrons can occupy the same orbital with the same spin

Electron-electron repulsions

Forces that cause splitting of orbital energies in multi-electron atoms

Energy difference between 2s and 2p orbitals

Caused by penetration and shielding

4s orbital filling

Fills before 3d because it has lower energy before filling

Z_eff for a 3p electron in sulfur

6 (calculated with Z = 16 and estimated shielding = 10)

Atomic radius across a period

Decreases as Z_eff increases

Atomic radius down a group

Increases due to additional energy levels

Element with highest Z_eff for outermost electron

Cl

Element with exception to expected electron configuration

Cr

Electron configuration for Mg

[Ne] 3s²

Electron configuration for Fe²⁺

[Ar] 3d⁶

Orbital most likely to be removed during ionization

4s

Lowest Z_eff electron

3s in Na

Most shielded orbital

3d

Orbital that penetrates the nucleus the most

3s

Additional energy levels (shells)

They are added, increasing the size of the atom.

Cations

They are smaller than their parent atoms because they have fewer electrons, reducing electron-electron repulsion and allowing the nucleus to pull remaining electrons closer.

Anions

They are larger than their parent atoms because adding electrons increases repulsion among them, expanding the electron cloud.

Effective nuclear charge (Z_eff)

The net positive charge felt by valence electrons after accounting for shielding by inner electrons.

Trend of Z_eff across a period

It increases because more protons are added while shielding remains roughly the same.

Ionization energy increase across a period

Electrons are more strongly attracted to the nucleus due to increasing Z_eff, making them harder to remove.

Ionization energy decrease down a group

Electrons are farther from the nucleus and more shielded, making them easier to remove.

Large jump in successive ionization energies

After all valence electrons are removed, the next electron comes from a core shell closer to the nucleus and is more tightly held.

Group with the highest electron affinities

Group 17 (halogens) because they need only one electron to complete their octet.

Noble gases and positive electron affinities

They already have full shells and do not want to gain electrons.

More negative electron affinity value

Indicates the atom more strongly wants to gain an electron, releasing more energy.

Comparison of Na and Na⁺

Na⁺ is smaller because it has fewer electrons and less repulsion.

Comparison of F and F⁻

F⁻ is larger because adding an electron increases electron-electron repulsion.

Metallic character increase down a group

Outer electrons are farther from the nucleus and easier to remove.

Metallic character decrease across a period

Atoms more strongly hold on to their electrons due to higher Z_eff.

Nonmetallic character trend down a group

It decreases because atoms are less likely to gain electrons.

Reason ionization energy differs between Group 2 and Group 13

Group 13 elements start removing p-electrons, which are easier to remove than the s-electrons of Group 2.

Similar chemical properties in the same group

They have the same number of valence electrons, leading to similar bonding behavior.

Shielding

It is when inner electrons block the attraction between the nucleus and outer electrons, reducing Z_eff and making electrons easier to remove.

Atomic radius increase down a group

It is due to the addition of energy levels (shells).

Trend in metallic character across a period

It decreases.