Master the Periodic Table with Groups for DAT Success

II. Orbital and Periodicity

Key focus of this chapter: periodic table with groups

This chapter focuses on periodic table with groups and gives concise summaries of the important things about orbital theory, shapes of orbitals, rules of filling up electrons in orbitals, quantum numbers, electron configuration, atomic radius, ionization, and electron affinitive in more detail.

A. Orbital theory

1. Bohr’s atomic model

• Bohr’s model accurately describes the hydrogen atom and other one-electron species.

• In Bohr’s model, electrons occupy quantized energy levels (shells) around the nucleus; they are not treated as classical particles moving in fixed circular paths in the modern model.

• Each electron shell is identified by the principal quantum number (n). For hydrogen, energy increases as n increases: K (n=1) < L (n=2) < M (n=3) < N (n=4).

Fig. 1 Bohr’s atomic model

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• Energy change when electrons move to other electron shells

Classification

Features

Ground state

electrons

• Ground state = lowest-energy electron arrangement.
• Promotion to a higher energy level requires absorption of energy (for example, n = 1 → n = 2).

Excited state

electrons

• Excited state = an electron arrangement above the ground-state energy.
• Relaxation to a lower energy level emits a photon with ΔE = hν (for example, n = 2 → n = 1).

Fig. 2 Ground and excited state electron s Ground state electron Excited state electron Absorb energy Electron shell Nucleus Emit energy K (n=1) L (n=2)

2. Lyman, Balmer, and Paschen series

Classification	Features
Lyman series	• When excited state electrons at a high shell go down to K (n=1) shell, energy is emitted. • Ultraviolet region
Balmer series	• When excited state electrons at a high shell go down to L (n=2) shell, energy is emitted. • Visible region
Paschen series	• When excited state electrons at a high shell go down to M (n=3) shell, energy is emitted. • Infrared region

3. Modern atomic model

: Limitations of Bohr’s model led to the modern quantum-mechanical model (orbital theory).

a. Orbital theory

• Electrons do not move in fixed paths. An orbital describes a region of high probability for finding an electron.

• Each principal shell contains one or more subshells (s, p, d, f as allowed by n).

b. Electron shells and orbitals

• As n increases, more subshell types become available: n = 1 has s; n = 2 has s and p; n = 3 has s, p, and d; n = 4 has s, p, d, and f.

Electron shells	Orbitals
n= 1	1s
n=2	2s 2p
n=3	3s 3p 3d

Fig. 3 Modern atomic model Authenticated embedded content preview

B. Shapes of orbitals

Classification

Features

s orbitals

• Spherical shape
• One s orbital per s subshell
• ml = 0
• Maximum 2 electrons in an s subshell

p orbitals

• Dumbbell-shaped orbitals
• Three orientations: px, py, pz
• ml = -1, 0, +1
• Maximum 6 electrons in a p subshell

d orbitals

• Five d orbitals; most are cloverleaf-shaped (dz² is an important exception)
• ml = -2, -1, 0, +1, +2
• Maximum 10 electrons in a d subshell
• The d-block contains the transition metals

f orbitals

• Seven f orbitals with complex shapes
• ml = -3, -2, -1, 0, +1, +2, +3
• Maximum 14 electrons in an f subshell

C. Rules of filling up electrons in orbitals

: Electrons are expressed by arrow ( Source symbol Symbol F0AF , Source symbol Symbol F0AD ) in the orbital filling diagram

Classification

Features

Pauli exclusion principle

• Pauli exclusion principle: an orbital holds at most 2 electrons, and paired electrons have opposite spins (ms = +1/2 and -1/2).

Hund’s rule

• Hund’s rule: electrons occupy degenerate orbitals singly with parallel spins before pairing.
• For p⁴: ↑↓ ↑ ↑

Filling order of different orbitals

• Aufbau filling order (DAT high-yield): 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4f < 5d < 6p < 7s.
• Common exceptions include Cr and Cu.

D. Quantum numbers

: Quantum numbers describe an electron’s shell, subshell, orbital orientation, and spin.

Classification

Features

The principal quantum

number (n), shell

• Principal quantum number n identifies the shell/energy level and relative orbital size.
• n = 1, 2, 3, ...; larger n generally means a larger orbital and, within a given atom, higher energy.

Angular-momentum quantum number, subshell (l)

• Angular momentum quantum number l identifies the subshell/orbital shape.
• l = 0 to n − 1; l = 0, 1, 2, 3 correspond to s, p, d, f.

Magnetic quantum

number (M_l)

• Magnetic quantum number ml identifies orbital orientation within a subshell.
• ml = −l, ..., 0, ..., +l. Example: l = 2 gives five d orbitals.

n (shell)	l (subshell)	Orbitals	M_l
1	0	1s	0
2	0 1	2s 2p	0 -1, 0, +1
3	0 1 2	3s 3p 3d	0 -1, 0, +1 -2, -1, 0, +1, +2

• Combined quantum numbers (n, l, m_l)

Principal

quantum number

Angular momentum quantum number

Orbital

notation

Magnetic quantum

number

m_l

Number of

orbital(s) in

subshell

Maximum electrons in shell (2n²)

1 Source symbol Symbol F0B4 2 = 2 or

(2n² = 2 Source symbol Symbol F0B4 1²=2)

-1, 0, +1,

(1+3) Source symbol Symbol F0B4 2 = 8 or (2n² = 22²=8)

-1, 0, +1

-2, -1, 0, +1, +2

(1+3+5) Source symbol Symbol F0B4 2 =18 or (2n² = 23²=18)

-1, 0, +1

-2, -1, 0, +1, +2

-3, -2, -1, 0, +1, +2, +3

(1+3+5+7) Source symbol Symbol F0B4 2 = 32

or (2n² = 2 Source symbol Symbol F0B4 4²=32)

E. Electron configuration

1. Filling order of electrons in orbitals (from low to high energy)

1s Source symbol Wingdings F0E0 2s 2p 3s 3p 4s 3d 4p 5s 4d 5p ….

Fig. 4 Position of orbitals in the periodic table

1s			1s
2s		2p
3s		3p
4s	3d	4p
5s	4d	5p
6s	5d	6p
7s	6d

2. Electron configuration

: Electron configuration describes how electrons occupy atomic orbitals.

• Ex/

	Ground state electron configuration	Shorthand electron configuration	Orbital filling diagrams
Be	1s²2s²	[He] 2s²	[He] 2s
Ar	1s²2s²2p⁶3s²3p⁶	[Ne] 3s²3p⁶	[Ne] 3s 3p
V	1s²2s²2p⁶3s²3p⁶4s²3d³	[Ar] 4s²3d³	[Ar] 4s 3d

Fig. 5 Ground state electron configuration in the periodic table

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** Exceptional atoms of electron configuration

Cr, Cu, Nb, Mo, Ru, Rh, Ag (Dark red color above)

Ex/ Cr: 1s²2s²2p⁶3s²3p⁶4s²3d⁴ (X) Source symbol Wingdings F0E0 1s²2s²2p⁶3s²3p⁶4s¹3d⁵ (O)

Cu: 1s²2s²2p⁶3s²3p⁶4s²3d⁹ (X) Source symbol Wingdings F0E0 1s²2s²2p⁶3s²3p⁶4s¹3d¹⁰ (O)

F. Application of electron configuration

1. Counting of unpaired electrons: half filling electrons in orbital

Si: 1s²2s²2p⁶3s²3p² Source symbol Wingdings F0E0 Source symbol Symbol F0AD Source symbol Symbol F0AF 1s 2s 2p 3s 3p (2 unpaired electrons in 3p orbital)

P: 1s²2s²2p⁶3s²3p³ Source symbol Wingdings F0E0 Source symbol Symbol F0AD Source symbol Symbol F0AF 1s 2s 2p 3s 3p (3 unpaired electrons in 3p orbital)

S: 1s²2s²2p⁶3s²3p⁴ Source symbol Wingdings F0E0 Source symbol Symbol F0AD Source symbol Symbol F0AF 1s 2s 2p 3s 3p (2 unpaired electrons in 3p orbital)

Cl: 1s²2s²2p⁶3s²3p⁵ Source symbol Wingdings F0E0 Source symbol Symbol F0AD Source symbol Symbol F0AF 1s 2s 2p 3s 3p (1 unpaired electron in 3p orbital)

2. Ground state and excited state

• Ground state: the lowest-energy electron configuration.

Si: 1s²2s²2p⁶3s²3p² Source symbol Wingdings F0E0 Source symbol Symbol F0AD Source symbol Symbol F0AF 1s 2s 2p 3s 3p (low energy state electron in 3s orbital)

• Excited state: one or more electrons are promoted to higher-energy orbitals.

Si: 1s²2s²2p⁶3s¹3p³ Source symbol Wingdings F0E0 Source symbol Symbol F0AD Source symbol Symbol F0AF 1s 2s 2p 3s 3p (high energy state electron in 3p orbital)

G. Electromagnetic spectrum

1. Light

• Classified by wavelength ()

• Speed of light: c = 3.00 × 10⁸ m/s in vacuum.

• Frequency (ν): number of wave cycles per second (Hz = s⁻¹).

C = Source symbol Symbol F06C Source symbol Symbol F0B4 Source symbol Symbol F06E Source symbol Wingdings F0E0 =

** Photoelectric effect: electrons are ejected from a metal only when incident light has a frequency above the threshold frequency; photon energy is E = hν.

2. Energy (E)

E = Authenticated embedded content preview = h Source symbol Symbol F06E (physical constant, h = 6.626 Source symbol Symbol F0B4 10^-34J Source symbol Symbol F0B7 s)

• Long wavelength low frequency low energy

• Short wavelength high frequency high energy

• Ex/ Gamma rays short wavelength high energy

3. Order of energy

Gamma rays > X ray > Ultraviolet light > Visible light > Infrared light > Microwaves > Radio waves

H. Periodicity

1. Periods

• Seven rows (n = 1, 2, 3, 4, 5, 6, 7)

• The period number corresponds to the highest occupied principal energy level for main-group elements.

2. Groups

• 18 groups (columns). Modern numbering is Groups 1–18; older A/B notation may also appear in study materials.

• Elements in the same group often have similar chemical properties because they have similar valence-electron configurations.

have the same outermost electrons)

Ex/ Be, Mg, and Ca are Group 2 elements and have two valence electrons.

electrons)

Fig. 6 Periods and groups

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a. Metal and nonmetal

Classification

Characteristics

Metal

• Metals tend to lose electrons and form cations.
• Generally good conductors of heat and electricity.
• Usually malleable, ductile, and lustrous; most are solids at room temperature.

Nonmetal

• Nonmetals tend to gain or share electrons; many form anions in ionic compounds.
• Generally poor electrical/thermal conductors as solids; solid nonmetals are often brittle.

b. Classification of groups

Fig. 7 Classification of groups in periodic table

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Classification

Sub classification

Features

Metal

groups

Alkali metals

• Group 1 (1A) except H (Li, Na, K, Rb, Cs, Fr)

• One electron at their outermost shell tending to

lose one electron to become (+1) cation (good conductors)

Source symbol Wingdings F0E0 the strongest reducing agent groups highly basic

• The lowest M.P, B.P, and density group

• Found as salt forms

Alkaline earth metals

• Group 2 (2A) (Be, Mg, Ca, Sr, Ba, Ra)

• Two electrons in their outermost shell tending to

lose two electrons to become (+2) cations Source symbol Wingdings F0E0 the second

strongest reducing agents Source symbol Wingdings F0E0 basic

• The second lowest M.P, B.P, and density group

• Found as salt forms

Transition

metals

• Position between 2A and 3A groups

• Many transition-metal atoms/ions have unpaired d electrons and can be paramagnetic

• Filling electrons in d orbitals multiple oxidation states

• May have incomplete inner shell (moving electron from

outer shell [4s orbital] to inner shell [3d orbital])

• Outermost 1 or 2 electron(s)

• Ex/ Mn: 1s²2s²2p⁶3s²3p⁶4s²3d⁵ 2 outermost electrons at

4s orbital and incomplete inner shell (3d orbital)

Cu: 1s²2s²2p⁶3s²3p⁶4s¹3d¹⁰ Source symbol Wingdings F0E0 1 outermost electron at 4s

orbital and completed inner shell (3d orbital)

Semi metals (metalloid)

• Intermediate characteristics between metals and

nonmetals

• B, Si, Ge, As, Sb, Te, At

Nonmetals

Halogen

group

• Group 17 (7A) (F, Cl, Br, I, At)

• Seven electrons in the outermost shell tending to gain

one electron to become (–1) anion Source symbol Wingdings F0E0 strong oxidizing

agent Source symbol Wingdings F0E0 acidic

• Corrosive and colorful nonmetals

• Found in binary compound

Noble gas

group

• Group 18 (8A) (He, Ne, Ar, Kr, Xe, Rn)

• Very low reactivity because of 8 electrons (except He) in their outermost shells

• Colorless and odorless

I. Atomic radius

: Atomic radius is a measure of atomic size; it generally decreases across a period and increases down a group.

1. Atomic radius in periodic table

• In the same period, when the atomic number is increased, the atomic radius is decreased

because the attraction force between the protons (+) and outer electrons (-) is increased.

Li > Be > B > C > N > O > F

• In the same group, when the atomic number is increased, the atomic radius is increased

because the electron shells are increased.

Li < Na < K < Rb < Cs < Fr

• In an isoelectronic series, radius decreases as nuclear charge (Z) increases.

metallic) radius is greater than its neutral atom.

Ca²⁺ < K⁺ < Ar < Cl^- < S^2-

2. Cationic (metallic) and anionic (nonmetallic) radius

Classification

Features

Cationic (metallic) atoms

• A cation is smaller than its neutral atom because electron loss reduces electron-electron repulsion and may remove the outer shell.
• In an isoelectronic series, radius decreases as nuclear charge (Z) increases. Example: Na⁺ > Mg²⁺.

Anionic(nonmetallic) atoms

• An anion is larger than its neutral atom because added electrons increase electron-electron repulsion.
• In an isoelectronic series, radius decreases as nuclear charge (Z) increases. Example: O²⁻ > F⁻ > Na⁺ > Mg²⁺.

J. Ionization energy (E_i)

: Ionization energy is the energy required to remove an electron from a gaseous atom or ion.

Atom + energy Cationic atom + e^-

Fig. 8 Ionization energy of Li

+ 520 kJ/mol energy Li + Li e -

Li + energy of (+520 kJ/mol) Li⁺ + e^-

Increasing ionization energy Increasing nonmetallic character Increasing attraction force b/w nucleus and electrons Decreasing atomic radius

• First ionization energy (IE₁): energy required to remove the first electron from a gaseous atom.

octet rule)

Ar > Be > Li

• Second ionization energy (IE₂): energy required to remove a second electron from the gaseous +1 ion.

Li > Ar > Be

	Ground state electron configuration	Outermost electron(s)	Ei 1 (kJ/mol)	Ei 2 (kJ/mol)	Ei 3 (kJ/mol)
Li	1s²2s¹	1	520	7,300	11,820
Be	1s²2s²	2	900	1,760	14,850
Ar	1s²2s²2p⁶3s²3p⁶	8	1,520	2,670	3,930

a. General trend

• In the same period, with the increasing atomic number, the E_i is increased.

O < F < Ne

• In the same group, with the increasing atomic number, the E_i is decreased.

He > Ne > Ar

• The half-filled or fully-filled electrons in each orbital have a strong E_ito remove an

outermost electron.

Ex/ Be: Source symbol Symbol F0AD Source symbol Symbol F0AF 1s 2s (1/1 fully-filled in 2s orbital)

B: Source symbol Symbol F0AD Source symbol Symbol F0AF 1s 2s 2p (1/3 half-filled in 2p orbital)

Therefore, ionization energy: Be > B

Ex/ C: Source symbol Symbol F0AD Source symbol Symbol F0AF 1s 2s 2p (2/3 half-filled in 2p orbitals)

N: Source symbol Symbol F0AD Source symbol Symbol F0AF 1s 2s 2p (3/3 half-filled in 2p orbitals)

O: Source symbol Symbol F0AD Source symbol Symbol F0AF 1s 2s 2p (1/3 fully-filled in 2p orbitals)

Therefore, ionization energy: N > O > C

Fig. 9 Ionization energy of the atoms

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K. Electron affinity (EA)

: Electron affinity describes the energy change when a gaseous atom gains an electron. For many nonmetals the process is exothermic; sign conventions vary by textbook, so focus on relative favorability/trends for DAT questions.

Atom + e^- anionic atom + energy

^{Fig. 10 Electron affinity of F}

e - - 340 kJ/mol energy F - F

F + e^- F^- + energy of (-340 kJ/mol)

Easily becoming anionic atom Increasing electron affinitive Increasing nonmetallic character

a. General trends

• Noble gases have very unfavorable electron affinities because an added electron must enter a higher-energy shell/subshell.

• Across a period, electron affinity generally becomes more favorable (more energy released), but the trend is irregular.

N < O < F

• Down a group, electron affinity generally becomes less favorable, with important exceptions (for example, Cl is more favorable than F).

Cl > Br > I ** exception (F < Cl)

• An atom that is 2/3 half-filled in orbitals has the stronger E_ea than 3/3 half-filled in orbitals.

Ex/ C: Source symbol Symbol F0AD Source symbol Symbol F0AF 1s 2s 2p (2/3 half-filled in 2p orbitals)

N: Source symbol Symbol F0AD Source symbol Symbol F0AF 1s 2s 2p (3/3 half-filled in 2p orbitals)

Therefore, electron affinity is more favorable for C than for N (N is relatively unfavorable because of its half-filled 2p subshell).

• An atom that is 2/3 fully-filled in orbitals, has the stronger E_ea than 3/3 fully-filled in

orbitals.

Ex/ F: Source symbol Symbol F0AD Source symbol Symbol F0AF 1s 2s 2p (2/3 fully-filled in 2p orbitals)

Ne: Source symbol Symbol F0AD Source symbol Symbol F0AF 1s 2s 2p (3/3 fully-filled in 2p orbitals)

Therefore, F has a much more favorable electron affinity than Ne; noble gases have very unfavorable electron addition.

Fig. 11 Electron affinity of the atoms

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L. Tendency of atoms in periodic table

• Moving left and down the periodic table, metallic character and atomic radius generally increase. London dispersion forces generally increase with electron count/polarizability, but are not a simple universal periodic trend.

• Moving right and up, electronegativity and first ionization energy generally increase. Electron affinity is less regular and has important exceptions.

Increasing

Decreasing

Going left and down

• Metallic character
• Atomic radius
• Polarizability (generally)

• Electronegativity
• First ionization energy
• Electron affinity generally becomes more favorable toward the upper right, with notable exceptions

Fig 12. Tendency of atoms in periodic table

• Metallic character • Atomic radius • Intermolecular forces (=London dispersion) He He F F • Electronegativity • Ionization energy • Electro affinity

** He: Strongest ionization energy

F: Strongest electronegativity

II. Orbital and Periodicity

Key focus of this chapter: periodic table with groups

A. Orbital theory

1. Bohr’s atomic model

2. Lyman, Balmer, and Paschen series

3. Modern atomic model

B. Shapes of orbitals

C. Rules of filling up electrons in orbitals

Features

D. Quantum numbers

E. Electron configuration

1. Filling order of electrons in orbitals (from low to high energy)

2. Electron configuration

F. Application of electron configuration

1. Counting of unpaired electrons: half filling electrons in orbital

2. Ground state and excited state

G. Electromagnetic spectrum

1. Light

2. Energy (E)

3. Order of energy

H. Periodicity

1. Periods

2. Groups

I. Atomic radius

1. Atomic radius in periodic table

2. Cationic (metallic) and anionic (nonmetallic) radius

J. Ionization energy (Ei)

K. Electron affinity (EA)

L. Tendency of atoms in periodic table

J. Ionization energy (E_i)