Periodic Classification Formula Sheet

Last-minute revision for Classification of Elements and Periodicity. This is a largely descriptive chapter, so it mixes the few genuine relations (periodic law, Zeff, Slater’s rules) with high-yield trends, reactions and must-know facts pulled straight from this chapter.

Periodic Law & Table Structure

$$\boxed{\text{Properties} = f(\text{Atomic Number})}$$

Feature	Value
Periods	7 (horizontal rows)
Groups	18 (vertical columns)
s-block elements	14 (Groups 1, 2)
p-block elements	36 (Groups 13-18)
d-block elements	40 (Groups 3-12)
f-block elements	28 (Lanthanoids + Actinoids)
Total elements	118

Period length pattern: $2, 8, 8, 18, 18, 32, 32$

Period	Elements	Orbitals filled
1	2	1s
2	8	2s + 2p
3	8	3s + 3p
4	18	4s + 3d + 4p
5	18	5s + 4d + 5p
6	32	6s + 4f + 5d + 6p
7	32*	7s + 5f + 6d + 7p (incomplete)

Block Classification & Position

Block	Last electron	Groups	Example
s	$ns^{1-2}$	1, 2	Na: [Ne] 3s¹
p	$np^{1-6}$	13-18	Cl: [Ne] 3s² 3p⁵
d	$(n-1)d^{1-10}$	3-12	Fe: [Ar] 3d⁶ 4s²
f	$(n-2)f^{1-14}$	Lanthanoids/Actinoids	Ce: [Xe] 4f¹ 5d¹ 6s²

Finding position from electronic configuration:

Quantity	Rule
Period	Highest principal quantum number $n$ (use $n$ of the s orbital, not d)
Group (s-block)	Number of valence electrons (1 or 2)
Group (p-block)	$10 + \text{total valence electrons (s + p)}$
Group (d-block)	$d \text{ electrons} + s \text{ electrons}$

Aufbau exceptions (half/fully-filled d stability):

Element	Expected	Actual	Reason
Cr (24)	[Ar] 3d⁴ 4s²	[Ar] 3d⁵ 4s¹	half-filled d⁵
Cu (29)	[Ar] 3d⁹ 4s²	[Ar] 3d¹⁰ 4s¹	full d¹⁰
Mo (42)	[Kr] 4d⁴ 5s²	[Kr] 4d⁵ 5s¹	half-filled d⁵
Ag (47)	[Kr] 4d⁹ 5s²	[Kr] 4d¹⁰ 5s¹	full d¹⁰

Periodic Trends Master Table

Property	Across Period (→)	Down Group (↓)	Reason
Atomic radius	Decreases	Increases	Zeff ↑ / new shells
Ionic radius	Decreases	Increases	same as atomic
Ionization energy	Increases*	Decreases	size ↓ / size ↑
Electron affinity	More negative†	Less negative	size ↓ / size ↑
Electronegativity	Increases	Decreases	size ↓ / size ↑
Metallic character	Decreases	Increases	IE ↑ / IE ↓

* Exceptions Be>B, N>O, Mg>Al, P>S. † Exception Cl>F.

Atomic & Ionic Radius

$$\boxed{\text{Cation} < \text{Neutral atom} < \text{Anion}}$$

• Cations smaller (lost e⁻, less repulsion): Na 186 pm → Na⁺ 102 pm
• Anions larger (gained e⁻, more repulsion): Cl 99 pm → Cl⁻ 181 pm
• Isoelectronic species (same e⁻ count): higher $Z$ → smaller radius.

Isoelectronic example (10 e⁻, Ne config): $N^{3-} > O^{2-} > F^- > Na^+ > Mg^{2+} > Al^{3+}$

Ion	Z	Radius (pm)
O²⁻	8	140
F⁻	9	136
Na⁺	11	102
Mg²⁺	12	72
Al³⁺	13	54

Radius types: covalent (½ internuclear distance, bonded), metallic, Van der Waals (non-bonded, largest — noble gases are measured this way and are NOT directly comparable to covalent radii).

Ionization Energy (IE)

$$\text{M(g)} \rightarrow \text{M}^+(g) + e^- \quad \Delta H = IE_1 \quad (\text{kJ/mol or eV})$$$$\boxed{IE_1 < IE_2 < IE_3 < \dots < IE_n}$$

Successive IE always rises (removing e⁻ from increasingly positive ion). A huge jump marks the break into a noble-gas core, e.g. Na: $IE_2/IE_1 = 4562/496 \approx 9.2$.

Anomalies (half/full-filled stability):

Comparison	IE (kJ/mol)	Why
Be > B	899 > 801	B removes from 2p¹ (easier) vs filled 2s²
N > O	1402 > 1314	N has stable half-filled 2p³
Mg > Al	738 > 578	Al removes from 3p¹ vs filled 3s²
P > S	1012 > 1000	P has stable half-filled 3p³

Reference IE₁ values: He 2372 (highest), Ar 1521, F 1681, H 1312, Li 520.

Electron Affinity (EA)

$$\text{M(g)} + e^- \rightarrow \text{M}^-(g) \quad \Delta H = EA$$

• Negative EA = energy released (favourable, most elements).
• Positive EA = energy absorbed (noble gases, e.g. He +48).
• Across period: more negative; down group: less negative.

$$\boxed{\text{EA: } Cl(-349) > F(-328) \text{ kJ/mol}}$$

Electronegativity (EN) — Pauling Scale

$$\boxed{F(4.0) > O(3.5) > N(3.0) \approx Cl(3.0)}$$

Relative scale (not a measurable energy). Across period EN ↑, down group EN ↓.

Element	EN		Element	EN
F	4.0		C	2.5
O	3.5		H	2.1
N, Cl	3.0		Cs, Fr	0.7

Bond type from EN difference:

ΔEN	Bond type
< 0.5	Non-polar covalent
0.5 – 1.7	Polar covalent
> 1.7	Ionic

Example: HCl, ΔEN = 3.0 − 2.1 = 0.9 → polar covalent.

Effective Nuclear Charge & Slater’s Rules

$$\boxed{Z_{eff} = Z - S}$$

where $Z$ = nuclear charge (protons), $S$ = shielding constant.

Slater shielding contributions:

Electron position	Contribution to S
Same shell ($n$)	0.35 each
$(n-1)$ shell	0.85 each
$(n-2)$ and lower	1.00 each

Worked example — Na 3s¹ (Z = 11): $S = (8 \times 0.85) + (2 \times 1.00) = 8.80$, so $Z_{eff} = 11 - 8.80 = 2.2$.

Trends: $Z_{eff}$ rises across a period (Z ↑, S ~constant); stays roughly constant down a group (Z and S both rise).

Diagonal Relationships

Similar charge/radius ratio (polarising power) → similar chemistry.

Pair	Shared behaviour
Li ~ Mg	carbonate decomposes, normal oxide, NO₂ on heating nitrate, covalent-ish LiCl
Be ~ Al	covalent chloride, amphoteric oxide, complex formation
B ~ Si	acidic oxide, covalent halide that hydrolyses, polymeric/chain structures

s-block Elements (Groups 1 & 2)

$$\boxed{\text{s-block valence config: } ns^{1-2}}$$

Group	Name	Config	Members
1	Alkali metals	ns¹	Li, Na, K, Rb, Cs, Fr
2	Alkaline earth metals	ns²	Be, Mg, Ca, Sr, Ba, Ra

Reactivity with water increases down each group (size ↑ → IE ↓): Li < Na < K < Rb < Cs and Be < Mg < Ca < Sr < Ba. Group 1 only +1, Group 2 only +2.

Key Reactions — Group 1

$$2M + 2H_2O \rightarrow 2MOH + H_2\uparrow$$$$4Li + O_2 \rightarrow 2Li_2O \quad (\text{oxide, O.S. } -2)$$$$2Na + O_2 \rightarrow Na_2O_2 \quad (\text{peroxide, O.S. } -1)$$$$K + O_2 \rightarrow KO_2 \quad (\text{superoxide, O.S. } -\tfrac{1}{2})$$$$2M + X_2 \rightarrow 2MX \quad (\text{ionic halide})$$$$2M + H_2 \xrightarrow{\Delta} 2MH \quad (\text{ionic hydride, } H^- \text{ at } -1)$$

Key Reactions — Group 2

$$M + 2H_2O \rightarrow M(OH)_2 + H_2\uparrow \quad (\text{Be no reaction even with steam})$$$$2M + O_2 \rightarrow 2MO \quad (\text{normal oxides})$$$$M + 2HCl \rightarrow MCl_2 + H_2\uparrow$$

Anomalous Beryllium (diagonal to Al)

Amphoteric oxide:

$$BeO + 2HCl \rightarrow BeCl_2 + H_2O$$

$$BeO + 2NaOH \rightarrow Na_2BeO_2 + H_2O$$

Be²⁺ is tiny (31 pm) → high charge density → covalent character; doesn’t react with water; forms complexes — all like Al.

Important s-block Compounds

Compound	Common name	Key equation
NaOH	Caustic soda	$2NaCl + 2H_2O \xrightarrow{elec.} 2NaOH + Cl_2 + H_2$
Na₂CO₃·10H₂O	Washing soda	Solvay process (see below)
NaHCO₃	Baking soda	$2NaHCO_3 \xrightarrow{\Delta} Na_2CO_3 + H_2O + CO_2$
CaO	Quick lime	$CaCO_3 \xrightarrow{\Delta,\,1200K} CaO + CO_2$
Ca(OH)₂	Slaked lime	$CaO + H_2O \rightarrow Ca(OH)_2$
CaCO₃	Limestone/marble	$CaCO_3 \xrightarrow{\Delta} CaO + CO_2$
CaSO₄·2H₂O	Gypsum	heat → PoP
CaSO₄·½H₂O	Plaster of Paris	$CaSO_4{\cdot}2H_2O \xrightarrow{393K} CaSO_4{\cdot}\tfrac{1}{2}H_2O + \tfrac{3}{2}H_2O$

Solvay process:

$$NaCl + NH_3 + CO_2 + H_2O \rightarrow NaHCO_3 + NH_4Cl$$

$$2NaHCO_3 \xrightarrow{\Delta} Na_2CO_3 + H_2O + CO_2$$

Lime-water test for CO₂ (milky → clear):

$$Ca(OH)_2 + CO_2 \rightarrow CaCO_3\downarrow + H_2O \quad (\text{milky})$$

$$CaCO_3 + H_2O + CO_2 \rightarrow Ca(HCO_3)_2 \quad (\text{excess CO}_2,\ \text{clears})$$

s-block Solubility & Flame Tests

Group 2 salt	Solubility down group
Hydroxides	Increases ↓ (hydration energy falls slower)
Sulphates	Decreases ↓ (BaSO₄ insoluble — barium meal)

Element	Flame colour		Element	Flame colour
Li	Crimson red		Cs	Blue
Na	Golden yellow		Ca	Brick red
K	Lilac/violet		Sr	Crimson red
Rb	Red-violet		Ba	Green

p-block Elements (Groups 13-18)

$$\boxed{\text{p-block valence config: } ns^2\,np^{1-6}}$$

Group	Family	Config	First element	Common O.S.
13	Boron	ns²np¹	B (metalloid)	+3, +1
14	Carbon	ns²np²	C (non-metal)	+4, +2
15	Nitrogen (pnictogens)	ns²np³	N (non-metal)	+5, +3, −3
16	Oxygen (chalcogens)	ns²np⁴	O (non-metal)	+6, +4, −2
17	Halogens	ns²np⁵	F (non-metal)	+7 to −1 (F only −1)
18	Noble gases	ns²np⁶ (He 1s²)	He	0 (Xe: +2,+4,+6)

Oxidation States

$$\boxed{\text{Maximum O.S.} = \text{Group number} - 10}$$

Minimum O.S.: Group 15 = −3, Group 16 = −2, Group 17 = −1, Group 18 = 0.

Inert Pair Effect

Reluctance of $ns^2$ electrons to bond, increasing down a group (poor d/f shielding → s² held tightly). Lower O.S. becomes more stable.

$$\text{Stable: } Tl^+ \gg Tl^{3+},\quad Pb^{2+} \gg Pb^{4+},\quad Bi^{3+} \gg Bi^{5+}$$

TlCl₃ and PbCl₄ are unstable strong oxidisers; TlCl and PbCl₂ are stable.

Catenation

$$\boxed{\text{Catenation: } C \gg Si > Ge > Sn > Pb}$$

Bond	Energy (kJ/mol)
C–C	348 (strongest)
Si–Si	226
Ge–Ge	188
Si–O	452 (why Si prefers silicates over chains)

C wins: strong small-atom overlap + resistance to oxidation. C radius ≈ 77 pm.

Allotropy

Element	Allotropes
Carbon	Diamond, Graphite, Fullerene (C₆₀), Graphene
Phosphorus	White (P₄), Red, Black
Sulfur	Rhombic (α-S), Monoclinic (β-S)
Oxygen	O₂ (dioxygen), O₃ (ozone)

Metalloids (6 total)

$$\boxed{B,\ Si,\ Ge,\ As,\ Sb,\ Te}$$

Semiconductors; properties intermediate between metals and non-metals.

First-Element Anomaly (Period 2: B, C, N, O, F)

No d orbitals in valence shell → cannot expand octet (max coordination number 4) → strong pπ-pπ multiple bonding.

• N forms NCl₃ only (no NCl₅); P forms PCl₅ (sp³d, has 3d).
• O cannot form OF₆; S forms SF₆.
• F shows only −1 O.S.; forms a single oxoacid (HOF).
• C: maximum catenation and stable C=C, C≡C.

Oxide & Hydride Trends

Oxide acidity — across a period Basic → Amphoteric → Acidic; down a group acidity decreases.

Period 3: Na₂O, MgO (basic) → Al₂O₃ (amphoteric) → SiO₂ (weakly acidic) → P₄O₁₀, SO₃, Cl₂O₇ (acidic).

$$SO_3 + H_2O \rightarrow H_2SO_4 \qquad Cl_2O_7 + H_2O \rightarrow 2HClO_4$$

Hydride stability decreases down a group (bond length ↑, bond strength ↓):

$$\boxed{NH_3 > PH_3 > AsH_3 > SbH_3 > BiH_3}$$

Halides by Group

Group	Typical halide
13	MX₃ (BCl₃, AlCl₃)
14	MX₄ (CCl₄, SiCl₄)
15	MX₃ and MX₅ (PCl₃, PCl₅)
16	MX₂, MX₄, MX₆ (SF₆)
17	Interhalogens (ClF₃, IF₅, IF₇)

Noble Gases

Monatomic, complete octet, very low boiling points (only weak Van der Waals forces). BP increases with size: He < Ne < Ar < Kr < Xe < Rn. Xenon fluorides XeF₂, XeF₄, XeF₆ and oxides XeO₃, XeO₄.

High-Yield Recap

Related Topics

• Modern Periodic Law
• Periodic Trends
• s-block Elements
• p-block Introduction