d- and f-Block Elements Formula Sheet

One-page revision of every must-know relation, reaction, and trend for the d- and f-block chapter. This is a largely descriptive chapter, so alongside the few quantitative formulas you will find high-yield reactions, oxidation-state charts, and periodic trends.

Core Formulas

The handful of genuinely quantitative relations in this chapter.

Quantity	Formula	Notes
Spin-only magnetic moment	$\mu_s = \sqrt{n(n+2)}$ BM	$n$ = number of unpaired electrons; BM = Bohr Magneton
Lanthanoid effective moment	$\mu_{eff} = g\sqrt{J(J+1)}$ BM	Includes orbital contribution; $J$ = total angular momentum
Octahedral crystal field split	$\Delta_o = E_{e_g} - E_{t_{2g}}$	Energy gap that determines color and spin state

$$\boxed{\mu_s = \sqrt{n(n+2)} \text{ BM}}$$

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Electronic Configurations

General

$$\boxed{\text{Transition elements: } (n-1)d^{1-10}\, ns^{0-2}}$$

A transition element has partially filled $(n-1)d^{1-9}$ orbitals in its ground state OR in a common oxidation state.

$$\boxed{\text{Lanthanoids: } [Xe]\,4f^{1-14}\,5d^{0-1}\,6s^2}$$$$\boxed{\text{Actinoids: } [Rn]\,5f^{0-14}\,6d^{0-2}\,7s^2}$$

3d series anomalies

Element	Expected	Actual	Reason
Cr (24)	[Ar]3d⁴4s²	[Ar]3d⁵4s¹	Half-filled d⁵ stability
Cu (29)	[Ar]3d⁹4s²	[Ar]3d¹⁰4s¹	Fully-filled d¹⁰ stability

f-block anomalies

Element	Expected	Actual	Reason
Ce (58)	[Xe]4f²6s²	[Xe]4f¹5d¹6s²	4f, 5d similar energy
Gd (64)	[Xe]4f⁸6s²	[Xe]4f⁷5d¹6s²	Half-filled 4f stability
Lu (71)	[Xe]4f¹⁴6s²	[Xe]4f¹⁴5d¹6s²	Fully-filled 4f stability

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Oxidation States (3d Series)

Element	Config	Common OS	Max OS
Sc	3d¹4s²	+3	+3
Ti	3d²4s²	+4	+4
V	3d³4s²	+4, +5	+5
Cr	3d⁵4s¹	+3, +6	+6
Mn	3d⁵4s²	+2, +4, +7	+7
Fe	3d⁶4s²	+2, +3	+6
Co	3d⁷4s²	+2, +3	+4
Ni	3d⁸4s²	+2	+4
Cu	3d¹⁰4s¹	+1, +2	+3
Zn	3d¹⁰4s²	+2	+2

Key rules: Maximum OS = group number up to Mn; after Mn it decreases. Higher oxidation states are more oxidizing. Mn shows the maximum number of states (+2 to +7).

Disproportionation reactions

$$2Mn^{3+} \rightarrow Mn^{2+} + Mn^{4+}$$$$2Cu^+ \rightarrow Cu^{2+} + Cu \quad \text{(Cu⁺ unstable in water)}$$$$3MnO_4^{2-} + 4H^+ \rightarrow 2MnO_4^- + MnO_2 + 2H_2O$$

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Color and Magnetism

Color rule

Color arises from d-d transitions. Compounds with d⁰ or d¹⁰ configuration are colorless (no d-d transition possible). Exceptions like KMnO₄ (Mn⁷⁺, d⁰) and K₂Cr₂O₇ (Cr⁶⁺, d⁰) are colored due to charge transfer, not d-d transitions.

Ion	d-electrons	Color
Sc³⁺	d⁰	Colorless
Ti⁴⁺	d⁰	Colorless
Ti³⁺	d¹	Purple
V³⁺	d²	Green
Cr³⁺	d³	Green
Mn²⁺	d⁵	Pale pink
Fe³⁺	d⁵	Yellow
Fe²⁺	d⁶	Pale green
Co²⁺	d⁷	Pink
Ni²⁺	d⁸	Green
Cu²⁺	d⁹	Blue
Zn²⁺	d¹⁰	Colorless

Spin state and ligand field

$$\Delta_o = E_{e_g} - E_{t_{2g}}$$

Field	Ligands	Spin	$\Delta$
Weak	I⁻, Br⁻, Cl⁻, F⁻, OH⁻	High spin (max unpaired)	Small
Strong	CN⁻, CO, NO₂⁻	Low spin (min unpaired)	Large

Ferromagnetism (Curie temperatures)

Fe, Co, Ni are ferromagnetic. Above the Curie temperature they become paramagnetic: Fe = 1043 K, Co = 1394 K, Ni = 631 K.

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Catalysts (Memorize the Process → Catalyst Pairs)

Reaction	Catalyst	Process
$N_2 + 3H_2 \rightarrow 2NH_3$	Fe	Haber process
$2SO_2 + O_2 \rightarrow 2SO_3$	V₂O₅	Contact process
$4NH_3 + 5O_2 \rightarrow 4NO + 6H_2O$	Pt	Ostwald process
Vegetable oil + H₂ → fat	Ni	Hydrogenation
$2H_2O_2 \rightarrow 2H_2O + O_2$	MnO₂	Decomposition
CO + hydrocarbons	Pt/Pd/Rh	Catalytic converter

Contact process mechanism (homogeneous):

$$V_2O_5 \rightarrow V_2O_4 + \tfrac{1}{2}O_2$$

$$V_2O_4 + SO_2 \rightarrow V_2O_5 + SO_3$$

Wilkinson’s catalyst (hydrogenation):

$$RCH{=}CH_2 + H_2 \xrightarrow{[RhCl(PPh_3)_3]} RCH_2CH_3$$

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Potassium Dichromate — K₂Cr₂O₇

Cr in +6; orange color from charge transfer ($O^{2-} \rightarrow Cr^{6+}$). Cr⁶⁺ is [Ar]3d⁰.

Preparation (chromite ore route)

$$4FeCr_2O_4 + 8Na_2CO_3 + 7O_2 \xrightarrow{\Delta} 8Na_2CrO_4 + 2Fe_2O_3 + 8CO_2$$

$$2Na_2CrO_4 + H_2SO_4 \rightarrow Na_2Cr_2O_7 + Na_2SO_4 + H_2O$$

$$Na_2Cr_2O_7 + 2KCl \rightarrow K_2Cr_2O_7 + 2NaCl$$

Chromate–dichromate equilibrium

$$\boxed{2CrO_4^{2-} + 2H^+ \rightleftharpoons Cr_2O_7^{2-} + H_2O}$$

Basic → yellow chromate (CrO₄²⁻); acidic → orange dichromate (Cr₂O₇²⁻).

Oxidizing half-reaction (acidic, n-factor = 6)

$$\boxed{Cr_2O_7^{2-} + 14H^+ + 6e^- \rightarrow 2Cr^{3+} + 7H_2O}$$

Color change: orange → green (Cr³⁺).

Key oxidations

$$Cr_2O_7^{2-} + 14H^+ + 6Fe^{2+} \rightarrow 2Cr^{3+} + 6Fe^{3+} + 7H_2O$$

$$Cr_2O_7^{2-} + 14H^+ + 6I^- \rightarrow 2Cr^{3+} + 3I_2 + 7H_2O$$

$$Cr_2O_7^{2-} + 8H^+ + 3H_2S \rightarrow 2Cr^{3+} + 3S + 7H_2O$$

$$Cr_2O_7^{2-} + 8H^+ + 3SO_3^{2-} \rightarrow 2Cr^{3+} + 3SO_4^{2-} + 4H_2O$$

Action of heat

$$\boxed{4K_2Cr_2O_7 \xrightarrow{\Delta} 4K_2CrO_4 + 2Cr_2O_3 + 3O_2}$$

Chromyl chloride test (chlorides only)

$$K_2Cr_2O_7 + 4NaCl + 6H_2SO_4 \rightarrow 2CrO_2Cl_2 + 2KHSO_4 + 4NaHSO_4 + 3H_2O$$

$$CrO_2Cl_2 + 4NaOH \rightarrow Na_2CrO_4 + 2NaCl + 2H_2O$$

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Potassium Permanganate — KMnO₄

Mn in +7; purple from charge transfer ($O^{2-} \rightarrow Mn^{7+}$). Mn⁷⁺ is [Ar]3d⁰.

Preparation (Baeyer’s process)

$$2MnO_2 + 4KOH + O_2 \xrightarrow{\Delta} 2K_2MnO_4 + 2H_2O$$

Then oxidize green manganate to purple permanganate (electrolytic, or acidic disproportionation):

$$MnO_4^{2-} - e^- \rightarrow MnO_4^-$$

$$3MnO_4^{2-} + 4H^+ \rightarrow 2MnO_4^- + MnO_2 + 2H_2O$$

Reduction depends on pH

$$\boxed{MnO_4^- + 8H^+ + 5e^- \rightarrow Mn^{2+} + 4H_2O} \quad \text{(acidic, n=5)}$$$$\boxed{MnO_4^- + 2H_2O + 3e^- \rightarrow MnO_2 + 4OH^-} \quad \text{(neutral, n=3)}$$$$\boxed{MnO_4^- + e^- \rightarrow MnO_4^{2-}} \quad \text{(strongly basic, n=1)}$$

Key oxidations (acidic medium)

$$2MnO_4^- + 5C_2O_4^{2-} + 16H^+ \rightarrow 2Mn^{2+} + 10CO_2 + 8H_2O$$

$$MnO_4^- + 5Fe^{2+} + 8H^+ \rightarrow Mn^{2+} + 5Fe^{3+} + 4H_2O$$

$$2MnO_4^- + 10I^- + 16H^+ \rightarrow 2Mn^{2+} + 5I_2 + 8H_2O$$

$$2MnO_4^- + 5H_2S + 6H^+ \rightarrow 2Mn^{2+} + 5S + 8H_2O$$

$$2MnO_4^- + 5SO_3^{2-} + 6H^+ \rightarrow 2Mn^{2+} + 5SO_4^{2-} + 3H_2O$$

$$2MnO_4^- + 5NO_2^- + 6H^+ \rightarrow 2Mn^{2+} + 5NO_3^- + 3H_2O$$

Neutral medium

$$2MnO_4^- + I^- + H_2O \rightarrow 2MnO_2 + IO_3^- + 2OH^-$$

$$8MnO_4^- + 3S_2O_3^{2-} + H_2O \rightarrow 8MnO_2 + 6SO_4^{2-} + 2OH^-$$

Action of heat and acids

$$\boxed{2KMnO_4 \xrightarrow{\Delta} K_2MnO_4 + MnO_2 + O_2}$$

$$2KMnO_4 + 16HCl \rightarrow 2KCl + 2MnCl_2 + 5Cl_2 + 8H_2O$$

$$4KMnO_4 + 6H_2SO_4 \rightarrow 2K_2SO_4 + 4MnSO_4 + 6H_2O + 5O_2$$

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K₂Cr₂O₇ vs KMnO₄ — Quick Compare

Property	K₂Cr₂O₇	KMnO₄
Color	Orange-red	Dark purple
Oxidation state	Cr⁶⁺	Mn⁷⁺
Oxidizing power	Moderate	Very strong
n-factor (acidic)	6 (Cr⁶⁺→Cr³⁺)	5 (Mn⁷⁺→Mn²⁺)
Self-indicator	No (orange→green)	Yes (purple→colorless)
Primary standard	Yes	No
Medium	Best in acidic	Works in all pH

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Other Important Compounds and Tests

Compound	Key fact	Test / reaction
K₄[Fe(CN)₆]	Fe²⁺, diamagnetic (low spin d⁶), yellow	Detects Fe³⁺ → Prussian blue
K₃[Fe(CN)₆]	Fe³⁺, paramagnetic (1 unpaired), red	Detects Fe²⁺ → Turnbull’s blue
CuSO₄·5H₂O	Cu²⁺ (d⁹), blue, paramagnetic	Anhydrous CuSO₄ white → blue (water test)
ZnSO₄·7H₂O	Zn²⁺ (d¹⁰), colorless, diamagnetic	—

$$Fe^{3+} + [Fe(CN)_6]^{4-} \rightarrow Fe_4[Fe(CN)_6]_3 \quad \text{(Prussian blue)}$$

$$Fe^{2+} + [Fe(CN)_6]^{3-} \rightarrow Fe_3[Fe(CN)_6]_2 \quad \text{(Turnbull's blue)}$$

Baeyer’s test (unsaturation): cold dilute neutral KMnO₄ turns purple → brown MnO₂ with C=C or C≡C.

$$3CH_2{=}CH_2 + 2MnO_4^- + 4H_2O \rightarrow 3CH_2OH{-}CH_2OH + 2MnO_2 + 2OH^-$$

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Lanthanoid Contraction

Steady decrease in atomic/ionic radii from La to Lu, caused by poor shielding of 4f electrons → effective nuclear charge rises → outer electrons pulled in.

$$\boxed{\text{Ionic size: } La^{3+} > Ce^{3+} > \dots > Yb^{3+} > Lu^{3+}}$$

Total decrease La³⁺ (106 pm) → Lu³⁺ (85 pm) = 21 pm (≈ 1.5 pm per element across 14 steps).

Consequences

graph TD
    A[Poor shielding by 4f electrons] --> B[Effective nuclear charge increases]
    B --> C[Atomic/ionic radius decreases: La to Lu]
    C --> D[Similar radii of 4d and 5d pairs
Zr-Hf, Nb-Ta, Mo-W]
    C --> E[Difficult separation of lanthanoids]
    C --> F[Basicity decreases: La OH 3 to Lu OH 3]
    C --> G[Density and hardness increase]

4d / 5d radii made nearly identical: Zr ≈ Hf (160 ≈ 159 pm), Nb = Ta (146 = 146 pm), Mo = W (139 = 139 pm).

$$\boxed{\text{Basic strength: } La(OH)_3 > Ce(OH)_3 > \dots > Lu(OH)_3}$$

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Lanthanoids (4f Series) — Key Facts

Elements Ce (58) to Lu (71), 14 elements. Predominant oxidation state is +3.

$$\boxed{\text{Stable state: } Ln^{3+}, \text{ config } [Xe]4f^{0-14}}$$

Notable non-+3 states (CEEBY)

Element	Extra OS	Reason
Ce	+4	Ce⁴⁺ = [Xe] noble gas config (good oxidizer)
Eu	+2	Eu²⁺ = f⁷ half-filled (good reducer)
Yb	+2	Yb²⁺ = f¹⁴ fully filled

Magnetism and color

• Magnetic moment uses $\mu_{eff} = g\sqrt{J(J+1)}$ (orbital contribution matters).
• Gd³⁺ (f⁷) has the highest moment (7 unpaired electrons).
• Diamagnetic ions: La³⁺ (f⁰), Lu³⁺ (f¹⁴). Color from weak f-f transitions → pale colors; colorless: La³⁺, Gd³⁺ (f⁷), Lu³⁺.
• Less tendency to form complexes (large size, no vacant d-orbitals); high coordination numbers (8, 9, 12).

Typical reactions

$$4M + 3O_2 \rightarrow 2M_2O_3$$

$$2M + 6H_2O \rightarrow 2M(OH)_3 + 3H_2$$

$$2M + 3X_2 \rightarrow 2MX_3$$

$$2M + 6HCl \rightarrow 2MCl_3 + 3H_2$$

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Actinoids (5f Series) — Key Facts

Elements Th (90) to Lr (103), 14 elements. ALL are radioactive — the defining difference from lanthanoids.

• Oxidation states +3 to +7 (much more variable than lanthanoids) because 5f, 6d, 7s are close in energy and all can bond.
• Early actinoids (Th–Pu) are variable; from Am onward +3 dominates. Common: Th = +4, U = +6 (as UO₂²⁺), Am onward = +3.
• Actinoid contraction parallels lanthanoid contraction but is more irregular.
• Deep, intense colors (5f-5f, 5f-6d, charge-transfer); form complexes more readily than lanthanoids.

Nuclear / breeding reactions

$$^{238}U \rightarrow {}^{234}Th + {}^4He \quad (\alpha\text{-decay})$$

$$^{238}U + n \rightarrow {}^{239}U \xrightarrow{\beta^-} {}^{239}Np \xrightarrow{\beta^-} {}^{239}Pu$$

$$^{235}U + n \rightarrow {}^{141}Ba + {}^{92}Kr + 3n + 200 \text{ MeV}$$

$$^{232}Th + n \rightarrow {}^{233}Th \xrightarrow{\beta^-} {}^{233}Pa \xrightarrow{\beta^-} {}^{233}U \quad \text{(fertile → fissile)}$$

Isotope facts: U-238 (99.3%, fertile), U-235 (0.7%, fissile). UO₂²⁺ is linear (O=U=O). Am-241 is used in smoke detectors.

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Lanthanoids vs Actinoids — Top 5 JEE Differences

Property	Lanthanoids (4f)	Actinoids (5f)
Radioactivity	Only Pm	ALL elements
Oxidation states	Mostly +3	+3 to +7 (variable)
Complex formation	Less	Greater (5f bonding)
Color intensity	Pale (f-f)	Deep, intense
Occurrence	Natural (except Pm)	Th, Pa, U natural; rest synthetic

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Related Topics

• Transition Elements — general characteristics
• Properties of d-Block — oxidation states, color, magnetism
• Important Compounds — K₂Cr₂O₇, KMnO₄ chemistry
• Lanthanoids — 4f series in detail
• Actinoids — 5f series and radioactivity