Class-12 Boards Formula Sheet
Solutions • Electrochemistry • Kinetics • d & f Block • Coordination Compounds
By Omtex Classes
Solutions
\[ \chi_1 = \frac{w_1}{w_1 + w_2} \quad \text{or} \quad \chi_2 = \frac{w_2}{w_1 + w_2} \] (v) Parts per million (ppm): \[ = \frac{w_2}{w_1 + w_2} \times 10^6 \]
(vii) Molality (m): \[ = \frac{w_B \times 1000}{M_B \times w_A(g)} \] (Unit = moles/kg)
Note: Molarity is inversely proportional to temperature.
\[ \chi_A = \frac{n_A}{n_A + n_B} \quad \chi_B = \frac{n_B}{n_A + n_B} \] \[ \chi_A + \chi_B = 1 \] (Mole fraction is a unitless quantity)
Gas Laws
Henry Law: Partial pressure of gas \( p = K_H \chi_g \)
Raoult's Law:
(i) For Volatile Solute:
\[ P_A \propto \chi_A \Rightarrow P_A = P_A^0 \times \chi_A \] \[ P_B \propto \chi_B \Rightarrow P_B = P_B^0 \times \chi_B \] \[ P_T = P_A + P_B \](ii) For Non-Volatile Solute:
\[ P_T = P_A = P_A^0 \times \chi_A \]Ideal vs Non-Ideal Solutions
Ideal Solution: Interaction of A-B are equal to interaction of A-A and B-B.
- \( P_T = P_A + P_B \)
- \( \Delta V_{mix} = 0 \)
- \( \Delta H_{mix} = 0 \)
- Example: n-Hexane + n-Heptane
Non-Ideal Solution:
| Positive Deviation | Negative Deviation |
|---|---|
| A-B interaction are weaker than A-A and B-B interaction | A-B interaction are stronger than A-A and B-B interaction |
| \( \Delta V_{mix} = +ve \) | \( \Delta V_{mix} = -ve \) |
| \( \Delta H_{mix} = +ve \) | \( \Delta H_{mix} = -ve \) |
| \( P_T > P_A + P_B \) | \( P_T < P_A + P_B \) |
| Form minimum boiling Azeotropes | Form Maximum boiling Azeotropes |
| e.g Acetone + Ethanol | e.g Acetone + Chloroform |
Colligative Properties
Depend on number of solute particles.
Unit of \(K_b\) and \(K_f\): K kg mol⁻¹
\( \Delta T_b = T_b - T_b^0 \)
\( \Delta T_f = T_f^0 - T_f \)
\( T_b \rightarrow \) B.Pt of Solution
\( T_b^0 \rightarrow \) B.Pt of pure Solvent
\( T_f^0 \rightarrow \) F.Pt of pure solvent
\( T_f \rightarrow \) F.Pt of solution
Van't Hoff Factor (i)
\[ i = \frac{\text{Normal Molecular Mass}}{\text{Observed Molecular Mass}} \]- (i) \( i > 1 \): Solute undergoes dissociation
- (ii) \( i < 1 \): Solute undergoes association
- (iii) \( i = 1 \): No association, no dissociation
\( \alpha_{\text{dissociation}} = \frac{i - 1}{n - 1} \)
\( \alpha_{\text{association}} = \frac{i - 1}{\frac{1}{n} - 1} \)
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Electrochemistry
Cell: Which convert one form of energy into another form of energy.
| Electrochemical Cell | Electrolytic Cell |
|---|---|
| Chemical \(\rightarrow\) Electrical | Electrical \(\rightarrow\) Chemical |
Representation of a Cell:
Oxidation Half Cell || Reduction Half Cell
\( M/M^{n+} \) || \( M^{n+}/M \)
(Two solution separation)
Nernst Equation
\( M^{n+} + ne^- \rightarrow M(s) \)
\[ E_{cell} = E_{cell}^0 - \frac{0.0591}{n} \log \frac{1}{[M^{n+}]} \] \[ E_{cell} = E_{cell}^0 - \frac{0.0591}{n} \log \frac{[\text{Ox}]}{[\text{Red}]} \quad (\text{at 298 K}) \]At Equilibrium: \( E_{cell} = 0 \)
\[ E_{cell}^0 = \frac{0.0591}{n} \log K_c \quad (\text{at 298 K}) \]Gibbs Free Energy
\[ \Delta G^0 = -nF E_{cell}^0 \] \[ \Delta G^0 = -2.303 RT \log K_c \]Conductivity of Ionic Solutions
Conductance (G): \( = \frac{1}{R} = \frac{1}{\rho} \frac{A}{l} = \kappa \frac{A}{l} \)
Unit = ohm⁻¹ or \(\Omega^{-1}\) or Siemens
Increases on dilution as larger no. of ions are produced.
Specific Conductance (Conductivity) \(\kappa\):
\[ \kappa = \frac{1}{\rho} \text{ or } G \times \frac{l}{A} \text{ or } G \times G^* \]Unit = ohm⁻¹ cm⁻¹ or S cm⁻¹
Decrease on dilution as number of ions per cm³ decrease.
Molar Conductivity (\(\Lambda_m\)):
\[ \Lambda_m = \kappa \times V \quad \text{or} \quad \frac{\kappa \times 1000}{M} \]Unit = \(\Omega^{-1}\) cm² mol⁻¹ or S m² mol⁻¹
Increase with dilution due to large increase in V.
Kohlrausch's Law
\[ \Lambda_m^0 \text{ Electrolyte} = \lambda_m^0 \text{ Cation} + \lambda_m^0 \text{ Anion} \]Degree of dissociation:
\[ \alpha = \frac{\Lambda_m}{\Lambda_m^0} = \frac{[\text{Molar conductivity at concentration C}]}{[\text{Molar conductivity at infinite dilution}]} \]Dissociation Constant (\(K_a\)): \( = \frac{C\alpha^2}{1-\alpha} \)
Faraday's Laws
First Law:
\[ w = Z \times I \times t = \frac{\text{Molar Mass}}{n \times F} \times I \times t \]Second Law:
\[ \frac{w_1}{w_2} = \frac{E_1}{E_2} \quad [\text{where E is equivalent weight}] \]Batteries & Corrosion
Fuel Cell:
- At Anode: \( 2H_2 + 4OH^- \rightarrow 4H_2O + 4e^- \)
- At Cathode: \( O_2 + 2H_2O + 4e^- \rightarrow 4OH^- \)
- Overall Reaction: \( 2H_2 + O_2 \rightarrow 2H_2O \)
Dry Cell:
- At Anode: \( Zn \rightarrow Zn^{2+} + 2e^- \)
- At Cathode: \( 2MnO_2 + 2NH_4^+ + 2e^- \rightarrow 2MnO(OH) + 2NH_3 \)
- Overall Rxn: \( Zn + 2NH_4^+ + 2MnO_2 \rightarrow Zn^{2+} + 2MnO(OH) + 2NH_3 \)
Lead-Storage Battery:
- At Anode: \( Pb + SO_4^{2-} \rightarrow PbSO_4 + 2e^- \)
- At Cathode: \( PbO_2 + SO_4^{2-} \rightarrow PbSO_4 + 2H_2O \)
Corrosion of Iron:
- At Anode: \( 2Fe \rightarrow 2Fe^{2+} + 4e^- \)
- At Cathode: \( O_2 + 4H^+ + 4e^- \rightarrow 2H_2O \)
- Overall Reaction: \( 2Fe + O_2 + 4H^+ \rightarrow 2Fe^{2+} + 2H_2O \)
- Formula of rust: \( Fe_2O_3 \cdot xH_2O \)
Chemical Kinetics
R \(\rightarrow\) P
Rate of Reaction: \( - \frac{\Delta [R]}{\Delta t} = + \frac{\Delta [P]}{\Delta t} \)
Unit of Rate: mol L⁻¹ sec⁻¹ or mol L⁻¹ min⁻¹
| Average rate | Instantaneous rate |
|---|---|
| change in concentration at large time interval | change in concentration at any instant of time |
| \( \frac{-\Delta [R]}{\Delta t} = + \frac{\Delta [P]}{\Delta t} \) | \( \frac{-d[R]}{dt} = + \frac{d[P]}{dt} \) |
Rate Law:
\( aA + bB \rightarrow \text{Product} \)
Rate of Reaction = \( k [A]^\alpha [B]^\beta \)
Order = \( \alpha + \beta \)
(where \(\alpha, \beta\) are actual used)
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Kinetics (Continued)
Half Life (\(t_{1/2}\)): \( t_{1/2} \propto \frac{1}{a^{n-1}} \); where \(n\) is order of rxn.
Rate = \( k[A]^0 \)
\( [A]_t = -kt + [A]_0 \)
\( t = \frac{R_0 - R}{k} \)
\( t_{1/2} = \frac{R_0}{2k} \)
Unit of K = mol L⁻¹ s⁻¹
Rate = \( k[A]^1 \)
\( \ln [A]_t = -kt + \ln [A]_0 \)
\( t = \frac{2.303}{k} \log \frac{R_0}{R_t} \)
\( t_{1/2} = \frac{0.693}{k} \)
Unit of K = sec⁻¹
Pseudo First Order: Those reaction which are not truly of first order but under certain conditions becomes of first order.
Inversion of sugar: \( C_{12}H_{22}O_{11} + H_2O \xrightarrow{H^+} C_6H_{12}O_6 + C_6H_{12}O_6 \)
Rate = \( K [C_{12}H_{22}O_{11}] \)
Arrhenius Equation
\[ k = A e^{-E_a/RT} \] \[ \log k = \log A - \frac{E_a}{2.303 RT} \]where \(k\) = Rate constant, \(A\) = Pre-exponential factor, \(E_a\) = Activation Energy.
\[ \log \frac{k_2}{k_1} = \frac{E_a}{2.303 R} \left[ \frac{T_2 - T_1}{T_1 T_2} \right] \]Role of Catalyst: A chemical substance which alters the rate of reaction without undergoing any chemical change.
Collision Theory
The number of collision b/w the reacting molecules taking place per second per unit volume is known as Collision Frequency.
\[ \text{rate} = P Z_{AB} e^{-E_a/RT} \]\(P\) (or \(\rho\)) called the probability or steric factor.
\(Z_{AB}\) = Collision Frequency for reactant A & B.
The d- and f- Block Elements
- General Electronic Confi: \( (n-1)d^{1-10} ns^{0-2} \)
- M.Pt & B.Pt: High due to strong metallic bond. Strength of metallic bond due to unpaired e⁻.
- Enthalpies of Atomisation: High due to strong interatomic interaction.
- Oxidation State: Variable oxidation state due to participation of ns and (n-1)d electrons [Highest in 3d series Mn +7].
- Atomic Radii: Decreases from left to right, in midway size remains same and in the end of series size increases.
- Complex Formation: Form complex due to high nuclear charge, small size metal ions and availability of empty d-orbital to accept lone pair of e⁻ donated by ligands.
- Coloured Compounds: Form coloured compounds due to d-d transition, due to unpaired e⁻.
- Alloy Formation: Due to small similar atomic radii, atom of one metal can easily replace the atom of other metal.
- Interstitial Compound: Due to empty space in their lattices, small atoms can be easily accommodated.
- Magnetic Properties: Transition metal ions and their compounds are paramagnetic due to presence of unpaired e⁻ in (n-1)d orbitals and it is calculated by using formula \( \mu = \sqrt{n(n+2)} \) where n is number of unpaired e⁻.
Oxides in higher oxidation state are acidic, lower oxidation state are Basic whereas in the intermediate oxi. state are amphoteric.
Example: \( MnO \) (Basic), \( Mn_3O_4, MnO_2 \) (Amphoteric), \( Mn_2O_7 \) (Acidic).
Potassium Dichromate [\(K_2Cr_2O_7\)]
Preparation:
\( FeCr_2O_4 \xrightarrow[O_2]{Na_2CO_3} Na_2CrO_4 \xrightarrow{H_2SO_4} Na_2Cr_2O_7 \xrightarrow{KCl} K_2Cr_2O_7 \)
(Ferrochromate) \(\rightarrow\) (Sod. Chromate) \(\rightarrow\) (Sod. Dichromate) \(\rightarrow\) (Pot. Dichromate)
\( CrO_4^{2-} \xrightleftharpoons[OH^-]{H^+} Cr_2O_7^{2-} \)
(Yellow) \(\rightleftharpoons\) (Orange)
Oxidising action in acidic medium:
\( Cr_2O_7^{2-} + 14H^+ + 6e^- \rightarrow 2Cr^{3+} + 7H_2O \)
- \( I^- \rightarrow I_2 \)
- \( S^{2-} \rightarrow S \)
- \( Sn^{2+} \rightarrow Sn^{4+} \)
- \( Fe^{2+} \rightarrow Fe^{3+} \)
Potassium Permanganate (\(KMnO_4\))
Deep purple crystalline solid, oxidising agent, having m.pt 240°C.
Preparation:
\( MnO_2 + KOH + O_2 \rightarrow K_2MnO_4 \xrightarrow{H^+ \text{ or } Cl_2} KMnO_4 \)
(Pyrolusite) \(\rightarrow\) (Pot. Manganate) \(\rightarrow\) (Pot. Permanganate)
Oxidising Action in Acidic Medium:
\( MnO_4^- + 8H^+ + 5e^- \rightarrow Mn^{2+} + 4H_2O \)
- \( Fe^{2+} \rightarrow Fe^{3+} \)
- \( H_2S \rightarrow S \)
- \( I^- \rightarrow I_2 \)
- Oxalic acid \(\rightarrow CO_2 + H_2O\)
- \( SO_2 \rightarrow H_2SO_4 \)
Oxidising Reactions in Neutral Medium:
\( MnO_4^- + 2H_2O + 3e^- \rightarrow MnO_2 + 4OH^- \)
- \( H_2S \rightarrow S \)
- \( MnSO_4 \rightarrow MnO_2 \)
- \( Na_2S_2O_3 \rightarrow Na_2SO_4 \) (Thiosulphate to Sulphate)
Inner Transition Elements (f-Block)
| Lanthanoids (4f) | Actinoids (5f) |
|---|---|
| last e⁻ enters in 4f orbital | last e⁻ enters in 5f orbital |
General electronic confi: \( \rightarrow (n-2)f^{1-14} (n-1)d^{0-1} ns^2 \)
Lanthanoid Contraction: decrease in atomic and ionic radii from Lanthanum to Lutetium. The gradual and steady decrease across the period is called lanthanoid contraction.
Cause: 4f orbitals have very poor shielding effect from left to right nuclear charge increases and radius decreases.
Consequences:
- Separation of 4f elements becomes difficult because there is small difference in their properties.
- 4d and 5d transition series have same atomic radii.
- Basic character of hydroxides of Lanthanoides decreases from left to right. \( La(OH)_3 \) most basic, \( Lu(OH)_3 \) least basic.
Difference b/w Lanthanoids and Actinoids
| Lanthanoids | Actinoids |
|---|---|
| They exhibit mainly +3 O.S in addition they show +2 and +4 oxidation state. | They exhibit +3 oxi. state in addition they show +4, +5, +6, +7 oxi. state. |
| They show Lanthanoid Contraction. | They show Actinoid Contraction. |
| Most ions of Lanthanoids are colourless. | Most ions of Actinoids are coloured. |
| Lesser tendency to form complex. | Greater tendency to form complex. |
| They do not give oxo cations. | They give oxo cations eg \( UO_2^{2+}, PuO_2^{2+} \) etc. |
| Their compounds are less Basic. | Their compounds are more Basic. |
| They are non-radioactive (except Promethium). | They are radioactive. |
Co-ordination Compounds
Addition Compounds: formed by combination of two or more simple compounds.
Double Salt: which dissociate completely into its ions. e.g \( K_2SO_4 \cdot Al_2(SO_4)_3 \cdot 24H_2O \) (Potash alum).
Co-ordination Compound: retain its identity both in solid state and in solution. e.g \( [Co(NH_3)_6]^{3+} \).
Werner Theory
Explain the nature of bonding in complexes, metal show two kind of valencies.
- Primary Valency: Non-directional and ionisable and equal to oxi. state of central metal ion.
- Secondary Valency: Directional and non-ionisable. It is equal to the coordin. number of metal.
Ligands: These are atom, ion or molecule which donate lone pair of e⁻ to central metal atom.
On the Basis of Charge:
- -ve ligands: \( CN^-, F^-, Cl^-, NO_2^-, OH^-, O^- \)
- +ve ligands: \( NO^+, NO_2^+, NH_4^+ \)
- Neutral ligands: \( H_2O, NH_3, CO, CH_3NH_2 \)
Basis of Number of Donor Sites:
- Monodentate: Only one donor site (e.g., \( H_2O, NH_3 \))
- Bidentate: Two donor site (e.g., \( COO^- - COO^- \) (oxalato), Ethylene diamine)
- Polydentate: More than two donor site (e.g., EDTA)
Chelating Ligand: a bidentate or polydentate ligand which form more than one coordinate bond in such a way that a ring is formed.
Ambidentate Ligand: Monodentate ligand which contain more than one coordinating atoms.
\( M \leftarrow SCN \) vs \( M \leftarrow NCS \); \( M \leftarrow CN \) vs \( M \leftarrow NC \); \( M \leftarrow NO_2 \) vs \( M \leftarrow ONO \)
Isomerism
Two or more substance have same molecular formula but different structure or spatial arrangement are called isomers.
Structural Isomerism:
- Ionisation: \( [CoBr(NH_3)_5]SO_4 \) vs \( [CoSO_4(NH_3)_5]Br \)
- Hydrate: \( [Cr(H_2O)_6]Cl_3 \) vs \( [Cr(H_2O)_5Cl]Cl_2 \cdot H_2O \)
- Linkage: \( [Co(NO_2)(NH_3)_5]Cl \) vs \( [Co(ONO)(NH_3)_5]Cl \)
- Co-ordination: \( [Co(NH_3)_6][Cr(CN)_6] \) vs \( [Co(CN)_6][Cr(NH_3)_6] \)
Stereo Isomerism
Geometrical:
- Cis: same ligands occupy adjacent position.
- Trans: same ligands occupy opposite position.
- Occurs in Square Planar (\( MA_2X_2 \)) and Octahedral (\( MA_4X_2, MA_3X_3 \)).
Optical:
- This isomerism arises due to non-super imposable mirror images.
- only in octahedral complexes with 2 or 3 bidentate ligands.
- \( Cis \) (Optically Active) vs \( Trans \) (Optically inactive).
Spectrochemical Series
Arrangement of ligands in the order of increasing field strength.
\( I^- < Br^- < SCN^- < Cl^- < S^{2-} < F^- < OH^- < OX < H_2O < NCS^- < edta^{4-} < NH_3 < en < NO_2 < CN^- < CO \)
Weak field ligands \(\rightarrow\) Strong field ligands.
Valence Bond Theory
| Inner Orbital Complex | Outer Orbital Complex |
|---|---|
| involve inner d-orbital i.e (n-1) d-orbital | involve outer d-orbital i.e nd-orbitals |
| Low Spin Complex | High Spin Complex |
| Have less or no unpaired e⁻ | Have large number of unpaired e⁻ |
| e.g \( [Co(NH_3)_6]^{3+}, [Co(CN)_6]^{4-} \) | e.g \( [MnF_6]^{3-}, [CoF_6]^{3-} \) |
Valence Bond Theory:
Acc. to this theory, metal-ligand bond arises due to the donation of electron pair from ligands to central metal atom. The metal atom or ion under the influence of ligands can use (n-1)d, ns, np, nd orbitals for hybridisation.
| Hybridisation | C.N | Geometry | Example |
|---|---|---|---|
| sp | 2 | Linear | \( [Ag(CN)_2]^- \) |
| sp² | 3 | Trigonal planar | \( [HgI_3]^- \) |
| sp³ | 4 | Tetrahedral | \( [Ni(CO)_4] \) |
| dsp² | 4 | Square planar | \( [Ni(CN)_4]^{2-} \) |
| dsp³ | 5 | Square pyramidal | \( Fe(CO)_5 \) |
| d²sp³ | 6 | Octahedral (inner) | \( [Cr(NH_3)_6]^{3+} \) |
| sp³d² | 6 | Octahedral (outer) | \( [FeF_6]^{3-} \) |
Paramagnetism \(\propto\) No. of unpaired e⁻
Magnetic Moment \( = \sqrt{n(n+2)} \) (n is no of unpaired e⁻)
Strong field ligands: like \( CN^-, NO_2, CN, NH_3 \), cause pairing of e⁻.
Weak field ligands: like \( Cl^-, Br^-, H_2O \), are unable to cause pairing of e⁻.
Crystal Field Theory
Metal-Ligand bond is ionic in nature. So, there is electrostatic force of attraction b/w metal and ligands.
For Octahedral Complex:
Splitting into \( t_{2g} \) (lower energy) and \( e_g \) (higher energy).
Gap is \( \Delta_o \).
For Tetrahedral Complex:
Splitting into \( e \) (lower energy) and \( t_2 \) (higher energy).
Gap is \( \Delta_t \).
\( \Delta_t = \frac{4}{9} \Delta_o \)
Filling Rules:
- If \( \Delta_o < P \), a high spin complex is formed.
- If \( \Delta_o > P \), a low spin complex is formed.