Unit 1

Atomic Structure

de-Broglie waves, Heisenberg uncertainty, Schrödinger equation, quantum numbers, orbital shapes, Aufbau & Pauli principles

1.1 Why We Need Quantum Mechanics

Prerequisites: Atoms are made of protons (+), neutrons (0), and electrons (−). Electrons exist around the nucleus. That's all you need!

Before 1900 (classical physics), scientists believed:

  • Electrons orbit the nucleus like planets orbit the sun
  • Energy can have any value (continuous)
  • We can know exactly where a particle is and how fast it moves
All three turned out to be WRONG at the atomic scale! Quantum mechanics was born to fix these failures.

1.2 de-Broglie Matter Waves (1924)

💡
The Big Idea: "If light (a wave) can behave like a particle (photon), then particles (like electrons) should also behave like waves." — This is wave-particle duality.
de-Broglie Wavelength
$$\lambda = \frac{h}{mv} = \frac{h}{p}$$
SymbolMeaningUnit
λ (lambda)Wavelength of the particlemetres (m)
hPlanck's constant = 6.626 × 10⁻³⁴ J·sJ·s
mMass of the particlekg
vVelocity of the particlem/s
pMomentum (= m × v)kg·m/s

Key Points for Exam

  • Heavier objects → smaller λ → wave nature is negligible
  • Lighter objects (electrons) → larger λ → wave nature is significant
  • de-Broglie waves are not electromagnetic waves — they are probability waves
  • Experimentally confirmed by Davisson and Germer (1927) — electron diffraction
📝 Example

An electron (m = 9.1 × 10⁻³¹ kg) moving at v = 10⁶ m/s:

$$\lambda = \frac{h}{mv} = \frac{6.626 \times 10^{-34}}{9.1 \times 10^{-31} \times 10^6} = 7.28 \times 10^{-10} \text{ m} \approx 0.728 \text{ Å}$$

This is comparable to atomic dimensions, so electron diffraction is observable!

1.3 Heisenberg Uncertainty Principle (1927)

💡
The Big Idea: "It is impossible to simultaneously determine both the exact position and the exact momentum of a microscopic particle with absolute precision."
Uncertainty Relation
$$\Delta x \cdot \Delta p \geq \frac{h}{4\pi}$$
$$\Delta x \cdot m \cdot \Delta v \geq \frac{h}{4\pi}$$

What This Means (Simply)

  • If you know exactly where an electron is (Δx → 0), you lose all information about its speed (Δv → ∞)
  • If you know exactly how fast it's going (Δv → 0), you have no idea where it is (Δx → ∞)
  • This is NOT a limitation of instruments — it is a fundamental law of nature
Why it matters: Destroys the concept of fixed orbits (Bohr's model). Replaces "orbit" with "orbital" — a region of probability.
Energy-Time Form
$$\Delta E \cdot \Delta t \geq \frac{h}{4\pi}$$

1.4 Atomic Orbitals

An orbital is a 3D region around the nucleus where the probability of finding an electron is maximum (typically 90-95% boundary).

FeatureOrbit (Bohr) ❌Orbital (Quantum) ✅
Dimension2D (circular path)3D (probability cloud)
ShapeAlways circulars, p, d, f — various
Max electrons2n²2 per orbital
PrecisionExact path knownOnly probability known

1.5 Schrödinger Wave Equation (1926)

Erwin Schrödinger combined de-Broglie's matter wave concept, classical wave equation, and energy conservation to create the most important equation in quantum mechanics.

Time-Independent Schrödinger Equation
$$\hat{H}\Psi = E\Psi$$
$$-\frac{h^2}{8\pi^2 m} \nabla^2\Psi + V\Psi = E\Psi$$
SymbolMeaning
ĤHamiltonian operator (total energy operator)
Ψ (psi)Wave function
ETotal energy of the system
VPotential energy
∇²Laplacian operator
Time-Dependent Schrödinger Equation
$$i\hbar \frac{\partial \Psi}{\partial t} = \hat{H}\Psi$$

1.6 Significance of Ψ and Ψ²

Ψ (Wave Function)

  • No direct physical meaning by itself
  • Contains all information about the quantum system
  • Can be positive, negative, or complex
  • Also called the amplitude of the electron wave

Ψ² (Probability Density) ⭐

  • Ψ² gives the probability of finding the electron at a particular point
  • Always positive (it's a square)
  • If Ψ² is large → high chance of finding electron
  • If Ψ² = 0 → node (zero probability)
Normalization Condition
$$\int_{-\infty}^{+\infty} |\Psi|^2 \, dV = 1$$

Total probability of finding electron somewhere = 1 (100%)

Conditions for a Valid Wave Function (ψ must be):

  1. Single-valued — one value of ψ per point
  2. Continuous — no sudden jumps
  3. Finite — ψ must not be infinity
  4. Normalizable — integral of |ψ|² must equal 1

1.7 Quantum Numbers ⭐

Quantum numbers are the "address" of an electron in an atom. There are four:

Quantum No.SymbolWhat it tellsValues
PrincipalnShell / energy level1, 2, 3, ...
AzimuthallSubshell / shape0 to (n−1)
MagneticmlOrientation−l to +l
SpinmsSpin direction+½ or −½

Subshell Names

l value0123
Subshellspdf
ShapeSphericalDumbbellCloverleafComplex
Orbitals1357

1.8 Radial & Angular Wave Functions

Separation of Wave Function
$$\Psi(r, \theta, \phi) = R(r) \times Y(\theta, \phi)$$

R(r) — Radial Part

  • How Ψ changes with distance from nucleus
  • Depends on n and l
  • Radial nodes = n − l − 1

Y(θ,φ) — Angular Part

  • How Ψ changes with direction
  • Depends on l and ml
  • Angular nodes = l
Total nodes = n − 1 = (Radial nodes) + (Angular nodes) = (n−l−1) + l
OrbitalnlRadial nodesAngular nodesTotal
1s10000
2s20101
2p21011
3s30202
3d32022

1.9 Probability Distribution Curves

Radial Distribution Function
$$P(r) = 4\pi r^2 R^2(r)$$

Key Features:

  • 1s orbital: Single peak at r = a₀ (Bohr radius = 0.529 Å)
  • 2s orbital: Two peaks with a node; outer peak is higher
  • s orbitals have non-zero probability at the nucleus
  • p, d, f orbitals have zero probability at the nucleus
  • s electrons have more "penetration" toward the nucleus

1.10 Shapes of s, p, d Orbitals ⭐

s Orbitals (l = 0)

🔵 Sphere
  • Spherical shape
  • Non-directional
  • 1s: no node
  • 2s: 1 radial node
  • 3s: 2 radial nodes

p Orbitals (l = 1)

🎯 Dumbbell
  • Two lobes
  • 3 orientations: px, py, pz
  • 1 nodal plane each
  • Lobes have opposite signs (+/−)
  • Degenerate in free atom

d Orbitals (l = 2)

🍀 Cloverleaf
  • 5 orientations
  • dxy, dxz, dyz: 4 lobes between axes
  • dx²−y²: 4 lobes along axes
  • dz²: unique doughnut shape
  • 2 nodal surfaces each

1.11 Aufbau Principle

💡
"Electrons fill orbitals starting from the lowest energy level to the highest."

Filling Order (n+l rule):

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

Lower (n+l) fills first. Same (n+l) → lower n fills first.

1.12 Pauli Exclusion Principle

💡
"No two electrons in an atom can have the same set of all four quantum numbers."
  • Each orbital: maximum 2 electrons with opposite spins (↑↓)
  • Max per subshell: s=2, p=6, d=10, f=14
  • Max per shell = 2n²

1.13 Hund's Rule of Maximum Multiplicity

💡
"Electrons occupy degenerate orbitals singly (with parallel spins) before pairing up."
📝 Example: Nitrogen (Z = 7) — 1s² 2s² 2p³
2p: [↑] [↑] [↑] ✅ CORRECT (all parallel spins first) 2p: [↑↓] [↑] [ ] ❌ WRONG (pairing before all singly occupied)

Reason: Less electron repulsion + exchange energy stabilization. Multiplicity = 2S + 1; higher multiplicity = more stable.

Unit 2

Elementary Quantum Mechanics

Black-body radiation, Planck's law, photoelectric effect, Bohr's model, Compton effect, Hamiltonian, particle in a box, H-atom, MO theory

2.1 Black-Body Radiation

A black body absorbs all radiation and emits based only on temperature.

TheoryPredictionResult
Wien's LawCorrect at short λ, fails at long λ❌ Partial
Rayleigh-JeansCorrect at long λ, predicts infinite energy at short λ!❌ "Ultraviolet Catastrophe"

2.2 Planck's Radiation Law (1900)

💡
Max Planck's solution: "Energy is emitted in discrete packets called quanta, not continuously."
Quantized Energy
$$E = nh\nu$$

n = integer, h = 6.626 × 10⁻³⁴ J·s, ν = frequency

This was the birth of quantum theory. Correctly predicts the entire black-body spectrum.

2.3 Photoelectric Effect ⭐

Light of sufficient frequency on a metal surface → electrons are ejected.

ObservationClassical?
Below threshold ν₀ → NO electrons regardless of intensity❌
Above ν₀ → electrons emitted instantly❌
KE depends on FREQUENCY, not intensity❌
More intensity → more electrons (not faster)❌
Einstein's Equation (1905)
$$KE_{max} = h\nu - h\nu_0 = h(\nu - \nu_0)$$

W = hν₀ = work function; one photon ejects one electron

2.4 Heat Capacity of Solids

Classical (Dulong-Petit): Cv = 3R ≈ 25 J/(mol·K) — works at high T, fails at low T.

Einstein's Model (1907): Atoms as quantum harmonic oscillators — at low T, atoms can't absorb even one quantum → Cv drops.

Debye's improvement: Cv ∝ T³ at very low temperatures (Debye T³ law).

2.5 Bohr's Model of Hydrogen Atom (1913)

Postulates:

  1. Electrons in fixed circular orbits (stationary states)
  2. Each orbit has definite energy — no radiation while in orbit
  3. Angular momentum quantized: $mvr = \frac{nh}{2\pi}$
  4. Energy emitted/absorbed during jumps: $\Delta E = h\nu$
Key Results
$$r_n = \frac{n^2 a_0}{Z} \qquad E_n = -\frac{13.6 \, Z^2}{n^2} \text{ eV}$$
SeriesTo n =Region
Lyman1Ultraviolet
Balmer2Visible
Paschen3Infrared
Brackett4Far infrared
Pfund5Far infrared
Defects of Bohr's Model ⚠️:
  1. Cannot explain multi-electron spectra
  2. Cannot explain fine structure
  3. Cannot explain Zeeman/Stark effects
  4. Violates Heisenberg uncertainty
  5. Cannot explain bonding
  6. Ignores wave nature of electron
  7. Cannot predict line intensities

2.6 Compton Effect (1923)

X-rays scattered by electrons have a longer wavelength than incident X-rays — photon transfers energy to electron.

Compton Shift
$$\Delta\lambda = \frac{h}{m_e c}(1 - \cos\theta)$$

Compton wavelength = h/(mec) = 0.02426 Å

  • Max shift at θ = 180°: Δλ = 0.0485 Å
  • Zero shift at θ = 0°
  • Proves particle nature of EM radiation

2.7 Hamiltonian Operator

The Hamiltonian represents total energy (kinetic + potential):

Hamiltonian
$$\hat{H} = -\frac{h^2}{8\pi^2 m}\nabla^2 + V$$

The Schrödinger equation is simply: $\hat{H}\Psi = E\Psi$ — an eigenvalue equation.

2.8 Postulates of Quantum Mechanics

1
Wave Function: Every system is completely described by Ψ.
2
Operators: Every observable is represented by a linear Hermitian operator.
3
Measurement: Results are eigenvalues of the operator.
4
Expectation Value: $\langle A \rangle = \int \Psi^* \hat{A} \Psi \, d\tau$
5
Time Evolution: $i\hbar \frac{\partial \Psi}{\partial t} = \hat{H}\Psi$

2.9 Particle in a 1D Box ⭐⭐

This is the simplest QM problem and a favourite in exams!

Setup: Mass m confined between x = 0 and x = L. V = 0 inside, V = ∞ outside.

Wave Functions
$$\Psi_n(x) = \sqrt{\frac{2}{L}} \sin\!\left(\frac{n\pi x}{L}\right)$$
Energy Levels
$$E_n = \frac{n^2 h^2}{8mL^2} \qquad n = 1, 2, 3, ...$$

Key Results:

FeatureClassicalQuantum
EnergyAny valueOnly specific values (quantized)
Min energyE = 0 possibleE₁ = h²/(8mL²) > 0 (zero-point energy!)
Position probabilityEqual everywhereDepends on n
  • E₂ = 4E₁, E₃ = 9E₁ (goes as n²)
  • n−1 nodes in the nth state
  • Boundary conditions: Ψ(0) = 0, Ψ(L) = 0

2.10 Schrödinger Equation for H-Atom

Separation into 3 Equations
$$\Psi_{n,l,m_l} = R_{n,l}(r) \cdot Y_{l,m_l}(\theta, \phi)$$
  1. Radial equation → R(r) → depends on n, l → introduces n
  2. Polar equation → Θ(θ) → depends on l, ml → introduces l
  3. Azimuthal equation → Φ(φ) → depends on ml → introduces ml

Example Wave Functions:

$$\Psi_{1s} = \frac{1}{\sqrt{\pi}} \left(\frac{Z}{a_0}\right)^{3/2} e^{-Zr/a_0}$$
$$\Psi_{2p_z} = \frac{1}{4\sqrt{2\pi}} \left(\frac{Z}{a_0}\right)^{5/2} r \cdot e^{-Zr/(2a_0)} \cdot \cos\theta$$

2.11 Molecular Orbital Theory ⭐

Basic Ideas

Atomic orbitals combine to form molecular orbitals (MOs) using the LCAO method (Linear Combination of Atomic Orbitals).

$$\Psi_b = \phi_A + \phi_B \quad \text{(bonding MO)}$$
$$\Psi_a = \phi_A - \phi_B \quad \text{(antibonding MO)}$$

Criteria for Forming MOs from AOs

  1. Similar energy — combining AOs must have comparable energies
  2. Maximum overlap — significant spatial overlap required
  3. Same symmetry about the molecular axis

Types of Molecular Orbitals

Propertyσσ*ππ*
FormationHead-on overlapHead-on (subtract)Lateral overlapLateral (subtract)
Node between nucleiNoYesNodal plane on axisYes
EnergyLow (bonding)High (antibonding)IntermediateHigh
StabilityStabilizingDestabilizingStabilizingDestabilizing
Bond Order
$$\text{Bond Order} = \frac{N_b - N_a}{2}$$

For H₂⁺: Bond order = (1−0)/2 = 0.5 (half bond, weakest but exists!)

Unit 3

Molecular Spectroscopy

EM radiation, Born-Oppenheimer approximation, rotational & vibrational spectra, Raman spectroscopy, electronic spectrum

3.1 Electromagnetic Radiation

$$c = \nu\lambda \qquad E = h\nu = \frac{hc}{\lambda} = hc\bar{\nu}$$
RegionWavelengthTransitionSpectroscopy
Radio> 1 mNuclear spinNMR
Microwave1 mm – 1 mMolecular rotationRotational
Infrared700 nm – 1 mmMolecular vibrationIR
Visible400 – 700 nmOuter electronUV-Vis
Ultraviolet10 – 400 nmOuter electronUV-Vis

3.2 Born-Oppenheimer Approximation

💡
"Because nuclei are much heavier (~1836×) than electrons, nuclear and electronic motion can be treated independently."
$$E_{total} = E_{electronic} + E_{vibrational} + E_{rotational}$$

$E_{electronic} \gg E_{vibrational} \gg E_{rotational}$

3.3 Degrees of Freedom

TypeTranslationalRotationalVibrational
Linear (N atoms)323N − 5
Non-linear (N atoms)333N − 6
Examples

HCl (2 atoms, linear): 3(2)−5 = 1 mode | H₂O (3, non-linear): 3(3)−6 = 3 modes | CH₄ (5, non-linear): 3(5)−6 = 9 modes

3.4 Rotational Spectrum — Rigid Rotor

Energy Levels
$$\tilde{E}_J = \tilde{B}J(J+1) \quad \text{cm}^{-1}$$

B = h/(8π²Ic), I = μr², μ = m₁m₂/(m₁+m₂)

Selection Rule: ΔJ = ±1 AND molecule must have permanent dipole moment (μ ≠ 0)

Absorption frequencies: $\tilde{\nu} = 2\tilde{B}(J+1)$ → equally spaced lines at 2B, 4B, 6B, 8B...

Population & Intensity (Maxwell-Boltzmann)

$N_J/N_0 = (2J+1) \cdot e^{-E_J/k_BT}$ → Intensity first increases, reaches max, then decreases.

Bond Length Determination

Spacing = 2B → B → I = h/(8π²cB) → r = √(I/μ)

Non-Rigid Rotor

Real bonds stretch: $\tilde{E}_J = \tilde{B}J(J+1) - \tilde{D}J^2(J+1)^2$

Isotope Effect

Bond length stays same, reduced mass changes → B changes → lines shift. Heavier isotope → smaller B → closer lines.

3.5 Vibrational Spectrum (IR)

Harmonic Oscillator Energy
$$E_v = \left(v + \tfrac{1}{2}\right)h\nu_0 \qquad \nu_0 = \frac{1}{2\pi}\sqrt{\frac{k}{\mu}}$$
Selection Rule: Δv = ±1 AND dipole moment must change during vibration

Force Constant: $k = (2\pi c\bar{\nu}_0)^2 \mu$

Bondk (N/m)
C−C~500
C=C~1000
C≡C~1500

Stronger bond → higher k → higher frequency

Anharmonicity Effects

  • Energy levels get closer at higher v
  • Overtones appear (Δv = ±2, ±3, ...)
  • Dissociation possible at high v

Isotope Effect

k stays same, μ changes → heavier isotope → lower frequency (e.g., HCl: 2886 cm⁻¹ vs DCl: 2091 cm⁻¹)

3.6 Raman Spectroscopy

Polarizability (α)

How easily the electron cloud is distorted by an electric field: $\vec{p}_{induced} = \alpha\vec{E}$

TypeFrequencyWhat Happens
Rayleigh (most)Same (ν₀)Elastic scattering
StokesLower (ν₀ − νvib)Molecule gains energy
Anti-StokesHigher (ν₀ + νvib)Molecule loses energy
Selection Rule: Polarizability must change (dα/dr ≠ 0). Rotational: ΔJ = 0, ±2

IR vs Raman

FeatureIRRaman
RequiresDipole changePolarizability change
H₂, O₂, N₂❌ Inactive✅ Active
Rot. selectionΔJ = ±1ΔJ = 0, ±2

Mutual Exclusion: For molecules with center of symmetry — IR active ↔ Raman inactive, and vice versa.

3.7 Electronic Spectrum

  • Bonding MO: PE curve has minimum at equilibrium bond length (stable)
  • Antibonding MO: PE increases monotonically (repulsive)
  • Electronic transitions need highest energy (UV-Vis region)

Selection Rules (Qualitative)

  1. Spin rule: ΔS = 0 (no change in spin multiplicity)
  2. Laporte rule: Change in parity required (g → u allowed)
Unit 4

UV-Visible Spectroscopy

Electronic transitions, chromophores, auxochromes, Woodward Rules, Beer-Lambert law

4.1 Origin & Beer-Lambert Law

UV-Vis light promotes electrons from lower to higher energy orbitals.

Beer-Lambert Law
$$A = \varepsilon c l = \log\!\left(\frac{I_0}{I}\right)$$

A = absorbance, ε = molar absorptivity (L mol⁻¹ cm⁻¹), c = concentration, l = path length

4.2 Types of Electronic Transitions

TransitionEnergyλ RangeExample
n → π*Lowest270-350 nmC=O in acetone (~280)
π → π*Moderate200-500 nmC=C, aromatics
n → σ*High150-250 nm−OH, −NH₂
σ → σ*Highest< 150 nmC−C, C−H

4.3 Chromophores, Auxochromes & Shifts

Chromophore

Group that absorbs UV-Vis radiation

Examples: C=C, C=O, N=N, NO₂, benzene

Auxochrome

Group that enhances absorption when attached to chromophore

Examples: −OH, −NH₂, −OR, −Cl

TermMeaningCause
Bathochromic (Red) Shiftλmax → longer λConjugation, auxochromes
Hypsochromic (Blue) Shiftλmax → shorter λRemoving conjugation
Hyperchromicε increases—
Hypochromicε decreases—

4.4 Woodward Rules for Conjugated Dienes ⭐

Diene TypeBase Value (nm)
Acyclic / Open-chain217
Homoannular (both C=C in same ring)253
Heteroannular (C=C in different rings)214

Increments:

FeatureAdd (nm)
Each additional conjugated C=C+30
Each alkyl substituent / ring residue on C=C+5
Each exocyclic double bond+5
−OR (alkoxy)+6
−Cl, −Br+5
−SR (thioether)+30
📝 Example: 2,3-dimethyl-1,3-butadiene

Base (acyclic): 217 + 2 alkyl groups (2 × 5) = 10 → λmax = 227 nm (Observed: 226 ✅)

4.5 cis vs trans Isomers

Propertytrans-Stilbenecis-Stilbene
λmax295 nm280 nm
εmax29,00013,500
PlanarityMore planarNon-planar (steric strain)
Rule: trans isomers absorb at longer wavelengths with higher intensity than cis (better conjugation due to planarity).
Unit 5

Infrared Spectroscopy

Molecular vibrations, Hooke's law, functional group frequencies, effects on IR positions, fingerprint region

5.1 Molecular Vibrations

Stretching (bond length changes)

  • Symmetric stretch
  • Asymmetric stretch

Bending (bond angle changes)

  • Scissoring & Rocking (in-plane)
  • Wagging & Twisting (out-of-plane)

Non-Fundamental Vibrations

Overtones (Δv = 2,3...), Combination bands (ν₁ + ν₂), Hot bands (from excited states)

5.2 IR Absorption Positions of Functional Groups ⭐

Carbonyl C=O (1650-1800 cm⁻¹) — Most Important!

Compoundν̃ (cm⁻¹)
Acid anhydride1800-1850 & 1740-1790 (two bands!)
Acid chloride1790-1815
Ester1735-1750
Aldehyde1720-1740
Ketone1705-1725
Carboxylic acid1700-1725
Amide1630-1690

Other Key Groups

Groupν̃ (cm⁻¹)Notes
O−H (free)3610-3640Sharp
O−H (H-bonded)3200-3550Broad
O−H (COOH)2500-3300Very broad
N−H (−NH₂)3350-3500Two bands
N−H (−NH)3310-3350One band
C−H (sp³)2850-3000Below 3000
C−H (sp²)3000-3100Above 3000
C≡N2200-2260Sharp, characteristic
C≡C2100-2260Weak-medium

5.3 Effects on IR Absorption

H-Bonding

Weakens O−H/N−H → lower ν̃, broader band

Conjugation

Delocalizes C=O electrons → lower ν̃

Acetone: 1715 → Acetophenone: 1690 cm⁻¹

Ring Size

Smaller ring → more strain → higher ν̃

6-ring: 1715 → 5-ring: 1745 → 4-ring: 1780

5.4 Fingerprint Region

The region below 1500 cm⁻¹ — unique pattern for each compound, like a fingerprint.

Functional Group Region (4000-1500)

O−H, N−H, C−H, C=O, C≡N → identify functional groups

Fingerprint Region (1500-400)

Complex pattern → identify specific compound

5.5 How to Interpret an IR Spectrum

1
3200-3600 cm⁻¹: O−H or N−H? (broad = O−H; two bands = −NH₂)
2
2850-3100 cm⁻¹: C−H stretches (below 3000 = sp³; above 3000 = sp²)
3
2100-2300 cm⁻¹: Triple bonds? (C≡N or C≡C)
4
1600-1800 cm⁻¹: C=O? (position → type of carbonyl)
5
Below 1500 cm⁻¹: Fingerprint region — match with reference
Unit 6

¹H-NMR Spectroscopy (PMR)

Nuclear spin, chemical shift, shielding/deshielding, spin coupling, coupling constant, Pascal's triangle, spectra interpretation

6.1 Introduction & Nuclear Spin

NMR studies atomic nuclei behavior in a strong magnetic field under radio-frequency radiation.

¹H has spin I = ½ → two orientations in a magnetic field → energy gap = hν

NMR Active?

ConditionIActive?Examples
Both p, n even0❌¹²C, ¹⁶O
Mass number odd½, 3/2...✅¹H, ¹³C

TMS — Internal Standard

Tetramethylsilane (CH₃)₄Si: 12 equivalent H's, single sharp peak, set to δ = 0, inert, volatile, soluble.

6.2 Chemical Shift (δ) ⭐

Chemical Shift
$$\delta = \frac{\nu_{sample} - \nu_{TMS}}{\nu_{spectrometer}} \times 10^6 \quad \text{ppm}$$
Proton Typeδ (ppm)
R−CH₃ (alkane)0.8-1.0
R₂CH₂ (alkane)1.2-1.4
−CO−CH₃ (ketone)2.0-2.5
−O−CH₃ (methoxy)3.3-3.5
C=C−H (vinyl)4.5-6.5
Ar−H (aromatic)6.5-8.0
R−CHO (aldehyde)9.0-10.0
R−COOH10.0-12.0

6.3 Shielding, Deshielding & Ring Current

⬆️ Upfield (shielded)

Towards TMS (right), lower δ

High electron density around proton

⬇️ Downfield (deshielded)

Away from TMS (left), higher δ

Electronegative neighbors, π bonds, ring current

Electronegativity Effect

Compoundδ of CH₃
CH₃−H0.23
CH₃−Cl3.05
CH₃−OH3.40
CH₃−F4.26

Ring Current Effect

Aromatic π-electrons circulate → create induced field → protons outside ring are deshielded (δ 6.5-8.0).

Anisotropic Effects

  • Alkenes: vinyl H deshielded (δ 4.5-6.5)
  • Alkynes: terminal H anomalously shielded (δ 1.8-3.0) — in shielding cone!
  • Aldehydes: −CHO very deshielded (δ 9-10)

6.4 Spin-Spin Coupling ⭐⭐

💡
n+1 Rule: If a proton has n equivalent neighbors, its signal splits into (n+1) lines.
LinesNameAbbrIntensity (Pascal's △)
1Singlets1
2Doubletd1:1
3Triplett1:2:1
4Quartetq1:3:3:1
5Quintetquin1:4:6:4:1
6Sextetsext1:5:10:10:5:1
7Septetsept1:6:15:20:15:6:1

Coupling Constant (J, in Hz)

Distance between lines in a multiplet. J is field-independent. JAB = JBA.

TypeJ (Hz)
Vicinal (free rotation)6-8
cis-alkene6-12
trans-alkene12-18
Aromatic (ortho)6-10
Peak Area (Integration): Area under each peak ∝ number of protons causing it.

6.5 NMR Spectra of Common Compounds ⭐⭐

Ethanol (CH₃−CH₂−OH)

GroupδSplittingH
−CH₃1.18Triplet3H
−CH₂−3.69Quartet2H
−OH2.61Singlet (variable)1H

Acetone (CH₃−CO−CH₃)

GroupδSplittingH
Both −CH₃2.17Singlet6H

Acetaldehyde (CH₃−CHO)

GroupδSplittingH
−CH₃2.21Doublet3H
−CHO9.80Quartet1H

Ethyl Acetate (CH₃COOCH₂CH₃)

GroupδSplittingH
−CO−CH₃2.04Singlet3H
−O−CH₂−4.12Quartet2H
−CH₃1.26Triplet3H

Toluene (C₆H₅−CH₃)

GroupδSplittingH
−CH₃2.36Singlet3H
Ar−H7.17Multiplet5H

Benzaldehyde (C₆H₅−CHO)

GroupδSplittingH
−CHO10.0Singlet1H
Ar−H7.4-7.9Multiplet5H

Phenol (C₆H₅−OH)

GroupδSplittingH
−OH~5-6Broad singlet1H
Ar−H6.7-7.3Multiplet5H

DMF ((CH₃)₂N−CHO)

GroupδSplittingH
−N(CH₃)₂2.88 & 2.97Two singlets (hindered rotation!)3H+3H
−CHO8.02Singlet1H
Unit 7

Mass Spectrometry

Principle, molecular ion, metastable ion, fragmentation, McLafferty rearrangement

7.1 Principle

Measures mass-to-charge ratio (m/z) of ions. Process:

Sample → Ionization (M → M⁺·) → Acceleration → Separation (magnetic field) → Detection

The Mass Spectrum

  • x-axis: m/z ratio
  • y-axis: Relative abundance (%)
  • Base peak: Most intense (= 100%)
  • Molecular ion (M⁺): Highest m/z peak = molecular weight

7.2 Molecular Ion & Nitrogen Rule

$M + e^- \rightarrow M^{+\cdot} + 2e^-$ (radical cation)

Nitrogen Rule: Even MW → even N atoms (0,2,4...). Odd MW → odd N atoms (1,3,5...).

Isotope patterns: M+2 peaks for Cl (³⁷Cl), Br (⁸¹Br), S (³⁴S) — helps identify these elements.

7.3 Metastable Ion (m*)

Broad, diffuse peaks at non-integer m/z. Ion fragments during flight.

Metastable Ion Formula
$$m^* = \frac{(m_2)^2}{m_1}$$

Confirms m₁⁺ directly produces m₂⁺. Example: 100 → 72: m* = 72²/100 = 51.84

7.4 Fragmentation Process

Common Losses

Loss (m/z)Fragment Lost
15CH₃·
17OH· or NH₃
18H₂O
28CO or C₂H₄
29CHO· or C₂H₅·
43CH₃CO· (acetyl)
44CO₂
77C₆H₅· (phenyl)

7.5 McLafferty Rearrangement ⭐

💡
Requirements: (1) C=O group, (2) γ-hydrogen (3rd carbon from C=O), (3) chain of ≥4 atoms from heteroatom.

γ-hydrogen migrates to C=O oxygen through a 6-membered cyclic transition state. The α−β bond breaks. A neutral alkene is lost.

📝 Example: 2-Pentanone (MW = 86)
γ β α | | | C----C----C=O--CH₃ /H ↗ H migrates to O through 6-ring → McLafferty fragment at m/z = 58 (loss of 28 = C₂H₄)

Occurs in: ketones, aldehydes, carboxylic acids, esters, amides.

Unit 8

Separation Techniques

Solvent extraction (batch, continuous, counter-current), chromatography (adsorption, partition, ion exchange, development methods)

8.1 Solvent Extraction

Separates compounds by different solubilities in two immiscible liquids.

Nernst Distribution Law
$$K_D = \frac{C_{organic}}{C_{aqueous}}$$
After n Extractions
$$q^n = \left(\frac{V_{aq}}{V_{aq} + K_D \cdot V_{org}}\right)^n$$

qn = fraction remaining in aqueous phase. Multiple small extractions > one large!

Mechanisms of Extraction

By Solvation

"Like dissolves like" — non-polar solutes into non-polar solvents

By Chelation

Metal ions form neutral chelates (complexes) → soluble in organic solvents

Agents: oxine, DMG, dithizone, acetylacetone

Techniques

TypeMethodAdvantage
BatchShake in separating funnel, repeatSimple, common
ContinuousSolvent recycled continuouslyWorks with small KD
Counter-currentMultiple stages, opposite flowExcellent separation

8.2 Chromatography — Principles

Separation based on differential distribution between stationary and mobile phases.

Retention Factor (TLC/Paper)
$$R_f = \frac{\text{Distance by solute}}{\text{Distance by solvent front}}$$

Mechanisms of Separation

Adsorption

Solute adsorbs on solid surface. More polar = stronger adsorption.

Activity: Al₂O₃ > SiO₂ > Charcoal

Partition

Solute distributes between two liquid phases based on solubility.

Used in paper chromatography.

Ion Exchange

Ions exchanged on charged resin.

Cation resin (−SO₃⁻) attracts cations. Anion resin attracts anions.

8.3 Development of Chromatograms ⭐

FeatureFrontalElution ✅Displacement
Sample amountLarge (continuous)SmallSmall
EluentSample itselfPure solventDisplacer
Pure components?Only firstAllDifficult (overlap)
Gaps between bandsNoYesNo
Most used?RarelyYes ✅Sometimes
📝

Key Formulas Cheat Sheet

All essential formulas in one place for quick revision

FormulaContext
$\lambda = h/mv$de-Broglie wavelength
$\Delta x \cdot \Delta p \geq h/4\pi$Heisenberg uncertainty
$E = h\nu$Energy of photon
$KE = h\nu - h\nu_0$Photoelectric effect
$E_n = -13.6\,Z^2/n^2$ eVBohr energy levels
$\Delta\lambda = \frac{h}{m_ec}(1-\cos\theta)$Compton effect
$\hat{H}\Psi = E\Psi$Schrödinger equation
$\Psi_n = \sqrt{2/L}\sin(n\pi x/L)$Particle in box — ψ
$E_n = n^2h^2/(8mL^2)$Particle in box — E
Bond Order $= (N_b - N_a)/2$MO theory
$\tilde{E}_J = \tilde{B}J(J+1)$Rotational energy
$E_v = (v+\tfrac{1}{2})h\nu_0$Vibrational energy
$\bar{\nu} = \frac{1}{2\pi c}\sqrt{k/\mu}$Vibrational frequency
$A = \varepsilon c l$Beer-Lambert law
$\delta = (\nu - \nu_{TMS})/\nu_0 \times 10^6$Chemical shift (NMR)
$m^* = m_2^2/m_1$Metastable ion
$K_D = C_{org}/C_{aq}$Distribution coefficient
$R_f = d_{solute}/d_{solvent}$Chromatography