Structure of Atom

1. Cathode Ray Experiment

• A discharge tube (cathode ray tube, CRT) is filled with gas at very low pressure (10⁻³–10⁻⁴ mm Hg).
• A high voltage (~10000 V) is applied between the cathode (−) and anode (+).
• Electrons emitted from the cathode (e⁻_M) strike gas atoms and knock out gas electrons (e⁻_g).

• Cathode rays produce a green glow when they strike the glass wall or a ZnS screen beyond the anode.

2. Properties of Cathode Rays

• Travel in straight lines from cathode with very high velocity — they cast shadows of objects placed in their path.
• Consist of material particles — they can rotate a light paddle wheel placed between the electrodes (mechanical effect).
• Negatively charged — deflected toward the positive plate in an electric field and also deflected by a magnetic field.
• Produce a green glow on glass; emit light when striking a ZnS screen.
• Penetrate through thin sheets of aluminium and metals.
• Affect photographic plates.
• Can ionize gases — the emitted electron knocks out electrons from gas atoms.
• When cathode rays strike metals of high atomic mass (Cu, W, Mo …), they produce X-rays.

3. Specific Charge of Cathode Rays

• The specific charge (e/m) of cathode rays is the same irrespective of the gas used in the tube — it is a universal constant.

4. Positive Rays (Canal / Anode Rays) — Discovery of Proton

• Discovered by E. Goldstein (1886) using a discharge tube with a perforated (pierced) cathode.
• At high potential difference (10000–20000 V), rays produced from the anode side travel toward the cathode and pass through its holes — called canal rays or anode rays.
• A red glow is observed near the cathode end.

Characteristics of Anode Rays :

• The e/m ratio is much smaller than that of electrons (heavier particles).
• e/m depends on the nature of gas used in the tube (unlike cathode rays).
• When hydrogen gas is used, the e/m value is maximum — since the charge (e) on the positive particle is the same for different gases, the mass (m) must be minimum for hydrogen. This lightest positive particle is called the proton.
• They can produce ionisation in gases.
• They can produce physical and chemical changes.

5. Millikan's Oil Drop Experiment

• An atomizer converts oil into very tiny droplets.
• A drop of oil falls through a tiny hole in the positively charged plate.
• A beam of electrons or X-rays ionizes the air inside the chamber; the released electrons are absorbed by oil drops, making them negatively charged.
• A potential difference is applied such that the net force on the oil drop becomes zero.

• Charge on oil drop is always an integral multiple of electronic charge:

• Electronic charge = −1.6 × 10⁻¹⁹ C

6. Discovery of Neutrons

• Discovered by Chadwick in 1932 by bombarding a thin sheet of Beryllium with α-particles (He²⁺).
• Electrically neutral particles with mass slightly greater than that of protons were emitted — named neutrons.

7. Properties of Charge

a) Quantisation of Charge :

• Charge is always quantised — always an integral multiple of e = 1.6 × 10⁻¹⁹ C.
• e.g. Al³⁺ has charge = 3 × 1.6 × 10⁻¹⁹ C (3 electronic charges lost).

b) Coulomb's Law :

• Like charges → repulsive ; unlike charges → attractive.

c) Potential Energy :

• PE = (+)ve → developed due to repulsion.
• PE = (−)ve → developed due to attraction.

d) Kinetic Energy of a Charged Particle Accelerated through Potential V :

• 1 eV = kinetic energy gained by an electron accelerated through 1 V = 1.6 × 10⁻¹⁹ J.

8. Thomson's Plum Pudding Model

• Atom is shaped like a sphere with a radius of approximately 10⁻¹⁰ m.
• Positive charge is uniformly distributed throughout the sphere.
• Electrons are embedded in this sphere (like seeds in a watermelon).

Limitations :

• Failed to explain the stability of atom.
• Did not mention anything about the nucleus.
• Did not mention the movement of electrons.

9. Rutherford's α-Particle Scattering Experiment

• α-particles (He⁺⁺) are bombarded on a thin gold foil.
• A fluorescent screen surrounding the foil detects scattered α-particles.
• Most α-particles pass through undeflected; a few are deflected at various angles θ; very few are deflected back (θ ≈ 180°).

10. Postulates of Rutherford's Atomic Model

• All the protons (+ve charge) and neutrons (neutral charge), i.e. nearly the total mass of an atom, is present in a very small region at the centre called the nucleus.
• The size of the nucleus is very small compared to the atom. Diameter of nucleus ≈ 10⁻¹³ cm; diameter of atom ≈ 10⁻⁸ cm. So the atom is 10⁵ times larger than the nucleus.
• Most of the space outside the nucleus is empty.
• Electrons (equal in number to the net nuclear positive charge) revolve around the nucleus in various circular orbits.
• The centripetal force needed is provided by the Coulombic force of attraction between the nucleus and the electron.

Z = atomic number, e = electronic charge, r = radius of orbit, m = mass of electron, v = speed of electron.

11. Size of the Nucleus

• 1 fm = 10⁻¹³ cm = 10⁻¹⁵ m
• Nucleus diameter ≈ 10⁻¹² to 10⁻¹³ cm; atom diameter ≈ 10⁻⁸ cm.
• Diameter of atom is 100,000 times the diameter of the nucleus.
• The radius of a nucleus is proportional to the cube root of the mass number.

R₀ ≈ 1.3 × 10⁻¹³ cm (or 1.3 × 10⁻¹⁵ m or 1.3 fm); A = mass number.

Volume ratio (atom to nucleus) :

Fraction of volume occupied by the nucleus ≈ 10⁻¹⁵.

12. Density of the Nucleus

• The mass number A cancels out, so nuclear density is independent of mass number — it is approximately the same for all nuclei.

13. Number of Deflected α-Particles

Z = atomic number of target, E = kinetic energy of α-particle, θ = scattering angle.

14. Closest Distance of Approach (CDA)

• An α-particle (charge 2e, mass m_α) moves with velocity v₀ towards a nucleus of charge Ze from infinity.
• At infinity: KE = ½ m_αv₀², PE = 0.
• At closest distance R₀: KE = 0 (momentarily at rest), PE = K (2e)(Ze)R₀.
• By conservation of energy: (TE)_A = (TE)_B

K = 9 × 10⁹ N m² / C², p = momentum of α-particle.

CDA for a proton :

• Proton has charge e and mass m_p.

Key relations :

• m_α ≈ 4 m_p  ;  q_α = 2 q_p
• CDA ∝ Z (for same particle & speed).
• CDA ∝ 1v₀² (for same particle & target).

15. Radiation

• Radiation is the emission and transfer of energy through space or a material medium in the form of waves or particles.
• Energy that travels from one place to another, either as electromagnetic waves (light) or as moving particles (alpha or beta particles).

16. Electromagnetic Waves

• A form of energy produced by the movement of electrically charged particles, by oscillating magnetic and electric fields.
• No medium is required.
• In vacuum they travel with the speed of light = 3 × 10⁸ m/s.
• Different EMRs have different frequencies and wavelengths.

17. Wave Characteristics

Wavelength (λ) :

• The distance between two consecutive crests or troughs.

Length unit conversions :

Frequency (ν) :

• Number of waves passing through a point in one second.
• Unit: s⁻¹ or Hz (cycle/sec).

Velocity :

• Linear distance travelled by the wave in one second.

• For electromagnetic radiation, speed = speed of light:

• For two different EMRs:   ν₁λ₁ = ν₂λ₂ = c

Amplitude (a) :

• Height of crest or depth of trough from the central baseline.

Wave number (ν̄) :

• Number of waves present in 1 cm length; reciprocal of wavelength.

18. The Electromagnetic Spectrum

• In order of increasing wavelength (decreasing frequency):

• Visible light — in order of increasing wavelength:

19. Particle Nature of EMR — Quantum Theory of Light

• Wave nature (diffraction, interference) does not explain:
  • (i) Black-body radiation
  • (ii) Photoelectric effect
• The smallest quantity of energy that can be emitted or absorbed in the form of EMR is called a quantum of light (also called quanta or photon).

E = energy of one photon, h = Planck's constant = 6.626 × 10⁻³⁴ J·s, c = 3 × 10⁸ m/s, λ = wavelength of EMR.

• Energy of one mole of photons = N_A × hcλ

20. Useful Energy–Wavelength Relation

21. Drawbacks of Rutherford's Model

• An accelerated charged particle radiates EMR continuously → the electron should spiral into the nucleus (energy loss), so atom should collapse.
• It cannot explain the line spectrum of hydrogen.

22. Bohr's Model

Postulate 1 — Stationary Orbits :

• Electrons revolve around the nucleus only in certain fixed orbits called stationary orbits (K, L, M … corresponding to n = 1, 2, 3 …).

Postulate 2 — Quantisation of Angular Momentum :

m = mass of electron, v = velocity, r = radius of orbit, n = principal quantum number, h = Planck's constant.

• For n = 1: mv₁r₁ = h/2π
• For n = 2: mv₂r₂ = 2h/2π
• For n = 3: mv₃r₃ = 3h/2π   … and so on.

Postulate 3 — Energy of Transition :

• Emission: When an electron jumps from a higher orbit n₂ to a lower orbit n₁, energy is emitted as a photon.
• Absorption: When an electron jumps from n₁ to n₂ (n₂ > n₁), energy is absorbed.

23. Bohr Model – Force Balance & Quantisation

• Electrostatic (centripetal) force on the electron revolving around a nucleus of charge +Ze:

• Centrifugal force (needed for circular motion):

• For a stable orbit: F₁ = F₂

• Bohr's quantisation condition (angular momentum is quantised):

24. Radius of n^th Orbit

• From (4): v = nh2πmr
• Substituting into (3):

h = 6.626 × 10⁻³⁴ J·s,   π = 3.14,   m = 9.1 × 10⁻³¹ kg,   e = 1.6 × 10⁻¹⁹ C,   K = 9 × 10⁹ N·m²/C²

25. Velocity of Electron in n^th Orbit

• v = nh2πmr, substituting the expression for r:

26. Energy of Electron in n^th Orbit

• From the force balance: mv² = K Ze²r

Key Relations :

• Substituting the expression for r into TE:

27. Applicability – Single Electron (H-like) Species Only

• All the derived formulae (r, v, E) are applicable only for single-electron (H-like) species.
• While deriving we considered only two forces on the electron — electrostatic attraction F₁ and centrifugal force F₂. In multi-electron atoms, inter-electron repulsions (F₃, F₄ …) also act, so these formulae do not apply.

28. Time Period, Frequency & Acceleration

Time Period (T) :

• Time taken by an electron to complete one revolution around the nucleus.
• 1 Å = 10⁻¹⁰ m

Frequency (f) :

• Number of revolutions per second.

Acceleration (a) :

29. Wavelength of Emitted Photon (Transition n₂ → n₁)

• When an electron falls from orbit n₂ to n₁ (n₂ > n₁) in a H-like species (atomic number Z), a photon is emitted.
• Energy of emitted photon = ΔE = E_n₂ − E_n₁

R_H = 109677 cm⁻¹ (Rydberg constant)

• Alternatively, using E in eV:

30. Energy Level Diagram

Energy becomes less negative (increases) as n increases; the electron is most tightly bound at n = 1.

31. Ground State, Excited States & Ionisation Energy

Ground State :

• Lowest energy state of any atom or ion — corresponds to n = 1.
• Ground state energy = −13.6 Z² eV
• For H-atom: −13.6 eV  ;   He⁺: −54.4 eV

Excited States :

• States other than the ground state.

Ionisation Energy (IE) :

• Minimum energy required to move an electron from the ground state (n = 1) to n = ∞.

• IE of H = 13.6 eV  ;   He⁺ = 54.4 eV  ;   Li²⁺ = 122.4 eV

32. Ionisation Potential

• Ionisation Potential (I.P.) : The potential difference through which a free electron must be accelerated from rest such that its kinetic energy becomes equal to the ionization energy of the atom.
• I.P. of H atom = 13.6 V, I.P. of He⁺ ion = 54.4 V.

33. Excitation Energy & Potential

• Excitation energy : Energy required to move an electron from the ground state of the atom to any other state.
• 1^st Excitation Energy = Excitation Energy of 2^nd state = Excitation Energy of 1^st excited state = 10.2 Z² eV.
• 2^nd Excitation Energy = Excitation Energy of 3^rd state = Excitation Energy of 2^nd excited state = 12.09 Z² eV.
  (Derived from 13.60 Z² − 1.51 Z² = 12.09 Z² eV)
• Excitation Potential : Potential difference through which an electron must be accelerated from rest so that its kinetic energy becomes equal to the excitation energy of a particular state.

34. Binding Energy (Separation Energy)

• Binding Energy : Energy required to move an electron from any state to n = ∞ is called the binding energy of that state.
• Binding energy of the ground state = I.E. of the atom or ion.
• B.E. of 2^nd state = +3.4 Z² eV.
• B.E. of 3^rd state = +1.51 Z² eV.

35. Atomic Spectra Types

• Continuous Spectra : When white light from a source (sun, bulb) passes through a prism, it splits into continuous bands of colors from violet to red merging into one another.
• Discrete Spectra (2 types) :
  • Band Spectrum : Contains colourful continuous bands separated by dark space. (Generally molecular spectrum).
  • Line Spectrum : Ordered arrangement of lines of particular wavelength separated by dark space. (Can be obtained from atoms, e.g. Hydrogen spectrum).
• Emission vs Absorption Spectra :
  • Emission : Radiation emitted from a source (heated substance, discharge tube) is passed directly through a prism.
  • Absorption : White light is first passed through a solution or vapour (absorbing certain wavelengths), and then analyzed. Shows dark lines in a continuous spectrum.

36. Hydrogen Emission Spectrum & Series

• Obtained when H₂ gas at low pressure is taken in a discharge tube and the light emitted on passing an electric discharge is examined.
• Maximum energy transition (Shortest wavelength line / Last line) corresponds to n = ∞ → n₁.
• Minimum energy transition (Longest wavelength line / First line) corresponds to n = (n₁ + 1) → n₁.
• For the last line (shortest λ) of Lyman (∞ → 1): 1λ = R(1)² [ 11² − 1∞² ] ⇒ λ = 1R.
• Energy ranges of emitted photons: Lyman (10.2 eV to 13.6 eV), Balmer (1.89 eV to 3.4 eV), Paschen (0.66 eV to 1.51 eV). Spaces between these are devoid of lines.

37. Calculation of Number of Unique Photons

Case 1 : Single Isolated Atom

• If an electron in a single atom makes a de-excitation transition from the n^th state to the ground state, it cascades down.
• Maximum number of unique spectral lines = (n − 1)

Case 2 : Sample with Enough Number of Atoms

• If the sample contains sufficient atoms and de-excitation occurs from the n^th level to the ground level (n = 1) :

• The total number of spectral lines follows the series distribution :
  n(n − 1)2 = (n − 1) + (n − 2) + ... + 3 + 2 + 1
  

  Where the terms correspond to lines in the Lyman, Balmer, Paschen, etc., series respectively.

• If de-excitation occurs between two specific levels, from n = n₂ to n = n₁ :

where Δn = n₂ − n₁

38. Dual Nature of Matter (de-Broglie Hypothesis)

• A particle in motion behaves as a wave as well as a material particle.
• The de-Broglie wavelength (λ) associated with a particle of mass m moving with velocity v is:

• Significant only for microscopic particles (e.g., electrons, protons). For macroscopic particles, λ is negligibly small (≈ 10⁻³⁷ m).
• For an electron in a Bohr orbit : An electron moving in the n^th orbit forms exactly 'n' waves along its circumference.
  nλ = 2πr ⇒ λ_dB = 2πrn
• In terms of Kinetic Energy (KE) :

h in J·s, m in kg, KE in Joules.

Charged Particle Accelerated Through Potential V

• When a particle of charge q is accelerated from rest through V volts, KE = qV.

39. Heisenberg's Uncertainty Principle

• The exact position and momentum of a fast-moving microscopic particle cannot be calculated precisely at the same time.
• If the position is measured precisely (Δx → 0), the uncertainty in momentum becomes infinite (Δp → ∞).
• Optics principle limit : To locate a particle's position, the maximum error is ±λ (Δx = ±λ). To make Δx small, a highly energetic (low λ) photon must be used, which significantly alters the particle's momentum upon collision.

• Other useful uncertainty forms :
  • Energy and time : ΔE · Δt ≥ h4π
  • Position and wavelength : Δx · Δλ ≥ λ²4π
  • Ratio of uncertainties : differentiating λ = h/mv gives ΔλΔv = |−λv|
• Special constraints for position uncertainty (Δx) :
  • If a particle is confined in a sphere : Δx = diameter
  • If a particle is confined in a cube of edge 'a' : Δx = body diagonal = a√3

40. Orbital

• An orbital is the region of space around the nucleus where the probability of finding an electron is maximum (90% to 95%).
• Orbitals do not define a definite path for the electron — they define only the probability of the electron being in various regions of space around the nucleus.

41. Orbits vs Orbitals

42. Shapes of Orbitals

s-orbital :

• Shape → Spherical.
• Non-directional; closest to the nucleus; has the lowest energy among subshells of the same shell.
• Can accommodate a maximum of 2 electrons.

p-orbital :

• Shape → Dumb-bell.
• The dumb-bell consists of two lobes separated by a region of zero probability called a node.
• Three orientations: p_x, p_y, p_z.
• Can accommodate a maximum of 6 electrons.

d-orbital :

• Shape → Double dumb-bell.
• Five orientations: d_xy, d_xz, d_yz, d_x²−y², d_z².
• d_z² has two lobes along the z-axis with a ring (torus) in the xy-plane.
• Can accommodate a maximum of 10 electrons.

43. Quantum Numbers

• A set of four numbers required to define an electron completely in an atom.
• The first three are derived from the Schrödinger wave equation.

(I) Principal Quantum Number (n) :

• Describes the size of the electron wave and the total energy of the electron.
• Values: n = 1, 2, 3, 4, … (denoted K, L, M, N, …).

(II) Azimuthal Quantum Number (ℓ) :

• Describes the shape of the electron cloud and the number of subshells in a shell.
• Values: ℓ = 0, 1, 2, … , (n − 1).

• s-orbital (ℓ = 0) → L = 0
• p-orbital (ℓ = 1) → L = √2 · h2π
• d-orbital (ℓ = 2) → L = √6 · h2π

(III) Magnetic Quantum Number (m) :

• Describes the orientations of the subshells.
• Values: m = −ℓ, …, 0, …, +ℓ → total (2ℓ + 1) values. Each value corresponds to one orbital.

Total orbitals in a principal shell = n².

(IV) Spin Quantum Number (s) :

• Describes the spin of the electron. Values: +½ (clockwise) and −½ (anticlockwise).

44. Pauli's Exclusion Principle

• No two electrons in an atom can have the same set of all four quantum numbers.
• An orbital cannot have more than 2 electrons; if two electrons occupy the same orbital, their spins must be in opposite directions.

45. Aufbau Principle

• Aufbau (German: "building up") — electrons are filled in orbitals in order of increasing energy. The lowest-energy orbital is filled first.

(n + ℓ) Rule :

• The energy of a subshell in a multi-electron atom is governed by its (n + ℓ) value.
• Lower (n + ℓ) → lower energy → filled first.
• If two subshells have the same (n + ℓ), the one with lower n has lower energy and is filled first.

e.g. 3s (n + ℓ = 3) < 5d (n + ℓ = 7) ;   4d (n + ℓ = 6) < 5p (n + ℓ = 6) since n is lower for 4d.

46. Hund's Rule of Maximum Multiplicity

• No electron pairing takes place in orbitals of a subshell until each orbital is occupied by one electron with parallel spin.
• Exactly half-filled and fully filled subshells give extra stability:
  p³, p⁶, d⁵, d¹⁰, f⁷, f¹⁴ are most stable configurations.

47. Electronic Configuration — Examples

Na (Z = 11) :

• Unpaired electrons n = 1 → μ = √(1 × 3) = √3 B.M.
• Maximum spin S = ½ × 1 = ½.

Fe (Z = 26) :

• 3d⁶ → 4 unpaired electrons.
• Fe²⁺ (remove 4s² first) → 1s² 2s² 2p⁶ 3s² 3p⁶ 3d⁶.

Cr (Z = 24) — Exception :

• Expected: …4s² 3d⁴ — ✗ Incorrect.
• Actual: … 4s¹ 3d⁵ — ✓ (half-filled d is more stable).

Exchange energy with 3d⁴ (4 parallel e⁻): 3 + 2 + 1 = 6 ; with 3d⁵ (5 parallel e⁻): 4 + 3 + 2 + 1 = 10 → higher exchange energy → more stable.

Cu (Z = 29) — Exception :

• Expected: …4s² 3d⁹ — ✗ Incorrect.
• Actual: … 4s¹ 3d¹⁰ — ✓ (fully filled d is more stable).

Other examples :

• B (Z = 5) : 1s² 2s² 2p¹
• Cl (Z = 17) : 1s² 2s² 2p⁶ 3s² 3p⁵

48. Quick Formula Revision