Structure of Atom: NEET notes

1. Discovery of sub-atomic particles

NCERT §2.1

• Discharge through gases at very low pressure and very high voltage produces cathode rays, a stream that travels from the cathode to the anode; it is made visible where it strikes a coating such as zinc sulphide.
• Cathode rays go straight in the absence of fields and bend in electric or magnetic fields as negative particles would, so they are negatively charged particles, the electrons.
• The properties of cathode rays do not change with the metal of the electrodes or the gas in the tube, so electrons are a common part of every atom.
• J.J. Thomson (1897) balanced perpendicular electric and magnetic fields to find e/mₑ = 1.758820 × 10¹¹ C kg⁻¹. Deflection grows with charge and field strength and falls as mass rises.
• Millikan's oil drop experiment (1906-14) showed every droplet charge is a whole-number multiple of e (q = ne). The accepted value is e = 1.602176 × 10⁻¹⁹ C; combined with e/mₑ it gives mₑ = 9.1094 × 10⁻³¹ kg.
• Canal rays are positive gaseous ions: unlike cathode rays, their mass and e/m depend on the gas, some carry more than one unit of charge, and they deflect the opposite way to electrons.
• The lightest positive ion, from hydrogen, is the proton (characterised in 1919). Chadwick (1932) found the neutral neutron, slightly heavier than the proton, by bombarding a thin beryllium sheet with α-particles.
• Masses in u: electron 0.00054, proton 1.00727, neutron 1.00867. Relative charges are −1, +1 and 0.

2. Thomson and Rutherford models

NCERT §2.2.1; §2.2.2; §2.2.5

• Thomson's model (1898): a sphere of radius about 10⁻¹⁰ m with positive charge spread evenly and electrons set in it (plum pudding, raisin pudding or watermelon); the mass is spread evenly too. It explains overall neutrality and little else.
• In Rutherford's scattering experiment, α-particles were fired at gold foil about 100 nm thick, with a zinc sulphide screen around it to catch each one as a flash.
• Most α-particles went straight through, a small fraction turned through small angles, and about 1 in 20,000 bounced back through nearly 180°.
• Conclusions: the atom is mostly empty space, and the positive charge with nearly all the mass sits in a tiny nucleus, radius about 10⁻¹⁵ m against about 10⁻¹⁰ m for the atom. If the nucleus were a cricket ball, the atom would be about 5 km in radius.
• Rutherford's nuclear model places electrons in circular orbits around the nucleus at high speed, held by electrostatic attraction, like planets round the sun.
• Drawback 1: an orbiting electron is accelerating, and by Maxwell's theory an accelerating charge radiates; it should lose energy and spiral into the nucleus in about 10⁻⁸ s. Atoms are stable, so the model fails.
• Drawback 2: the model says nothing about how electrons are arranged around the nucleus or what energies they have. A stationary electron is no fix, since it would simply be pulled into the nucleus.

3. Atomic number, isotopes and isobars

NCERT §2.2.3; §2.2.4

• Atomic number Z = number of protons in the nucleus, which equals the number of electrons in a neutral atom (H: Z = 1; Na: Z = 11).
• Protons and neutrons together are nucleons; mass number A = protons + neutrons, so neutrons = A − Z whether the species is neutral or an ion.
• An atom is written with A as a left superscript and Z as a left subscript on the symbol, e.g. ⁸⁰₃₅Br has 35 protons, 35 electrons and 45 neutrons.
• For an ion, compare protons and electrons first: 16 protons, 16 neutrons and 18 electrons is ³²₁₆S²⁻, an anion carrying the two extra electrons as its charge.
• Isotopes have the same Z and different A, so they differ only in neutron count. Hydrogen: protium ¹H (99.985%), deuterium ²H or D (0.015%), and tritium ³H, found in trace amounts.
• Other isotopes: carbon with 6, 7 and 8 neutrons (¹²C, ¹³C, ¹⁴C) and chlorine with 18 and 20 neutrons (³⁵Cl, ³⁷Cl).
• Isobars have the same A and different Z, such as ¹⁴₆C and ¹⁴₇N.
• Chemical behaviour is set by the electrons, which the proton count fixes; neutrons hardly affect it, so isotopes of an element react alike.

4. Wave nature of electromagnetic radiation

NCERT §2.3.1

• Maxwell (1870) showed that an accelerating charge produces oscillating electric and magnetic fields that travel as electromagnetic waves; the two fields are perpendicular to each other and to the direction of travel.
• Electromagnetic waves need no medium and all travel through vacuum at c = 3.0 × 10⁸ m s⁻¹ (2.997925 × 10⁸ m s⁻¹ more exactly).
• Frequency ν (unit hertz, Hz = s⁻¹) is the number of waves passing a point per second; c = νλ.
• Wavenumber ν̄ = 1/λ is the number of wavelengths per unit length; its SI unit is m⁻¹ but cm⁻¹ is the common unit in spectroscopy.
• The regions differ only in wavelength: radio about 10⁶ Hz (broadcasting), microwave about 10¹⁰ Hz (radar), infrared about 10¹³ Hz (heating), visible about 10¹⁵ Hz, ultraviolet about 10¹⁶ Hz (part of sunlight).
• Visible light runs from violet at 400 nm (7.5 × 10¹⁴ Hz) to red at 750 nm (4.0 × 10¹⁴ Hz); longer wavelength means lower frequency.
• Interference and diffraction are wave behaviours, and the wave picture explains them.

5. Planck's quantum theory and photoelectric effect

NCERT §2.3.2

• Classical wave theory failed for black-body radiation, the photoelectric effect, the heat capacity of solids at different temperatures and the line spectra of atoms.
• A black body is an ideal absorber and emitter of all frequencies; its emission depends only on temperature. Intensity rises with wavelength to a peak and then falls, and the peak moves to shorter wavelength as temperature rises, which is why heated iron glows dull red, then brighter red, then white and then blue.
• Planck (1900): energy is emitted or absorbed only in packets called quanta, with E = hν and h = 6.626 × 10⁻³⁴ J s. Allowed energies are 0, hν, 2hν, 3hν … and nothing in between, like standing on steps of a staircase.
• Photoelectric effect (Hertz, 1887): light ejects electrons from metals such as potassium, rubidium and caesium. Ejection is instant, the number of electrons grows with brightness, and below a threshold frequency ν₀ no electron comes out at any brightness.
• Einstein (1905): light is a stream of photons; one photon gives all its energy to one electron. Energy balance: hν = hν₀ + ½mₑv², where hν₀ = W₀, the work function.
• Above the threshold, the kinetic energy of the electrons rises with frequency and does not depend on brightness; brighter light only means more photons and so more electrons.
• NCERT's work functions (eV): Li 2.42, Na 2.3, K 2.25, Mg 3.7, Cu 4.8, Ag 4.3. NCERT's text also quotes the threshold for potassium as 5.0 × 10¹⁴ Hz, so in a numerical use the value the question gives.
• Light is dual: it shows wave behaviour (interference, diffraction) while travelling and particle behaviour when it interacts with matter.

6. Atomic spectra and the hydrogen spectrum

NCERT §2.3.3

• A prism spreads white light into a continuous spectrum: red bends least, violet most, and each colour merges into the next, as in a rainbow.
• An excited atom returns to lower energy by emitting radiation. The emission spectrum records the emitted wavelengths; the absorption spectrum is its photographic negative, with dark lines where the sample took in light.
• Gaseous atoms give line spectra, bright lines at specific wavelengths with dark gaps. Each element's pattern is unique, like a fingerprint, and the study of these spectra is spectroscopy.
• Spectroscopy found rubidium, caesium, thallium, indium, gallium and scandium, and helium was discovered in the sun this way; Robert Bunsen was an early user of line spectra to identify elements.
• An electric discharge splits H₂ and excites the H atoms, which emit a line spectrum. Balmer (1885) fitted the visible lines with ν̄ = 109,677 (1/2² − 1/n²) cm⁻¹ for n = 3, 4, 5 …
• Rydberg's general form: ν̄ = 109,677 (1/n₁² − 1/n₂²) cm⁻¹ with n₂ > n₁; 109,677 cm⁻¹ is the Rydberg constant for hydrogen.
• Series by n₁: Lyman 1 (ultraviolet), Balmer 2 (visible, the only visible series), Paschen 3, Brackett 4 and Pfund 5 (all infrared).
• Hydrogen has the simplest line spectrum; every element's spectrum is unique and shows regularity, which points to its electronic structure.

7. Bohr's model of the hydrogen atom

NCERT §2.4; §2.4.1; §2.4.2

• Bohr (1913) postulates: the electron moves in fixed circular orbits (stationary states) of fixed energy; energy is absorbed or emitted only in a jump between orbits, with ν = ΔE/h = (E₂ − E₁)/h (Bohr's frequency rule).
• Angular momentum is quantised: mₑvr = nh/2π with n = 1, 2, 3 … (the principal quantum number), so only certain orbits are allowed.
• Radii: rₙ = n²a₀ with a₀ = 52.9 pm, the radius of the first (Bohr) orbit, where the hydrogen electron normally sits.
• Energies: Eₙ = −2.18 × 10⁻¹⁸ (1/n²) J, so E₁ = −2.18 × 10⁻¹⁸ J and E₂ = −0.545 × 10⁻¹⁸ J. The free electron at rest (n = ∞) is zero; the negative sign means the bound electron is more stable than a free one, and n = 1 is the ground state.
• Hydrogen-like ions (He⁺, Li²⁺, Be³⁺, one electron each): Eₙ = −2.18 × 10⁻¹⁸ (Z²/n²) J and rₙ = 52.9 n²/Z pm. Larger Z binds the electron more tightly in a smaller orbit; the electron's speed rises with Z and falls as n rises.
• For a jump from nᵢ to n_f: ΔE = 2.18 × 10⁻¹⁸ (1/nᵢ² − 1/n_f²) J and ν = 3.29 × 10¹⁵ (1/nᵢ² − 1/n_f²) Hz; in wavenumbers this is Rydberg's formula with 109,677 cm⁻¹. Positive ΔE is absorption (n_f > nᵢ), negative is emission.
• Many atoms making many different jumps give the many lines; a line's brightness depends on how many photons of that wavelength are emitted or absorbed.
• Where Bohr fails: it cannot account for the closely spaced doublets seen in hydrogen's lines under fine instruments, for the spectrum of helium or any atom with more than one electron, for lines splitting in a magnetic field (Zeeman effect) or in an electric field (Stark effect), or for the way atoms join by chemical bonds.

8. Dual behaviour of matter and uncertainty principle

NCERT §2.5; §2.5.1; §2.5.2

• de Broglie (1924): matter, like radiation, is dual; a particle of mass m and speed v has wavelength λ = h/mv = h/p.
• Electron beams diffract, a wave property; the electron microscope uses this and magnifies about 15 million times.
• Every moving object has a wavelength, but for ordinary masses it is far too short to detect: a 0.1 kg ball at 10 m s⁻¹ has λ = 6.626 × 10⁻³⁴ m, while an electron with kinetic energy 3.0 × 10⁻²⁵ J (v ≈ 812 m s⁻¹) has λ ≈ 897 nm.
• Heisenberg (1927): exact position and exact momentum of an electron cannot both be known at once; Δx·Δp ≥ h/4π, equivalently Δx·Δv ≥ h/4πm.
• To locate an electron you must hit it with very short-wavelength light, whose high-momentum photons change the electron's velocity; the act of measuring disturbs it.
• So an electron has no definite path or trajectory, and exact statements about it are replaced by probabilities.
• The principle matters only for tiny masses: locating an electron within 0.1 Å leaves Δv ≈ 5.79 × 10⁶ m s⁻¹, but for a 40 g ball at 45 m s⁻¹ measured to 2%, Δx ≈ 1.46 × 10⁻³³ m, a meaningless limit.
• Bohr's model fails because it ignores the wave nature of the electron and assumes a fixed orbit, a path that needs exact position and velocity together.

9. Quantum mechanical model and quantum numbers

NCERT §2.6; §2.6.1

• Quantum mechanics (Heisenberg and Schrödinger, 1926) treats particles with wave-particle duality. Schrödinger's equation, Ĥψ = Eψ, gives the allowed energies E and wave functions ψ; it can be solved exactly only for one-electron systems.
• An atomic orbital is a one-electron wave function ψ. ψ itself has no physical meaning; |ψ|² is the probability density, always positive, and |ψ|² times a small volume gives the chance of finding the electron there.
• An orbit (Bohr's fixed path) and an orbital are not the same thing; an orbit cannot be observed, while an orbital describes where the electron is likely to be found.
• Features of the model: electron energies are quantised as a direct result of the electron's wave nature; only probabilities of location can be given; an orbital holds at most two electrons.
• Principal quantum number n = 1, 2, 3 … sets the shell (K, L, M, N …), the size and, largely, the energy. Shell n has n² orbitals; for H and hydrogen-like ions energy depends on n alone.
• Azimuthal quantum number l = 0 to n − 1 sets the subshell and shape: l = 0, 1, 2, 3 are s, p, d, f. Shell n has n subshells.
• Magnetic quantum number mₗ = −l … 0 … +l, so 2l + 1 values: one s, three p, five d, seven f orbitals; it sets the orientation. n = 3 has 1 + 3 + 5 = 9 orbitals.
• Spin quantum number mₛ = +½ or −½ (proposed in 1925 by Uhlenbeck and Goudsmit), needed to explain doublets and triplets in spectra; two electrons in one orbital must have opposite spins.

10. Shapes and energies of orbitals

NCERT §2.6.2; §2.6.3

• For 1s, |ψ|² is highest at the nucleus and falls away steadily; for 2s it falls to zero, rises to a small maximum and falls again. A surface where the probability density is zero is a node.
• A boundary surface diagram encloses the region with about 90% probability; 100% is impossible because |ψ|² never becomes exactly zero at any finite distance. Every s orbital is a sphere, and size grows 1s < 2s < 3s < 4s.
• Each p orbital has two lobes on either side of a nodal plane through the nucleus; the three (px, py, pz) are equal in size, shape and energy and lie along mutually perpendicular axes. There is no simple match between mₗ values and x, y, z.
• The five d orbitals (from n = 3 onwards) are dxy, dyz, dxz, dx²−y² and dz²; the first four share a shape, dz² looks different, and all five are equal in energy in the isolated atom.
• Nodes: angular nodes = l, radial nodes = n − l − 1, total = n − 1. So 2s has one radial node, 3p has one radial and one angular, and pz has the xy-plane as its nodal plane.
• In hydrogen, energy depends on n alone, so all subshells of one shell tie: 2s = 2p; 3s = 3p = 3d; and 4s, 4p, 4d and 4f are equal too. Orbitals of equal energy are called degenerate.
• In multi-electron atoms, electron-electron repulsion and shielding make energy depend on n and l: within a shell s < p < d < f, because an s electron penetrates closer to the nucleus and feels a larger effective nuclear charge (Zeff).
• The (n + l) rule: lower n + l means lower energy; for equal n + l, lower n is lower. This gives 4s < 3d, 6s < 5d and 4f < 6p. The same orbital drops in energy as Z rises: E₂ₛ(H) > E₂ₛ(Li) > E₂ₛ(Na) > E₂ₛ(K).

11. Filling of orbitals and electronic configuration

NCERT §2.6.4; §2.6.5; §2.6.6

• Aufbau principle: in the ground state, electrons enter the lowest-energy orbital available. The working order is 1s, 2s, 2p, 3s, 3p, then 4s, 3d, 4p, then 5s, 4d, 5p, then 6s, 4f, 5d, 6p, then 7s. It is a guide, and exceptions occur.
• Pauli exclusion principle: no two electrons in an atom share all four quantum numbers, so an orbital holds two electrons of opposite spin. Subshells hold 2 (s), 6 (p), 10 (d), 14 (f); a shell holds 2n².
• Hund's rule of maximum multiplicity: in a set of degenerate orbitals, electrons pair only after each orbital has one, so pairing starts with the 4th electron in p, the 6th in d and the 8th in f.
• Configurations are written as spdf notation (1s² 2s² 2p⁶ …) or as box diagrams with arrows; the box form shows all four quantum numbers.
• Examples: H 1s¹, He 1s², Li 1s² 2s¹, C 1s² 2s² 2p², N 1s² 2s² 2p³, Ne 1s² 2s² 2p⁶, Na [Ne] 3s¹, Ar [Ne] 3s² 3p⁶. Electrons of the filled inner shells are core electrons; those in the outermost shell are valence electrons.
• K and Ca put electrons into 4s before 3d; from Sc to Zn the 3d orbitals fill, then 4p from Ga to Kr.
• Exceptions: Cr is 3d⁵ 4s¹ and Cu is 3d¹⁰ 4s¹, not 3d⁴ 4s² and 3d⁹ 4s², because half-filled and completely filled subshells (p³, p⁶, d⁵, d¹⁰, f⁷, f¹⁴) are extra stable.
• That extra stability comes from the symmetrical electron distribution (small mutual shielding, stronger pull of the nucleus), smaller repulsion and the largest exchange energy, which is greatest when a subshell is half or completely filled and is also the basis of Hund's rule.

Must-know facts

1. e/mₑ = 1.758820 × 10¹¹ C kg⁻¹; e = 1.602176 × 10⁻¹⁹ C; mₑ = 9.1094 × 10⁻³¹ kg.
2. Masses in u: electron 0.00054, proton 1.00727, neutron 1.00867; the neutron is slightly heavier than the proton.
3. Chadwick (1932) found the neutron by hitting beryllium with α-particles.
4. In the gold-foil experiment about 1 in 20,000 α-particles bounced back; nucleus about 10⁻¹⁵ m, atom about 10⁻¹⁰ m.
5. Neutrons = A − Z for atoms and ions alike; isotopes share Z, isobars share A.
6. Hydrogen isotopes: protium 99.985%, deuterium 0.015%, tritium in traces.
7. c = νλ with c = 3.0 × 10⁸ m s⁻¹; visible light spans 400-750 nm, or 7.5 × 10¹⁴ to 4.0 × 10¹⁴ Hz.
8. E = hν, h = 6.626 × 10⁻³⁴ J s; one mole of photons at 5 × 10¹⁴ Hz carries about 199.5 kJ.
9. hν = W₀ + ½mₑv²; kinetic energy depends on frequency, number of electrons on brightness.
10. Balmer series (n₁ = 2) is the only visible hydrogen series; Lyman (n₁ = 1) is ultraviolet; Paschen, Brackett, Pfund are infrared.
11. Rydberg constant for hydrogen: 109,677 cm⁻¹; in energy form 2.18 × 10⁻¹⁸ J.
12. Bohr: rₙ = 52.9 n²/Z pm and Eₙ = −2.18 × 10⁻¹⁸ Z²/n² J; He⁺ n = 1 has E = −8.72 × 10⁻¹⁸ J and r = 26.45 pm.
13. The n = 5 to n = 2 hydrogen line has ν ≈ 6.91 × 10¹⁴ Hz (about 434 nm).
14. λ = h/mv; Δx·Δp ≥ h/4π.
15. Shell n has n subshells, n² orbitals and at most 2n² electrons; a subshell has 2l + 1 orbitals.
16. Radial nodes = n − l − 1, angular nodes = l, total nodes = n − 1.
17. (n + l) rule: lower sum first; tie goes to lower n (so 4s before 3d, 4f before 5d).
18. Cr 3d⁵ 4s¹ and Cu 3d¹⁰ 4s¹ are the exceptions to learn.
19. Hund's rule: pairing begins with the 4th p, 6th d and 8th f electron.

Common traps

Formulas

Key terms

Cathode rays

The stream of electrons that flows from cathode to anode in a low-pressure discharge tube.

Canal rays

Positive gaseous ions formed in a discharge tube, whose mass depends on the gas used.

Isotopes

Atoms of one element with the same atomic number but different mass numbers.

Isobars

Atoms of different elements that share the same mass number.

Quantum

The smallest packet of energy that can be emitted or absorbed as radiation, equal to hν.

Work function

The least energy a photon must bring to free an electron from a given metal surface.

Threshold frequency

The lowest light frequency that can eject electrons from a given metal.

Line spectrum

A spectrum of separate bright (or dark) lines at wavelengths characteristic of an element.

Stationary state

In Bohr's model, an allowed orbit in which the electron keeps a fixed energy.

Atomic orbital

A one-electron wave function ψ in an atom, labelled by n, l and mₗ.

Probability density

|ψ|², the probability per unit volume of finding the electron at a point.

Node

A surface on which the probability density of an orbital is zero.

Degenerate orbitals

Orbitals that have exactly the same energy.

Effective nuclear charge

The net positive pull felt by an electron after shielding by the other electrons.

Exchange energy

The stabilisation gained when electrons of the same spin in degenerate orbitals can swap places.