"Structure of Atom"

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What is inside an atom? This chapter traces the journey from Rutherford's gold foil experiment that discovered the nucleus to Bohr's quantum model that explained the stability of atoms and their characteristic spectral lines — a pivotal moment in the birth of modern physics.

Key Concepts

24.1 Rutherford's Scattering Experiment

Bombarded thin gold foil with α-particles (not β or γ rays)
Most α-particles passed straight through → atom is mostly empty space
Some deflected at large angles → presence of a positively charged hard core (nucleus) at the centre
Nucleus is surrounded by electrons

Rutherford's nuclear model: Atom consists of a small, dense, positively charged nucleus with electrons revolving around it (like planets around the Sun).

24.2 Limitations of Rutherford's Model

Stability of atom: According to classical EM theory, an accelerating charged particle (electron in circular orbit) must radiate energy and spiral into the nucleus. Atoms should be unstable — but they ARE stable.
Line spectrum vs continuous spectrum: The spiralling electron should emit a continuous range of frequencies. But experiments show atoms emit only discrete (specific) frequencies — line spectra, not continuous spectra.

24.3 Bohr's Model of the Hydrogen Atom

Bohr combined classical and quantum concepts:

Postulate 1 (Classical): Electrons revolve in circular orbits with centripetal force provided by Coulomb attraction: $\frac{m v ^{2}}{r} = \frac{Z e ^{2}}{4 π ε _{0} r ^{2}}$

Postulate 2 (Quantum): Angular momentum is quantised: $L = m v r = n \frac{h}{2 π}, n = 1, 2, 3, \dots$

Postulate 3 (Stationary states): Electrons in allowed orbits do NOT radiate energy — these are stationary states.

Postulate 4 (Transitions): Energy is emitted/absorbed only when an electron jumps between levels: $Δ E = E_{f} - E_{i} = h ν$

24.4 Energy Levels and Spectral Series

Energy of nth orbit: $E_{n} = - \frac{13.6}{n ^{2}} eV (for hydrogen)$

Radius of nth orbit: $r_{n} = n^{2} a_{0}, a_{0} = 0.53 \overset{˚}{A} (Bohr radius)$

Spectral series:

Series	Transition to	Region
Lyman	$n = 1$	Ultraviolet
Balmer	$n = 2$	Visible
Paschen	$n = 3$	Infrared
Brackett	$n = 4$	Infrared
Pfund	$n = 5$	Infrared

INTEXT QUESTIONS 24.1

Q1. In Rutherford's scattering experiment, the target was bombarded with (a) β-rays (b) γ-rays (c) α-rays.

Ans: (c) α-rays.

Q2. Name two experimental observations that could not be explained by Rutherford's model.

Ans:

Stability of the atom: Accelerating electrons should radiate energy and spiral into the nucleus — yet atoms are stable.
Line spectra: Model predicted continuous spectra; experiments showed discrete line spectra.

INTEXT QUESTIONS 24.2

Q1. Which of Bohr's postulates fit with classical physics and which support quantum physics?

Ans:

Classical: First postulate — circular orbits with Coulomb centripetal force.
Quantum: Angular momentum quantisation ( $L = nh /2 π$ ), stationary states (no radiation), photon emission/absorption during transitions ( $Δ E = h ν$ ).

Q2. According to Bohr, why did an atom not collapse while its electrons revolved around the nucleus?

Ans: Bohr's third postulate — electrons in allowed (stationary) orbits do NOT radiate energy, contradicting classical EM theory. They can exist indefinitely in these orbits. Energy is exchanged only during transitions between levels.

Q3. According to Bohr, what happens when a photon is (i) emitted, (ii) absorbed?

Ans: (i) Emission: Electron falls from higher ( $E_{f}$ ) to lower ( $E_{i}$ ) energy level — photon emitted with $h ν = E_{f} - E_{i}$ .

(ii) Absorption: Electron jumps from lower ( $E_{i}$ ) to higher ( $E_{f}$ ) energy level by absorbing a photon of exactly $h ν = E_{f} - E_{i}$ .

Terminal Exercise

Describe Rutherford's α-particle scattering experiment. State the conclusions drawn.
State the limitations of Rutherford's nuclear model.
State Bohr's postulates for the hydrogen atom.
Derive the expression for the radius of the nth orbit: $r_{n} = \frac{n ^{2} h ^{2} ε _{0}}{π m Z e ^{2}}$ .
Derive the expression for energy of electron in nth orbit: $E_{n} = - \frac{m e ^{4}}{8 ε _{0}^{2} h ^{2}} \frac{Z ^{2}}{n ^{2}}$ .
Calculate the radius of the first Bohr orbit of hydrogen. ( $h = 6.63 \times 1 0^{- 34}$ J⋅s, $m = 9.1 \times 1 0^{- 31}$ kg, $e = 1.6 \times 1 0^{- 19}$ C)
The ground state energy of hydrogen is −13.6 eV. Find the kinetic and potential energies of the electron in this state.
Name the spectral series of hydrogen. Which series lies in the visible region?
Calculate the wavelength of the first line of the Balmer series. ( $R = 1.097 \times 1 0^{7}$ m⁻¹)
Explain why Bohr's model could not explain the spectra of multi-electron atoms.

Quick Revision

Concept	Formula / Value
Bohr radius $a_{0}$	0.53 Å
Ground state energy (H)	−13.6 eV
Energy of nth level	$E_{n} = - 13.6/ n^{2}$ eV
Angular momentum	$L = nh /2 π$
Transition energy	$Δ E = h ν = E_{f} - E_{i}$
Rydberg formula	$\frac{1}{λ} = R (\frac{1}{n _{1}^{2}} - \frac{1}{n _{2}^{2}})$
Rutherford: nucleus discovered by	α-particle scattering

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