Atoms

Our understanding of atomic structure evolved from the classical "plum-pudding" model to the quantised Bohr model, ultimately leading to quantum mechanics. This chapter traces that journey and focuses on the Bohr model and atomic spectra.

Thomson's Model (Plum Pudding Model)

J.J. Thomson (1897) proposed that an atom is a sphere of uniformly distributed positive charge with electrons embedded in it like plums in a pudding. This model explained the neutral atom but could not explain the results of Rutherford's scattering experiment.

Rutherford's Nuclear Model (1911)

Alpha particle scattering experiment: Alpha particles were fired at a thin gold foil. Observations:
Most passed straight through (atom is mostly empty space).
A few were deflected at large angles.
Very few (1 in 8000) bounced back.

Conclusions: Most mass and all positive charge of an atom are concentrated in a tiny, dense nucleus. The electrons revolve around the nucleus in orbits.

Drawback: According to classical electrodynamics, a charged particle undergoing circular motion should continuously emit radiation, lose energy, and spiral into the nucleus within 10^-8 s. Atoms would be unstable — contradicting reality.

Bohr's Model of Hydrogen Atom (1913)

1.Bohr proposed three postulates:
2.Electrons revolve in certain allowed circular orbits without radiating energy (stationary states).
3.An electron can jump from one orbit to another by absorbing or emitting a photon: E_photon = E₂ - E₁ = h nu.
4.The angular momentum is quantised: L = n h / (2 pi) = n h-bar, where n = 1, 2, 3, ...

Bohr's Formulae for Hydrogen:
Radius of n-th orbit: r_n = n² a₀, where a₀ = 0.529 Angstrom (Bohr radius, n=1)
Speed in n-th orbit: v_n = v₁ / n, where v₁ = 2.18 x 10⁶ m/s
Energy of n-th orbit: E_n = -13.6 / n² eV
Wavelength of emitted photon: 1/lambda = R_H (1/n₁² - 1/n₂²), R_H = 1.097 x 10⁷ m^-1

Rydberg constant R_H = 1.097 x 10⁷ m^-1

Hydrogen Spectral Series

| Series | n₁ | Region |
|---|---|---|
| Lyman | 1 | Ultraviolet |
| Balmer | 2 | Visible |
| Paschen | 3 | Infrared |
| Brackett | 4 | Infrared |
| Pfund | 5 | Far infrared |

Ionisation energy of hydrogen = 13.6 eV (energy to remove electron from n=1 to infinity).

Bohr Model for Hydrogen-Like Ions

For an ion with atomic number Z:
r_n = n² a₀ / Z
E_n = -13.6 Z² / n² eV

Limitations of Bohr's Model

Works only for single-electron atoms/ions (H, He+, Li²+, etc.)
Cannot explain fine structure of spectral lines or Zeeman effect
Contradicts Heisenberg uncertainty principle

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Example 1

Find the radius of the third Bohr orbit of hydrogen.
r₃ = 3² x a₀ = 9 x 0.529 = 4.76 Angstrom.

Example 2

Find the energy of electron in n = 2 state of hydrogen.
E₂ = -13.6 / 2² = -13.6 / 4 = -3.4 eV.

Example 3

Find wavelength of light emitted when electron in hydrogen jumps from n = 3 to n = 2 (Balmer series).
1/lambda = R_H (1/2² - 1/3²) = 1.097 x 10⁷ (1/4 - 1/9) = 1.097 x 10⁷ x 5/36 = 1.524 x 10⁶ m^-1. lambda = 656 nm (red light — H-alpha line).

Example 4

The ionisation energy of hydrogen is 13.6 eV. Find the ionisation energy of He+ (Z = 2).
E_ionisation = 13.6 x Z² / 1² = 13.6 x 4 = 54.4 eV.

Example 5

An electron in hydrogen absorbs a photon and jumps from n = 1 to n = 4. Find energy of photon.
Delta E = 13.6 (1/1² - 1/4²) = 13.6 (1 - 1/16) = 13.6 x 15/16 = 12.75 eV.

Example 6

How many spectral lines can be emitted when an electron drops from n = 4 level in hydrogen?
Number of lines = n(n-1)/2 = 4 x 3/2 = 6 lines (transitions: 4→3, 4→2, 4→1, 3→2, 3→1, 2→1).

Example 7

Find the ratio of velocities of electrons in orbits n = 1 and n = 2 of hydrogen.
v_n = v₁/n. v₁ : v₂ = 1/1 : 1/2 = 2:1.

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Common mistakes

Taking energy as positive — remember E_n = -13.6/n² eV (negative means bound state).
Confusing Rydberg formula: 1/lambda = R_H(1/n₁² - 1/n₂²) with n₂ > n₁ for emission.
Assuming Bohr model works for multi-electron atoms — it only applies to hydrogen-like (single-electron) species.

Summary

Rutherford discovered the nuclear model; Bohr quantised electron orbits. Key result: E_n = -13.6/n² eV. Hydrogen spectra arise from electron transitions between levels. Bohr radius a₀ = 0.529 A for n = 1. Limitations led to quantum mechanics.