JEE Physics Atoms Notes Set 01

Early Models of the Atom

Thomson pictured the atom as a sphere of positive charge with electrons embedded in it like seeds in a watermelon. This plum-pudding model could not explain how atoms emit light of definite wavelengths, and it failed completely when alpha particles were fired at thin gold foil.

In the Rutherford scattering experiment most alpha particles passed straight through, a few were deflected through large angles and about one in 8000 came almost straight back. Rutherford concluded that the positive charge and nearly all the mass are packed in a tiny nucleus of size about 10⁻¹⁵ m, while the electrons move around it in the empty space of an atom of size about 10⁻¹⁰ m.

When an alpha particle of charge 2e and kinetic energy K is fired head on at a nucleus of charge Ze, it stops momentarily at the distance of closest approach r₀, where all its kinetic energy has become electrostatic potential energy: \( K = \frac{1}{4\pi\varepsilon_0}\frac{(2e)(Ze)}{r_0} \). Hence r₀ is proportional to Ze²/K, so a heavier nucleus or a lower energy gives a larger r₀. The impact parameter b is the perpendicular distance between the line of motion and the nucleus: the smaller b is, the larger the scattering angle, and b = 0 gives the head-on case with a deflection of 180°.

Example. A 5.3 MeV alpha particle approaches a gold nucleus (Z = 79) head on. Using \( r_0 = \frac{9\times10^{9}\times 2\times 79 \times (1.6\times10^{-19})^2}{5.3\times1.6\times10^{-13}} \) gives r₀ ≈ 4.3 × 10⁻¹⁴ m.

Rutherford's model has two serious defects. An electron moving in a circle is accelerated, so it should radiate energy and spiral into the nucleus within about 10⁻¹⁰ s, which would make atoms unstable. The model also predicts a continuous spectrum, whereas atoms give line spectra.

Bohr's Model of the Hydrogen Atom

Bohr kept the nuclear atom but added three postulates.

1. An electron moves in certain circular orbits without radiating energy; these are the stationary orbits.
2. The allowed orbits are those in which the angular momentum is a whole-number multiple of h/2π: \( mvr = \frac{nh}{2\pi} \), where n = 1, 2, 3, ... is the principal quantum number.
3. Radiation is emitted or absorbed only when the electron jumps between two stationary orbits, and the photon has energy equal to the difference of the two energies: hν = E_i − E_f.

Combining the centripetal condition \( \frac{mv^2}{r} = \frac{1}{4\pi\varepsilon_0}\frac{Ze^2}{r^2} \) with the quantisation rule gives the following results for a hydrogen-like atom (one electron, nuclear charge Z).

The negative total energy means that the electron is bound. The energy needed to remove it from the level n is +13.6 Z²/n² eV. Bohr's radius a₀ = 0.529 Å is the radius of the first orbit of hydrogen.

Example. The speed in the third orbit of He⁺ is 2.18 × 10⁶ × 2/3 = 1.46 × 10⁶ m/s, and its energy is −13.6 × 4/9 = −6.04 eV.

Energy Levels and Excitation

The first excitation energy of hydrogen (n = 1 to n = 2) is 10.2 eV, the ionisation energy is 13.6 eV, and an excited atom in the level n needs 13.6/n² eV to be ionised. An atom can absorb a photon only if its energy exactly matches a level difference, whereas a bombarding electron can give away any part of its energy that equals an allowed excitation. When atoms are excited to the level n, the number of different lines they can emit is n(n − 1)/2.

Hydrogen Spectrum

The wavelength of the line emitted when an electron falls from n₂ to n₁ is given by \( \frac{1}{\lambda} = RZ^2\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right) \), with the Rydberg constant R = 1.097 × 10⁷ m⁻¹. Also hc = 12400 eV Å, which is useful for converting energies to wavelengths.

The longest wavelength of a series comes from the jump from n₁ + 1 to n₁ and the shortest from the jump from infinity to n₁. For every series the shortest wavelength is n₁²/R. For the same transition in a hydrogen-like ion the wavelength is smaller by the factor Z². The ratio of the longest wavelengths of the Lyman and Balmer series is 5 : 27, and the ratio of the shortest wavelengths of the Balmer and Lyman series is 4 : 1.

Example. A hydrogen atom in the ground state is hit by light of wavelength 975 Å. The photon energy is 12400/975 = 12.7 eV, which raises the electron to n = 4 (the gap E₄ − E₁ is 12.75 eV). The atom can then emit 4 × 3/2 = 6 lines, with the longest wavelength 18750 Å (4 to 3) and the shortest 973 Å (4 to 1).

Other Useful Points

Bohr's model works well for hydrogen and for one-electron ions such as He⁺ and Li²⁺, but it cannot explain the spectra of atoms with more electrons, the relative intensities of lines, the fine structure of lines or the splitting of lines in a magnetic field. It also treats the electron as having an exact position and momentum, which is not allowed by the uncertainty principle.

The nucleus is not completely at rest, since the electron and nucleus orbit their common centre of mass. This is allowed for by using the reduced mass μ = m_eM/(m_e + M) in place of m_e, which changes the energies by about 0.05% for hydrogen. When an atom emits a photon it recoils with energy p²/2M where p = h/λ, which is negligible for light but matters for γ-rays.