Bohr Model of the Hydrogen Atom

Introduction

In 1913, Niels Bohr expanded upon Rutherford's atomic model by introducing quantum theory to explain the discrete energy levels of electrons in atoms. Bohr’s model of the atom resembles a "planetary model," where electrons revolve around a dense central nucleus in fixed, circular orbits, similar to planets orbiting the Sun. However, Bohr’s model introduced the revolutionary idea that the orbits of electrons are quantized.

Bohr's theory successfully described the hydrogen atom and other hydrogen-like atoms (such as the singly ionized helium atom and doubly ionized lithium atom) by explaining the discrete spectral lines they emitted. His model was based on four key postulates that combined classical mechanics with the new idea of quantization:

Electrons move in circular orbits around the nucleus without radiating energy.
The angular momentum of an electron in these orbits is quantized and given by:

m v r = n h / 2π

Where:

$m$ = mass of the electron,
$v$ = velocity of the electron,
$r$ = radius of the orbit,
$n$ = an integer (the quantum number),
$h$ = Planck’s constant,
$2π$ = constant factor (2π).

This postulate implies that only certain orbits with specific radii and energies are allowed for the electron. The quantization of angular momentum leads to the concept of discrete energy levels in the atom.

Energy Levels of the Hydrogen Atom

The energy associated with each allowed orbit (energy level) is also quantized and is given by the following expression:

E = - \frac{2π}{n} m e r

Where:

$E$ = energy of the electron in orbit,
$n$ = principal quantum number (n = 1, 2, 3, ...),
$m$ = mass of the electron,
$e$ = charge of the electron,
$r$ = radius of the orbit.

The energy levels are negative, indicating that the electron is bound to the nucleus. The closer the orbit is to the nucleus (i.e., the lower the quantum number n), the more negative the energy, and thus the more tightly bound the electron is to the nucleus.

Radiation Emission and Spectral Lines

When an electron jumps from a higher orbit (higher energy) to a lower orbit (lower energy), it emits electromagnetic radiation corresponding to the difference in energy between the two levels. This emitted radiation forms the characteristic spectral lines of hydrogen, and the wavelength of this radiation is given by the following equation:

1/λ = \frac{R}{n ₁ ²}

Where:

$λ$ = wavelength of emitted radiation,
$R$ = Rydberg constant,
$n₁$ and $n₂$ = initial and final quantum numbers, respectively.

Conclusion

The Bohr model was a revolutionary step in understanding atomic structure and successfully explained the hydrogen atom’s discrete spectral lines. Though modern quantum mechanics has supplanted Bohr's model, his ideas provided a foundational understanding of atomic behavior and the development of later, more complex theories in atomic physics.

Illustration: Bohr's atomic model : images of electron orbits, the atom's structure