7.4.3. Bohr Model

Table of Contents

A New Picture of the Atom

The Bohr model was an important step in the development of atomic physics. It was created to explain why atoms emit light at specific wavelengths instead of a continuous range of colors. Earlier models of the atom could not explain this behavior well. The Bohr model introduced the idea that electrons in an atom can exist only in certain allowed orbits, each with a definite energy.

This model works especially well for hydrogen, the simplest atom, which has one proton and one electron. It does not fully describe more complicated atoms, but it was a major success and helped lead to quantum mechanics.

The Basic Idea

In the Bohr model, the electron moves around the nucleus in circular orbits. Unlike in classical physics, the electron is not allowed to orbit at just any distance from the nucleus. Only certain orbits are possible.

Each allowed orbit has a fixed energy. When the electron stays in one of these orbits, it does not radiate energy. This was a radical idea, because according to classical electrodynamics an accelerating charge should continuously emit radiation and spiral into the nucleus.

Bohr avoided this problem by proposing special rules for atomic motion.

In the Bohr model, electrons can occupy only certain allowed energy levels. They do not radiate energy while remaining in one of these allowed states.

Bohr's Postulates

Bohr's model is based on a few key postulates. These are not derived from classical mechanics, they are assumptions introduced to match experimental observations.

First, the electron can move only in certain stable circular orbits around the nucleus.

Second, each stable orbit has a definite energy.

Third, radiation is emitted or absorbed only when the electron jumps from one orbit to another.

If an electron moves from a higher-energy orbit to a lower-energy orbit, the atom emits a photon. If it moves from a lower-energy orbit to a higher-energy orbit, the atom absorbs a photon.

The photon energy is given by

$$
E_{\gamma} = h f
$$

where $h$ is Planck's constant and $f$ is the frequency of the light.

The energy of the emitted or absorbed photon equals the difference between the two atomic energy levels:

$$
h f = E_i - E_f
$$

for emission, where $E_i$ is the initial energy and $E_f$ is the final energy.

Light is emitted or absorbed only during transitions between energy levels, with
$$
h f = |E_i - E_f|
$$

Quantized Angular Momentum

Bohr introduced a rule for the electron's angular momentum. He proposed that only orbits satisfying

$$
L = m_e v r = n \hbar
$$

are allowed, where $m_e$ is the electron mass, $v$ is its speed, $r$ is the orbit radius, $n = 1, 2, 3, \dots$ is the principal quantum number, and

$$
\hbar = \frac{h}{2\pi}
$$

This means angular momentum is quantized. The integer $n$ labels the allowed orbits.

The smallest orbit corresponds to $n = 1$, the next to $n = 2$, and so on.

Bohr's quantization rule for hydrogen is
$$
m_e v r = n \hbar, \quad n = 1, 2, 3, \dots
$$
Only these discrete values are allowed.

Allowed Orbits in Hydrogen

For a hydrogen atom, the electron is held in orbit by the electric attraction between the negatively charged electron and the positively charged proton. The electric force provides the centripetal force:

$$
\frac{k e^2}{r^2} = \frac{m_e v^2}{r}
$$

where $k = \frac{1}{4\pi \varepsilon_0}$ and $e$ is the magnitude of the electron charge.

Combining this with the angular momentum condition gives the allowed radii of the electron orbits:

$$
r_n = a_0 n^2
$$

where $a_0$ is the Bohr radius,

$$
a_0 = \frac{4\pi \varepsilon_0 \hbar^2}{m_e e^2}
$$

Numerically,

$$
a_0 \approx 5.29 \times 10^{-11}\,\text{m}
$$

So the first orbit has radius $a_0$, the second has radius $4a_0$, the third has radius $9a_0$, and so on.

Energy Levels

The total energy of the electron in the hydrogen atom is the sum of kinetic and electric potential energy. In the Bohr model, the allowed energies are

$$
E_n = -\frac{13.6\,\text{eV}}{n^2}
$$

for $n = 1, 2, 3, \dots$

The negative sign means the electron is bound to the nucleus. An electron with zero energy would be free, infinitely far from the nucleus.

The lowest energy state, $n=1$, is called the ground state. Higher values of $n$ are excited states.

Level	$n$	Energy
Ground state	1	$-13.6\,\text{eV}$
First excited state	2	$-3.4\,\text{eV}$
Second excited state	3	$-1.51\,\text{eV}$
Third excited state	4	$-0.85\,\text{eV}$

As $n$ becomes very large, the energy approaches $0$ from below.

For hydrogen, the Bohr energy levels are
$$
E_n = -\frac{13.6\,\text{eV}}{n^2}
$$
The ground state is $n=1$.

Emission and Absorption of Light

When an electron changes from one level to another, the atom exchanges a photon whose energy matches the energy difference.

If the electron drops from $n_i$ to $n_f$, with $n_i > n_f$, the photon energy is

$$
h f = E_{n_i} - E_{n_f}
$$

Since $E_n$ is negative, this gives a positive photon energy.

Using $E = h f = \frac{hc}{\lambda}$, the wavelength of the light is determined by the transition.

For hydrogen, this leads to the Rydberg formula:

$$
\frac{1}{\lambda} = R \left( \frac{1}{n_f^2} - \frac{1}{n_i^2} \right)
$$

where $R$ is the Rydberg constant and $n_i > n_f$ for emission.

This explained the observed spectral lines of hydrogen, which was one of the great successes of the Bohr model.

A Visual Picture

Rendering scientific sketch…

Bohr model of hydrogen atom

This drawing shows the nucleus at the center and several allowed circular orbits. The electron can jump from a larger orbit to a smaller one, releasing a photon.

Example of a Transition

Suppose an electron in hydrogen falls from $n=3$ to $n=2$.

The energies are

$$
E_3 = -\frac{13.6}{9}\,\text{eV} \approx -1.51\,\text{eV}
$$

and

$$
E_2 = -\frac{13.6}{4}\,\text{eV} = -3.4\,\text{eV}
$$

So the emitted photon has energy

$$
E_{\gamma} = E_3 - E_2 = (-1.51) - (-3.4) = 1.89\,\text{eV}
$$

Then

$$
\lambda = \frac{hc}{E_{\gamma}}
$$

This transition belongs to the visible part of the hydrogen spectrum.

Why the Model Was Important

The Bohr model was important because it explained several facts that older atomic models could not explain. It explained why hydrogen has discrete spectral lines. It also gave a simple picture of atomic energy levels and introduced quantization directly into atomic structure.

It showed that atomic energies are not continuous. This idea became one of the foundations of modern quantum physics.

Limitations of the Bohr Model

Although the Bohr model was a great advance, it is not the final theory of the atom. It has important limitations.

It works well mainly for hydrogen and hydrogen-like ions, such as $\text{He}^+$ or $\text{Li}^{2+}$, which have only one electron.

It cannot accurately explain atoms with many electrons.

It does not explain fine details of spectral lines.

It assumes definite circular orbits, which is not the modern quantum picture.

It does not fully agree with the wave nature of electrons.

Modern quantum mechanics replaced the idea of fixed electron orbits with orbitals and wave functions.

The Bohr model is historically important and useful for hydrogen, but it is not a complete description of atoms. Modern quantum mechanics gives the more accurate theory.

Summary Relations

The most important formulas of the Bohr model for hydrogen are collected here.

Quantity	Formula
Angular momentum quantization	$m_e v r = n\hbar$
Allowed radii	$r_n = a_0 n^2$
Bohr radius	$a_0 = \dfrac{4\pi\varepsilon_0\hbar^2}{m_e e^2}$
Energy levels	$E_n = -\dfrac{13.6\,\text{eV}}{n^2}$
Photon energy in transition	$hf =	E_i - E_f	$
Spectral wavelength	$\dfrac{1}{\lambda} = R\left(\dfrac{1}{n_f^2} - \dfrac{1}{n_i^2}\right)$

Final Perspective

The Bohr model is a bridge between classical and quantum physics. It keeps the familiar image of an electron circling a nucleus, but adds the crucial idea that only certain orbits and energies are allowed. Even though modern physics has gone beyond it, the Bohr model remains one of the clearest first explanations of atomic energy levels and atomic spectra.

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