Bohr's Model of the Atom

To understand where Bohr's model came from, you need to see the chain of ideas he was building on:

Thomson (1897–1904): Discovered the electron and proposed the plum-pudding model — electrons embedded in a diffuse positive sphere. No nucleus. No empty space.

Rutherford (1911): Blew up the plum-pudding model with the gold-foil experiment. Showed that almost all the atom's mass is concentrated in a tiny, dense, positively charged nucleus, and electrons occupy the vast empty space around it. But his model could not explain how electrons stayed in orbit without radiating energy and crashing.

Planck (1900) + Einstein (1905): Energy comes in discrete packets (quanta). Light is not a continuous wave — it consists of photons, each carrying energy $E = h\nu$.

Bohr's synthesis (1913): Took Rutherford's nuclear structure + Planck–Einstein quantisation + the experimental data on hydrogen's line spectrum, and wove them into a self-consistent model. The key leap: he simply assumed that only certain orbits are allowed — those where the electron's angular momentum is a whole-number multiple of $\frac{h}{2\pi}$. This assumption had no classical justification. But it worked.

Postulate i — Stationary Orbits (Allowed States)

An electron in a hydrogen atom can move around the nucleus in circular paths of fixed radius and fixed energy. These paths are called orbits, stationary states, or allowed energy states. The orbits are arranged concentrically around the nucleus, like rings. As long as the electron stays in one orbit, it neither gains nor loses energy — which is why these are called stationary states (the electron itself still moves, but its energy is stationary).

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Postulate ii — No Radiation While in Orbit; Quanta at Transitions

The energy of an electron in an allowed orbit does not change with time — it does not continuously radiate (defying classical electrodynamics). However:

• If the electron absorbs a quantum of energy, it jumps from a lower orbit to a higher orbit (excitation).
• If it falls from a higher orbit to a lower orbit, it emits that exact energy difference as a photon.

Crucially, the energy change does not happen gradually — it is instantaneous and comes in a single quantum. There is no in-between state.

Postulate iii — Bohr's Frequency Rule

When an electron transitions between two stationary states that differ in energy by $\Delta E$, the frequency $\nu$ of the photon absorbed or emitted is given by:

Postulate iv — Quantised Angular Momentum (The Key Assumption)

This is the quantum heart of Bohr's model. An electron can only move in orbits where its angular momentum is an integer multiple of $\frac{h}{2\pi}$.

Recall: for an electron of mass $m_e$ moving in a circular orbit of radius $r$ with speed $v$, the moment of inertia is $I = m_e r^2$ and the angular velocity is $\omega = v/r$, so:

$\text{angular momentum} = I \times \omega = m_e r^2 \times \frac{v}{r} = m_e v r$

Bohr's quantisation condition says this must equal a whole-number multiple of $h/2\pi$:

The electron stays in a circular orbit because the electrostatic attraction toward the nucleus exactly provides the centripetal force needed for circular motion:

$\frac{ke^2}{r^2} = \frac{m_e v^2}{r} \quad \Rightarrow \quad ke^2 = m_e v^2 r \quad \cdots (i)$

where $k = \frac{1}{4\pi\varepsilon_0}$ is Coulomb's constant and $e$ is the electron charge. From Bohr's quantisation (Eq. 2.11):

$m_e v r = \frac{nh}{2\pi} \quad \Rightarrow \quad v = \frac{nh}{2\pi m_e r} \quad \cdots (ii)$

Substituting (ii) into (i) and solving for $r$ gives:

Now substitute the result $r_n = n^2 a_0$ back into the quantisation condition $v = \frac{nh}{2\pi m_e r}$:

$v_n = \frac{nh}{2\pi m_e \cdot n^2 a_0} = \frac{h}{2\pi m_e n a_0}$

Substituting the known values of $h$, $m_e$, and $a_0$:

The total mechanical energy of the electron is the sum of its kinetic and potential energies. The potential energy is negative (attractive force):

$E = KE + PE = \frac{1}{2}m_e v^2 + \left(-\frac{ke^2}{r}\right)$

From the force balance, $ke^2 = m_e v^2 r$, so $\frac{1}{2}m_e v^2 = \frac{ke^2}{2r}$:

$E = \frac{ke^2}{2r} - \frac{ke^2}{r} = -\frac{ke^2}{2r}$

Substituting $r_n = n^2 a_0$ and defining $R_H = \frac{ke^2}{2a_0}$:

Extension to Hydrogen-Like Ions

Bohr's model also applies to any ion with just one electron — called hydrogen-like species: $\ce{He+}$ ($Z=2$), $\ce{Li^{2+}}$ ($Z=3$), $\ce{Be^{3+}}$ ($Z=4$), etc. The nuclear charge $Ze$ simply replaces $e$ in the Coulomb force:

• Energy gets more negative as $Z$ increases (electron more tightly bound to a higher-charge nucleus)
• Orbit radius gets smaller as $Z$ increases (stronger pull → smaller orbit)
• Velocity increases with $Z$ (stronger pull → faster orbit to maintain balance)

This is Bohr's greatest triumph. The line spectrum — which Balmer and Rydberg described with empirical formulas but could not explain — follows directly from Bohr's energy levels.

When an electron transitions from an initial orbit $n_i$ to a final orbit $n_f$, the energy difference is:

$\Delta E = E_f - E_i = -\frac{R_H}{n_f^2} - \left(-\frac{R_H}{n_i^2}\right) = R_H\left(\frac{1}{n_i^2} - \frac{1}{n_f^2}\right)$

• Emission ($n_i > n_f$): electron falls to a lower orbit, $\Delta E$ is negative, energy is released as a photon.
• Absorption ($n_f > n_i$): electron jumps to a higher orbit, $\Delta E$ is positive, energy is absorbed from a photon.

The frequency and wavenumber of the photon are:

Every spectral series of hydrogen is now understood as transitions into a particular final orbit:

• Lyman ($n_f = 1$): transitions from $n_i \geq 2$ into the ground state → high-energy UV photons
• Balmer ($n_f = 2$): transitions from $n_i \geq 3$ into $n=2$ → visible photons (the four coloured lines Balmer saw)
• Paschen ($n_f = 3$), Brackett ($n_f = 4$), Pfund ($n_f = 5$): infrared photons

The brightness of each spectral line depends on the number of atoms making that particular transition — more atoms making the same jump means a more intense line.

The fact that Bohr's formula gives exactly Rydberg's empirical constant ($1.09677 \times 10^7$ m$^{-1}$) was electrifying. A formula derived from quantum postulates perfectly matched a constant measured in the lab decades earlier.

Bohr's model was a brilliant breakthrough, but it came with serious limitations:

1. Fails for multi-electron atoms
Bohr's model works for hydrogen ($Z=1$) and hydrogen-like ions (one electron), but breaks down completely for atoms with two or more electrons. Electron–electron repulsions create interactions the model ignores entirely.

2. Cannot explain fine structure of spectral lines
When instruments of higher resolution examine hydrogen's spectral lines, each 'line' is actually a cluster of closely-spaced lines (doublets and triplets). Bohr's model predicts single lines and cannot account for this fine structure.

3. Cannot explain the Zeeman and Stark effects
In a magnetic field, spectral lines split into multiple components (Zeeman effect). In an electric field, they split differently (Stark effect). Bohr's model has no mechanism to explain these splittings.

4. Contradicts Heisenberg's Uncertainty Principle
The concept of a fixed, well-defined circular orbit implies that you know both the electron's position and momentum simultaneously with perfect precision. Heisenberg proved this is impossible — you cannot know both at once. An orbit is fundamentally a classical concept that doesn't apply to quantum particles.

5. Ignores the wave nature of the electron
de Broglie showed (1924) that electrons have wave properties. A particle with wave nature cannot be confined to a well-defined circular trajectory. Bohr's model treats electrons purely as particles.

6. Cannot explain chemical bonding or molecular shapes
Bohr's model says nothing about how atoms share electrons to form molecules, or why molecules have specific shapes.