Bohr's Model of Atom

Rutherford's nuclear atom had a fatal flaw: an orbiting electron should spiral into the nucleus in an instant, radiating away all its energy, and yet atoms are stable and hydrogen glows only at a handful of sharp, specific wavelengths, never a smear of colour. In 1913 Niels Bohr resolved both puzzles in one stroke by proposing that the electron is not free to circle at any radius it likes — only certain fixed, quantised orbits are allowed, and while sitting in one the electron simply does not radiate. This article builds up Bohr's four postulates, the resulting energy and radius formulas for the hydrogen atom, and the Rydberg formula that turns an orbit-to-orbit jump into a spectral line you can calculate the wavelength of.

The core insight

Bohr quantised the electron's angular momentum into fixed orbits, giving discrete energies $E_n = -13.6/n^2$ eV and radii $r_n = 0.529\,n^2$ Å, so that light is absorbed or emitted only when the electron jumps between these levels, exactly as the Rydberg formula predicts.

The core idea

Bohr's four postulates. The electron revolves around the nucleus only in certain fixed circular orbits — called stationary states — without radiating energy, contrary to what classical electromagnetism demands of any accelerating charge. Each stationary state has a definite, fixed energy, and the orbits are labelled outward from the nucleus by a principal quantum number $n = 1, 2, 3, \dots$ . The angular momentum of the electron in a stationary state is quantised in whole-number multiples of $h/2\pi$ :

mvr = n\,\frac{h}{2\pi}

where: $m$ is the electron's mass, $v$ its orbital speed, $r$ the orbit radius, $h$ is Planck's constant, and $n$ is the same principal quantum number that labels the orbit. Finally, the electron can jump between stationary states: it absorbs energy to jump to a higher orbit, and emits a photon on falling to a lower one, the photon carrying away exactly the energy difference between the two levels,

\Delta E = h\nu = E_{n_2} - E_{n_1}.

Energy and radius of an orbit. Working through the quantisation condition for the hydrogen atom ( $Z = 1$ ) gives two clean formulas that CBSE expects you to use directly, without re-deriving them from scratch each time:

E_n = -\frac{13.6}{n^2}\ \text{eV} \qquad r_n = 0.529\,n^2\ \text{Å}

where: $n$ is the principal quantum number of the orbit, $E_n$ is the electron's total energy in that orbit (in electron-volts), and $r_n$ is the orbit radius (in ångströms, $1\ \text{Å} = 10^{-10}\ \text{m}$ ). The minus sign is not a typo — it says the electron is bound to the nucleus, with energy measured relative to a free, motionless electron infinitely far away, which is defined as zero. As $n$ climbs, $E_n$ climbs too (becomes less negative), meaning the electron is more loosely held and its orbit swells outward, exactly as the $n^2$ in $r_n$ shows.

The Rydberg formula. When an electron falls from a higher orbit $n_2$ to a lower orbit $n_1$ , it releases a photon whose wavelength is fixed by the energy gap between the two orbits:

\frac{1}{\lambda} = R_H\left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right)

where: $\lambda$ is the wavelength of the emitted or absorbed light, $R_H = 1.097\times10^{7}\ \text{m}^{-1}$ is the Rydberg constant, $n_1$ is the lower (final) orbit and $n_2$ is the higher (initial) orbit, with $n_2 > n_1$ . Transitions that land on $n_1 = 1$ form the Lyman series (ultraviolet), those landing on $n_1 = 2$ form the Balmer series (visible light), and those landing on $n_1 = 3$ form the Paschen series (infrared). The picture below sketches the first few orbits and the transitions that build each series.

Figure

Concentric electron orbits around a nucleus labelled n equals 1 to 4 with downward arrows marking Lyman Balmer and Paschen transitions

Derivation AThe Rydberg formula, from Bohr's quantisation

The goal is to show that the Rydberg formula is not a separate law to memorise but a direct consequence of the two things Bohr's postulates already fix — the quantised energy $E_n = -13.6/n^2\ \text{eV}$ of each orbit and the rule $\Delta E = h\nu$ for a photon released on a jump — so that predicting the exact wavelength of any hydrogen spectral line becomes routine algebra rather than a fact to recall.

Step 1. Start from the two results already fixed by Bohr's postulates for the hydrogen atom: $E_n = -13.6/n^2\ \text{eV}$ for the energy of orbit $n$ , and the transition rule $\Delta E = h\nu = E_{n_2} - E_{n_1}$ whenever the electron falls from a higher orbit $n_2$ to a lower orbit $n_1$ .

Step 2. Substitute the energy formula into the transition rule. The energy released is $\Delta E = E_{n_2} - E_{n_1} = -\dfrac{13.6}{n_2^2} - \left(-\dfrac{13.6}{n_1^2}\right) = 13.6\left(\dfrac{1}{n_1^2} - \dfrac{1}{n_2^2}\right)\ \text{eV}$ , which is positive whenever $n_2 > n_1$ , exactly as it must be for an emitted (energy-releasing) transition.

Step 3. Convert energy to wavelength using $\Delta E = h\nu$ and $\nu = c/\lambda$ , so $\Delta E = hc/\lambda$ , giving $\dfrac{1}{\lambda} = \dfrac{\Delta E}{hc}$ . Rolling the constants $13.6\ \text{eV}$ , $h$ and $c$ together into a single number reproduces exactly the Rydberg constant, $R_H = 1.097\times10^{7}\ \text{m}^{-1}$ , so the formula collapses to $\dfrac{1}{\lambda} = R_H\left(\dfrac{1}{n_1^2} - \dfrac{1}{n_2^2}\right)$ .

Step 4. Read off the structure: the formula is nothing more than "energy gap between two quantised levels, converted to a wavelength." Every spectral line of hydrogen, in every series, is generated by picking a pair $(n_1, n_2)$ and plugging into this one equation — there is no separate formula per series, only a different choice of $n_1$ .

Worked example

Question. Calculate the energy and orbit radius of an electron in the $n = 2$ level of a hydrogen atom, then find the wavelength of light emitted when that same electron later falls from $n = 3$ to $n = 2$ , and name the spectral series.

Step 1. Energy of the $n = 2$ orbit: $E_2 = -\dfrac{13.6}{2^2}\ \text{eV} = -\dfrac{13.6}{4} = -3.4\ \text{eV}$ .

Step 2. Radius of the $n = 2$ orbit: $r_2 = 0.529 \times 2^2\ \text{Å} = 0.529 \times 4 = 2.116\ \text{Å}$ , roughly four times the ground-state radius $r_1 = 0.529\ \text{Å}$ , consistent with a more loosely bound, higher-energy electron.

Step 3. Apply the Rydberg formula to the $n = 3 \to n = 2$ fall, with $n_1 = 2$ (lower, final) and $n_2 = 3$ (upper, initial): $\dfrac{1}{\lambda} = R_H\left(\dfrac{1}{2^2} - \dfrac{1}{3^2}\right) = 1.097\times10^{7}\left(\dfrac{1}{4} - \dfrac{1}{9}\right) = 1.097\times10^{7} \times \dfrac{5}{36} = 1.524\times10^{6}\ \text{m}^{-1}$ .

Step 4. Invert to get the wavelength: $\lambda = \dfrac{1}{1.524\times10^{6}} \approx 6.56\times10^{-7}\ \text{m} = 656\ \text{nm}$ . Since the fall lands on $n_1 = 2$ , this is the Balmer series, and the emitted $656\ \text{nm}$ line is the well-known $H_\alpha$ line, in the red part of the visible spectrum.

Common traps

Dropping or mishandling the negative sign on $E_n$ : a higher $n$ gives a higher (less negative) energy, meaning a more loosely bound electron, not a "lower energy" electron.
Swapping $n_1$ and $n_2$ in the Rydberg formula — the larger $n$ (the initial, higher orbit) must be the one subtracted, otherwise $1/\lambda$ comes out negative.
Assuming Bohr's model applies to any atom: it is derived for, and strictly valid only for, hydrogen and hydrogen-like one-electron species, not multi-electron atoms such as helium.
Confusing absorption with emission: energy is absorbed only when the electron jumps to a higher orbit, and a photon is emitted only when it falls to a lower one.

Ask the tutor: want to work through a few more $E_n$ , $r_n$ , or Rydberg-formula transitions together, or see how Bohr's model compares with the wave-mechanical picture that replaced it?

Why doesn't the electron just spiral into the nucleus, the way Rutherford's model predicted?

Classical physics says any charge that accelerates — and an orbiting electron is constantly changing direction, so it is always accelerating — must radiate electromagnetic energy continuously, losing energy and spiralling inward within a fraction of a second. Bohr sidestepped this by simply postulating that stationary states do not radiate: as long as the electron sits in one of the allowed quantised orbits, no energy is lost, full stop. It is a rule imposed by hand rather than derived from classical electromagnetism, which is exactly why Bohr's model, while brilliantly successful for hydrogen, needed quantum mechanics to properly justify it later.

Why is the energy of an orbit negative instead of just being some positive number?

The zero of energy is a choice, and Bohr's model (like most of physics) sets it at the point where the electron is completely free — infinitely far from the nucleus, at rest, feeling no attraction. Any electron actually bound to the nucleus, sitting in a real orbit, therefore has less energy than that free state, which shows up as a negative number. The more negative $E_n$ is, the more tightly the electron is held; $E_1 = -13.6\ \text{eV}$ (the ground state) is the most tightly bound level, and $E_n \to 0$ as $n \to \infty$ corresponds to the electron just barely escaping the atom.

If Bohr's model gives the right answers for hydrogen's spectrum, why do we say it has limitations?

Getting the right numbers for one specific system does not mean the underlying picture is correct in general, and Bohr's model fails in several concrete ways once you push it further. It cannot be extended to work for atoms with more than one electron, where electron-electron repulsion complicates everything. It has nothing to say about the fine splitting of spectral lines seen in a magnetic field (Zeeman effect) or an electric field (Stark effect). And at a deeper level it treats the electron as a tiny particle tracing a fixed circular path, ignoring both its wave nature (de Broglie) and the fundamental limit on knowing position and momentum simultaneously (Heisenberg's uncertainty principle) — both of which the fuller quantum-mechanical model of the atom accounts for.

Bohr's Model of Atom

The core idea

Worked example

Common traps

Before this

Where this leads