Two Developments Behind the Model
Historically, the structure of atoms and molecules was deduced not by looking at atoms directly but by studying how radiation interacts with matter. Niels Bohr used the results of such studies in 1913 to improve upon Rutherford's nuclear model. According to NCERT Section 2.3, two developments played the major role in the formulation of Bohr's model.
| Development | What it established | Phenomena it explained |
|---|---|---|
| Dual character of electromagnetic radiation | Radiation behaves both as a wave and as a stream of particles | Interference and diffraction (wave); black-body radiation and photoelectric effect (particle) |
| Experimental results on atomic spectra | Atoms emit and absorb light only at discrete wavelengths | The line spectrum of hydrogen and other elements |
The common thread running through both developments is quantisation — the restriction of a physical property to discrete values. Before Bohr could model the electron's energy as fixed, allowed levels, physicists first had to accept that radiation itself is quantised. We therefore begin with the nature of electromagnetic radiation.
Wave Nature of Electromagnetic Radiation
In the mid-nineteenth century, physicists studied the absorption and emission of radiation by heated objects — the so-called thermal radiations. James Clerk Maxwell (1870) was the first to explain that when an electrically charged particle moves under acceleration, alternating electric and magnetic fields are produced and transmitted as electromagnetic waves. Heinrich Hertz later confirmed this experimentally. Light, supposed by Newton to be made of corpuscles, was thus shown in the nineteenth century to have a wave nature.
Maxwell revealed that a light wave carries oscillating electric and magnetic components. These two components have the same wavelength, frequency, speed and amplitude, but they vibrate in two mutually perpendicular planes, and both are perpendicular to the direction in which the wave travels. Unlike sound waves or water waves, electromagnetic waves do not require a medium and can travel through vacuum.
Electromagnetic radiation is characterised by two related properties: frequency and wavelength. In vacuum, every type of radiation, regardless of wavelength, travels at the same speed — the speed of light, $c = 3.0 \times 10^{8}\ \text{m s}^{-1}$ (more precisely $2.997925 \times 10^{8}\ \text{m s}^{-1}$). Frequency $(\nu)$, wavelength $(\lambda)$ and speed are connected by the fundamental relation:
$$ c = \nu\,\lambda $$
A third quantity, used especially in spectroscopy, is the wavenumber $(\bar{\nu})$, defined as the number of wavelengths per unit length — that is, the reciprocal of the wavelength:
$$ \bar{\nu} = \frac{1}{\lambda} $$
| Parameter | Symbol | Meaning | SI unit |
|---|---|---|---|
| Frequency | $\nu$ | Number of waves passing a point per second | hertz (Hz, s⁻¹) |
| Wavelength | $\lambda$ | Distance between two consecutive crests (or troughs) | metre (m) |
| Wavenumber | $\bar{\nu}$ | Number of wavelengths per unit length | m⁻¹ (cm⁻¹ common, non-SI) |
| Speed of light | $c$ | Speed of all EM radiation in vacuum | m s⁻¹ |
Wavenumber is not frequency.
Both look like "number of waves", but wavenumber $\bar{\nu} = 1/\lambda$ counts waves per unit length (unit cm⁻¹ or m⁻¹), whereas frequency $\nu = c/\lambda$ counts waves per second (unit Hz). They differ by a factor of $c$: $\nu = c\,\bar{\nu}$.
Watch the units. If the answer is in cm⁻¹ it is a wavenumber; if in Hz or s⁻¹ it is a frequency.
All India Radio, Delhi (Vividh Bharati) broadcasts at a frequency of 1,368 kHz. Calculate the wavelength of the radiation, and identify the region of the spectrum.
Rearranging $c = \nu\lambda$ gives $\lambda = c/\nu$. Here $\nu = 1368\ \text{kHz} = 1368 \times 10^{3}\ \text{s}^{-1}$ and $c = 3.0 \times 10^{8}\ \text{m s}^{-1}$.
$$ \lambda = \frac{3.0 \times 10^{8}\ \text{m s}^{-1}}{1.368 \times 10^{6}\ \text{s}^{-1}} \approx 219.3\ \text{m} $$
A wavelength of about 219 m is a characteristic radiowave wavelength, so the radiation lies in the radio-frequency region of the electromagnetic spectrum.
The Electromagnetic Spectrum
There are many types of electromagnetic radiation, differing from one another only in wavelength (and hence frequency). Arranged in order, they constitute the electromagnetic spectrum. Different regions are given different names, but the underlying physics is identical — all are EM waves obeying $c = \nu\lambda$. The small band our eyes can detect, around $10^{15}$ Hz, is called visible light; everything else requires special instruments.
| Region | Approx. frequency | Common use / source |
|---|---|---|
| Radio frequency | ~10⁶ Hz | Broadcasting |
| Microwave | ~10¹⁰ Hz | Radar |
| Infrared | ~10¹³ Hz | Heating |
| Visible light | ~10¹⁵ Hz | Detected by the eye (4.0–7.5 × 10¹⁴ Hz) |
| Ultraviolet | ~10¹⁶ Hz | A component of the Sun's radiation |
Within the visible band itself, NCERT notes that the spectrum runs from violet at $7.50 \times 10^{14}$ Hz (400 nm) to red at $4.0 \times 10^{14}$ Hz (750 nm). Red light has the longest wavelength and is deviated least by a prism; violet has the shortest and is deviated most.
Particle Nature: Planck's Quantum Theory
Wave theory comfortably explained interference and diffraction, but several observations defied even Maxwell's nineteenth-century (classical) physics: the emission of radiation from hot bodies (black-body radiation), the photoelectric effect, the variation of heat capacity of solids with temperature, and the line spectra of atoms. As NCERT puts it, these phenomena indicate that a system can take up or radiate energy only in discrete amounts, not in a continuous manner.
A perfect emitter and absorber of radiation of all frequencies is called a black body; the radiation it emits is black-body radiation. A good practical approximation is a cavity with a tiny hole. Experiment showed that, at a given temperature, the intensity of emitted radiation rises with wavelength, peaks at a particular wavelength, then falls again — and as temperature increases, this maximum shifts towards shorter wavelengths. (This is why a heated iron rod glows dull red, then white, then bluish as it gets hotter.) Wave theory could not reproduce this curve.
In 1900, Max Planck supplied the answer. He assumed that the atoms in the walls of a black body behave as oscillators that can emit or absorb energy only in discrete chunks. He named the smallest such quantity a quantum, and proposed that the energy of a quantum of radiation is proportional to its frequency:
$$ E = h\nu $$
The proportionality constant $h$ is Planck's constant, with the value $h = 6.626 \times 10^{-34}\ \text{J s}$. Because energy is exchanged only in whole quanta, the allowed energies form a discrete set:
$$ E = 0,\ h\nu,\ 2h\nu,\ 3h\nu,\ \ldots,\ nh\nu,\ \ldots $$
Quantisation is like standing on a staircase. A person can stand on any step, but never in between two steps. The energy can take any one of the allowed values, but none of the values between them.
Planck's quantum and the line spectrum feed straight into the next step: see how Bohr turned these clues into fixed orbits in Bohr's Model of the Hydrogen Atom.
The Photoelectric Effect
In 1887, H. Hertz observed that when certain metals — potassium, rubidium, caesium and the like — are exposed to a beam of light, electrons are ejected from the surface. This is the photoelectric effect. The experimental facts were striking and could not be reconciled with classical physics.
| Observation | Classical prediction | Resolved by |
|---|---|---|
| Electrons ejected instantly, with no time lag | Energy should accumulate over time | One photon, one electron |
| Number of electrons ∝ intensity (brightness) | Consistent — more energy ejects more | More photons in a brighter beam |
| A threshold frequency ν₀ exists; below it, nothing happens | Any frequency should work if bright enough | Each photon needs energy ≥ work function |
| Kinetic energy of electrons rises with frequency, not brightness | KE should depend on intensity | $KE = h\nu - h\nu_0$ |
Einstein resolved the puzzle in 1905 by treating the beam as a stream of particles called photons, each carrying energy $E = h\nu$, building directly on Planck's idea. When a photon strikes an electron in the metal, it transfers all its energy instantaneously. A minimum energy, the work function $W_0 = h\nu_0$, is needed merely to free the electron; any surplus appears as kinetic energy:
$$ \tfrac{1}{2} m_e v^{2} = h\nu - h\nu_0 = h(\nu - \nu_0) $$
Here $m_e$ is the mass of the electron and $v$ its ejection velocity. If $\nu < \nu_0$, no electron is freed however intense the light — this is exactly why red light of any brightness fails to eject electrons from potassium (threshold $\nu_0 = 5.0 \times 10^{14}$ Hz), while even a faint yellow light succeeds.
The threshold frequency ν₀ for a metal is $7.0 \times 10^{14}\ \text{s}^{-1}$. Calculate the kinetic energy of an electron emitted when radiation of frequency $\nu = 1.0 \times 10^{15}\ \text{s}^{-1}$ strikes the metal.
By Einstein's photoelectric equation, $KE = h(\nu - \nu_0)$.
$$ KE = (6.626 \times 10^{-34}\ \text{J s})\,(10.0 \times 10^{14}\ \text{s}^{-1} - 7.0 \times 10^{14}\ \text{s}^{-1}) $$
$$ = (6.626 \times 10^{-34}\ \text{J s})\,(3.0 \times 10^{14}\ \text{s}^{-1}) = 1.988 \times 10^{-19}\ \text{J} $$
The ejected electron carries a kinetic energy of $1.988 \times 10^{-19}$ J.
Intensity changes the count, frequency changes the energy.
A brighter beam (higher intensity) carries more photons, so it ejects more electrons — but it does not raise the kinetic energy of any one electron. The kinetic energy depends only on the photon's frequency through $KE = h(\nu - \nu_0)$. Increasing intensity below the threshold frequency still ejects nothing.
More intensity ⇒ more electrons. Higher frequency ⇒ faster electrons.
Dual Behaviour of Radiation
The particle picture explained black-body radiation and the photoelectric effect, yet it clashed with the well-established wave behaviour that accounts for interference and diffraction. The only way out was to accept that light possesses both particle and wave properties — light has dual behaviour. Depending on the experiment, light reveals one face or the other: it shows particle-like properties whenever it interacts with matter, and wave-like properties (interference, diffraction) when it propagates.
This idea was alien to classical intuition and took time to gain acceptance. As we shall see later in the chapter, even microscopic particles such as electrons share this wave–particle duality — the foundation of the quantum mechanical view of the atom.
Atomic Spectra and Quantised Energy Levels
When a ray of white light passes through a prism, shorter-wavelength waves bend more than longer ones, so the light fans out into a series of coloured bands called a spectrum. Because white light contains every wavelength in the visible range, this spectrum is unbroken — violet merges into blue, blue into green, and so on. Such a spectrum is called a continuous spectrum, and a rainbow is a familiar example.
When radiation interacts with matter, atoms and molecules absorb energy and jump to a higher, unstable energy state. To return to their stable, lower-energy state, they re-emit radiation. The recorded spectrum of this emitted radiation is an emission spectrum, and the excited species are said to be "excited". An absorption spectrum is its photographic negative: a continuum of light is passed through a sample, which removes certain wavelengths, leaving dark lines in the bright background. The study of both is called spectroscopy.
| Feature | Continuous spectrum | Line (atomic) spectrum |
|---|---|---|
| Source | White light through a prism; a hot solid | Excited atoms in the gas phase |
| Appearance | Unbroken band of all wavelengths | Discrete bright (or dark) lines with gaps |
| What it tells us | Visible light spans 4.0–7.5 × 10¹⁴ Hz | Energy levels are quantised; identity of element |
| Uniqueness | Same for all incandescent sources | Unique to each element — a "fingerprint" |
Crucially, the emission spectra of gaseous atoms are not continuous: atoms emit light only at specific wavelengths, with dark gaps between them. These are line spectra or atomic spectra. Each element has a unique line emission spectrum, used in chemical analysis like a fingerprint. Elements such as rubidium, caesium, thallium, indium, gallium and scandium were discovered through spectroscopy, and helium was first detected in the Sun by this method.
Line Spectrum of Hydrogen
Of all elements, hydrogen has the simplest line spectrum. When an electric discharge passes through gaseous hydrogen, $\ce{H2}$ molecules dissociate and the energetically excited atoms emit radiation of discrete frequencies. In 1885 Balmer showed that the visible lines obey a simple wavenumber formula, and Johannes Rydberg later generalised it to cover every series in the hydrogen spectrum:
$$ \bar{\nu} = 109677 \left( \frac{1}{n_1^{2}} - \frac{1}{n_2^{2}} \right)\ \text{cm}^{-1} $$
where $n_1 = 1, 2, 3, \ldots$ and $n_2 = n_1 + 1, n_1 + 2, \ldots$. The constant $109{,}677\ \text{cm}^{-1}$ is the Rydberg constant for hydrogen. The first five series — corresponding to $n_1 = 1, 2, 3, 4, 5$ — are named after their discoverers, and only the Balmer series falls in the visible region.
| Series | n₁ | n₂ | Spectral region |
|---|---|---|---|
| Lyman | 1 | 2, 3, … | Ultraviolet |
| Balmer | 2 | 3, 4, … | Visible |
| Paschen | 3 | 4, 5, … | Infrared |
| Brackett | 4 | 5, 6, … | Infrared |
| Pfund | 5 | 6, 7, … | Infrared |
Two features are common to all line spectra: the spectrum of each element is unique, and it shows a striking regularity. These regularities cried out for explanation in terms of electronic structure — and it is precisely this regularity that Bohr's model was built to account for, by allowing the electron only certain quantised energy states.
Developments Leading to Bohr's Model
- Two developments underpin Bohr's model: the dual nature of EM radiation and the line spectra of atoms.
- EM radiation is characterised by $\nu$, $\lambda$ and $\bar{\nu}$, related by $c = \nu\lambda$ and $\bar{\nu} = 1/\lambda$; all radiation travels at $c = 3.0 \times 10^{8}$ m s⁻¹ in vacuum.
- Planck (1900): energy is exchanged in quanta, $E = h\nu$, with $h = 6.626 \times 10^{-34}$ J s — explaining black-body radiation.
- Photoelectric effect: a threshold frequency ν₀ exists; $KE = h(\nu - \nu_0)$, where the work function $W_0 = h\nu_0$. Intensity controls electron count, frequency controls electron energy.
- Light shows dual behaviour — wave (interference, diffraction) and particle (black-body, photoelectric).
- Atoms give line spectra, not continuous ones; the hydrogen spectrum follows the Rydberg formula with $R = 109677\ \text{cm}^{-1}$ and series Lyman → Pfund.