2.2 Light, Quantisation and Bohr's Model
Light as a wave and as a stream of photons: Planck's quanta, the photoelectric effect, the line spectrum of hydrogen and Bohr's model of an atom whose energy comes in steps.
A firework bursts red, green or blue according to the metal salts packed inside it: strontium gives crimson, barium green, copper blue-green, sodium a fierce yellow. Each colour is a message from the atoms themselves — and Section 2.1 left us with an atom that could not send it. Rutherford's nuclear atom should collapse in a hundred-millionth of a second, and while collapsing it should glow with every colour at once, not with a few sharp lines.
The way out came from studying light. Around 1900 it became clear that light, long understood as a wave, also behaves as a stream of particles whose energy is fixed by their frequency. In 1913 Niels Bohr applied the same idea to the electron in a hydrogen atom: its energy, too, can take only certain values. This section develops both ideas — the wave and particle nature of radiation, the photoelectric effect, the hydrogen spectrum and Bohr's model — and shows how one formula predicts the wavelength of every line hydrogen can emit.
The Wave Nature of Electromagnetic Radiation
In the 1870s James Clerk Maxwell showed that light is an electromagnetic wave: an electric field and a magnetic field oscillating at right angles to each other and to the direction of travel. Unlike sound, these waves need no medium; in a vacuum all of them travel at the same speed, $c=3.0\times10^{8}\ \mathrm{m\,s^{-1}}$. Visible light is a tiny part of a vast electromagnetic spectrum that runs from radio waves (wavelengths of metres to kilometres) to gamma rays (shorter than $10^{-12}\ \mathrm{m}$).
The wavelength $\lambda$ is the distance between two successive crests of a wave (unit $\mathrm{m}$, or $\mathrm{nm}$ for light). The frequency $\nu$ is the number of waves passing a given point per second (unit hertz, $1\ \mathrm{Hz}=1\ \mathrm{s^{-1}}$). The wavenumber $\bar{\nu}$ is the number of wavelengths per unit length (unit $\mathrm{m^{-1}}$, or more commonly $\mathrm{cm^{-1}}$). For any electromagnetic wave in a vacuum,
$$c=\nu\lambda,\qquad\bar{\nu}=\frac{1}{\lambda}=\frac{\nu}{c}.$$The Particle Nature of Radiation: Planck and the Photoelectric Effect
The wave picture explains interference and diffraction perfectly. But three observations made around 1900 refused to fit it: the radiation given out by hot bodies, the ejection of electrons from metals by light (the photoelectric effect), and the variation of the heat capacity of solids with temperature. Line spectra (the next subsection) were a fourth puzzle.
Black-body radiation
An ideal body that absorbs all the radiation falling on it, and emits radiation of every frequency when heated, is called a black body (a small hole in a hot, hollow furnace behaves very like one). The radiation it emits depends only on its temperature. As an iron rod is heated it glows dull red, then orange, then white: the intensity of the emitted radiation rises steeply and the wavelength of maximum intensity shifts to shorter wavelengths. Classical wave theory predicted that the intensity should keep growing at short wavelengths — without limit — which is plainly false.
In 1900 Max Planck found that he could reproduce these curves exactly, but only by making a revolutionary assumption: the atoms of the hot body can give out or take in energy not continuously but only in small, fixed packets.
Atoms and molecules emit or absorb energy not continuously but in discrete amounts called quanta (singular: quantum). The energy of one quantum of radiation is proportional to its frequency:
$$E=h\nu=\frac{hc}{\lambda},\qquad h=6.626\times10^{-34}\ \mathrm{J\,s}\ \ (\text{Planck's constant}).$$The energy emitted or absorbed can only be a whole number of quanta, $nh\nu$ ($n=1,2,3,\ldots$). A quantum of light is called a photon.
The photoelectric effect
In 1887 Heinrich Hertz noticed that sparks jump more easily when ultraviolet light falls on the electrodes. The cause turned out to be electrons ejected from a metal surface by light — the photoelectric effect. Experiments with clean metal surfaces in a vacuum (alkali metals such as potassium, rubidium and caesium respond even to visible light) showed four features:
- electrons are ejected the instant the light strikes the surface — there is no measurable time lag, even in dim light;
- the number of electrons ejected per second is proportional to the intensity (brightness) of the light;
- for each metal there is a minimum frequency, the threshold frequency $\nu_{0}$, below which no electrons are ejected at all, however intense the light and however long it shines;
- above $\nu_{0}$, the kinetic energy of the ejected electrons increases with the frequency of the light but does not depend on its intensity.
On the wave picture, brighter light carries more energy and should eject faster electrons, at any frequency, after enough time to "build up" energy. Features 1, 3 and 4 flatly contradict this. In 1905 Albert Einstein explained all four using Planck's idea: light is a stream of photons, and each electron is ejected by one photon.
Dual behaviour of electromagnetic radiation
So is light a wave or a stream of particles? It is both — or rather, it is something that shows wave-like behaviour in some experiments (interference, diffraction) and particle-like behaviour in others (black-body radiation, the photoelectric effect). This dual behaviour was hard to accept, but every later experiment has confirmed it. In Section 2.3 we shall see that electrons, which we think of as particles, have the same dual nature.
Atomic Spectra and the Spectrum of Hydrogen
When white light passes through a prism, waves of different wavelength are bent by different amounts and spread into a band of colours, violet to red, with no gaps: a continuous spectrum. The rainbow is a natural example. Atoms behave very differently.
When atoms are given energy (by heating, or by an electric discharge through the gas) they emit radiation; the spectrum of this radiation is an emission spectrum. If white light is passed through a sample and the transmitted light is analysed, the wavelengths the sample has absorbed are missing; this is an absorption spectrum — dark lines on a bright, continuous background, a photographic negative of the emission spectrum.
Gaseous atoms give line spectra (atomic spectra): only certain sharp wavelengths, separated by dark gaps. The lines in the emission and absorption spectra of an element are at exactly the same wavelengths, and they are characteristic of the element.
Because each element has its own pattern of lines, spectroscopy identifies elements as surely as fingerprints identify people. Robert Bunsen and Gustav Kirchhoff discovered caesium and rubidium (1860–61) from new lines in their spectra; thallium, indium and gallium followed, and helium was found in the spectrum of the Sun before it was isolated on Earth.
The line spectrum of hydrogen
Hydrogen, the simplest atom, has the simplest spectrum. In 1885 the Swiss schoolteacher Johann Balmer found that the four visible lines of hydrogen ($656.3$, $486.1$, $434.0$ and $410.2\ \mathrm{nm}$) obey a simple formula. Johannes Rydberg then showed that all the series of lines of hydrogen, in the ultraviolet and infrared as well, fit one expression.
Every line in the spectrum of the hydrogen atom has a wavenumber given by
$$\bar{\nu}=\frac{1}{\lambda}=R_{H}\left(\frac{1}{n_{1}^{2}}-\frac{1}{n_{2}^{2}}\right),\qquad R_{H}=109\,677\ \mathrm{cm^{-1}},$$where $n_{1}=1,2,3,\ldots$ and $n_{2}=n_{1}+1,\,n_{1}+2,\ldots$ are whole numbers. Each value of $n_{1}$ gives one series of lines:
| Series | $n_{1}$ | $n_{2}$ | Region | Wavelength range |
|---|---|---|---|---|
| Lyman | 1 | 2, 3, 4, … | ultraviolet | $91$–$122\ \mathrm{nm}$ |
| Balmer | 2 | 3, 4, 5, … | visible | $365$–$656\ \mathrm{nm}$ |
| Paschen | 3 | 4, 5, 6, … | infrared | $821$–$1876\ \mathrm{nm}$ |
| Brackett | 4 | 5, 6, 7, … | infrared | $1459$–$4052\ \mathrm{nm}$ |
| Pfund | 5 | 6, 7, 8, … | infrared | $2279$–$7460\ \mathrm{nm}$ |
The line with $n_{2}=n_{1}+1$ has the longest wavelength in a series; as $n_{2}\to\infty$ the lines crowd together towards the series limit, $\bar{\nu}=R_{H}/n_{1}^{2}$ (the shortest wavelength).
Bohr's Model of the Hydrogen Atom
Why should hydrogen obey such a simple rule? In 1913 Niels Bohr combined Rutherford's nucleus with Planck's quantum idea. He simply postulated that the classical rule — an orbiting charge must radiate — does not apply to certain special orbits.
- The electron moves round the nucleus in circular paths of fixed radius and fixed energy, called orbits, stationary states or allowed energy states. While it stays in one orbit its energy does not change and it emits no radiation.
- The energy of the electron changes only when it moves from one stationary state to another: it absorbs energy to move to a higher state and emits energy when it falls to a lower one. The frequency of the radiation absorbed or emitted is given by (Bohr's frequency rule) $$\nu=\frac{\Delta E}{h}=\frac{E_{2}-E_{1}}{h}.$$
- Only orbits in which the angular momentum of the electron is a whole-number multiple of $h/2\pi$ are allowed: $$m_{e}vr=n\,\frac{h}{2\pi},\qquad n=1,2,3,\ldots$$
Applying ordinary mechanics and electrostatics to an electron in such an orbit (a calculation you will meet in Class 12 Physics) gives the radius and energy of each orbit.
For hydrogen ($Z=1$) and for hydrogen-like species, which have a nucleus of charge $+Ze$ and only one electron ($\mathrm{He^{+}}$, $\mathrm{Li^{2+}}$, $\mathrm{Be^{3+}}$, …), the $n$th orbit has
$$r_{n}=52.9\,\frac{n^{2}}{Z}\ \mathrm{pm},\qquad E_{n}=-2.18\times10^{-18}\,\frac{Z^{2}}{n^{2}}\ \mathrm{J}\quad(n=1,2,3,\ldots).$$$n$ is the principal quantum number. The radius $a_{0}=52.9\ \mathrm{pm}$ of the first hydrogen orbit is called the Bohr radius. The lowest-energy state ($n=1$) is the ground state; states with $n\gt1$ are excited states.
The zero of energy is chosen as the electron at rest and completely separated from the nucleus ($n=\infty$, $r=\infty$). A bound electron has less energy than this, so $E_{n}$ is negative. As $n$ increases, $E_{n}$ becomes less negative (higher) and the levels crowd together towards $0$. The energy needed to remove the electron from the ground state of hydrogen — the ionisation energy — is $0-E_{1}=2.18\times10^{-18}\ \mathrm{J}$ per atom, or $2.18\times10^{-18}\times6.022\times10^{23}=1.31\times10^{6}\ \mathrm{J\,mol^{-1}}$ ($1.31\times10^{3}\ \mathrm{kJ\,mol^{-1}}$).
Students often write $E_{n}=+2.18\times10^{-18}/n^{2}$ and get the direction of a transition wrong. Always check: a fall to a lower $n$ releases energy (emission), a rise to a higher $n$ needs energy (absorption).
How Bohr's model explains the hydrogen spectrum
When the electron falls from an outer orbit $n_{i}$ to an inner orbit $n_{f}$, the energy released is carried off by one photon:
$$\Delta E=E_{i}-E_{f}=2.18\times10^{-18}\left(\frac{1}{n_{f}^{2}}-\frac{1}{n_{i}^{2}}\right)\ \mathrm{J},\qquad\bar{\nu}=\frac{\Delta E}{hc}=\frac{2.18\times10^{-18}}{(6.626\times10^{-34})(3.0\times10^{10})}\left(\frac{1}{n_{f}^{2}}-\frac{1}{n_{i}^{2}}\right)\ \mathrm{cm^{-1}}.$$The constant works out to $1.097\times10^{5}\ \mathrm{cm^{-1}}$ — the Rydberg constant $R_{H}$. Bohr's model therefore derives the Rydberg formula: $n_{1}$ is the orbit the electron lands in, $n_{2}$ the orbit it falls from. All Lyman lines end in $n=1$, all Balmer lines in $n=2$, all Paschen lines in $n=3$. A sample of hydrogen contains vast numbers of atoms in different excited states, so all these transitions happen at once and all the lines appear together. An electron in level $n$ can reach the ground state by many routes, so the number of different lines that atoms excited to level $n$ can produce is $\dfrac{n(n-1)}{2}$.
Bohr's model was a triumph for hydrogen and hydrogen-like ions, but it
- fails to explain the spectra of atoms with more than one electron, even helium;
- cannot explain the fine structure of hydrogen's lines (each line, seen with a high-resolution spectroscope, is a close group of lines), nor the splitting of lines in a magnetic field (Zeeman effect) or an electric field (Stark effect);
- cannot explain how atoms combine to form molecules — chemical bonding and molecular shapes;
- treats the electron as a particle moving in a definite orbit with a definite position and speed, which contradicts the dual nature of matter and the uncertainty principle (Section 2.3).
Exercises
- 1 Find the wavelength, in nm, of radiation of frequency $6.0\times10^{14}\ \mathrm{Hz}$.
- 2 Find the frequency and the wavenumber (in $\mathrm{cm^{-1}}$) of the yellow light of wavelength $589\ \mathrm{nm}$.
- 3 A microwave oven works at $2.45\ \mathrm{GHz}$. Find the wavelength of the microwaves in cm.
- 4 Arrange in order of increasing frequency: X-rays, radio waves, visible light, ultraviolet, infrared, microwaves.
- 5 Find the energy of one photon of radiation of wavenumber $2.0\times10^{4}\ \mathrm{cm^{-1}}$.
- 6 Find the energy of one mole of photons of ultraviolet light of wavelength $250\ \mathrm{nm}$, in $\mathrm{kJ\,mol^{-1}}$. A $\mathrm{C{-}C}$ bond needs about $348\ \mathrm{kJ\,mol^{-1}}$ to break. Comment.
- 7 A green laser pointer emits $5.0\ \mathrm{mW}$ at $532\ \mathrm{nm}$. How many photons does it emit per second?
- 8 By what factor is the energy of a $400\ \mathrm{nm}$ photon greater than that of a $700\ \mathrm{nm}$ photon?
- 9 Applied. Sodium-vapour street lamps emit mainly $589\ \mathrm{nm}$ light. Find the energy of one mole of these photons, and the wavelength of a photon of energy $3.00\ \mathrm{eV}$.
- 1 An iron rod glows dull red, then orange, then white as it is heated. Explain this in terms of black-body radiation.
- 2 State two observations about the photoelectric effect that the wave theory of light cannot explain, and explain each using photons.
- 3 The work function of caesium is $2.14\ \mathrm{eV}$. Find its threshold frequency and threshold wavelength.
- 4 Light of wavelength $300\ \mathrm{nm}$ falls on caesium. Find the maximum kinetic energy of the ejected electrons in eV and in J.
- 5 Find the maximum speed of the electrons in part 4.
- 6 Electrons ejected from a metal by light of wavelength $250\ \mathrm{nm}$ have a maximum kinetic energy of $3.0\times10^{-19}\ \mathrm{J}$. Find the work function and the threshold wavelength of the metal.
- 7 Light above the threshold frequency falls on a metal. What happens to (a) the number and (b) the maximum kinetic energy of the ejected electrons if the intensity is doubled? (c) Show that doubling the frequency more than doubles the maximum kinetic energy.
- 8 Sketch, on one set of axes, the graph of maximum kinetic energy against frequency for sodium ($W_{0}=2.75\ \mathrm{eV}$) and zinc ($W_{0}=4.33\ \mathrm{eV}$). What is the same about the two lines, and what is different?
- 1 Distinguish between (a) a continuous spectrum and a line spectrum; (b) an emission spectrum and an absorption spectrum.
- 2 Find the wavelength of the second line of the Balmer series ($n_{2}=4$).
- 3 Find the wavenumber and wavelength of the first line of the Lyman series.
- 4 Find the shortest wavelength in the Balmer series.
- 5 Which transition in the hydrogen atom produces the infrared line at $1282\ \mathrm{nm}$? Name the series.
- 6 Calculate the energy of the electron in the $n=3$ orbit of a hydrogen atom.
- 7 How much energy is needed to excite a hydrogen atom from $n=1$ to $n=3$? What wavelength of light would do this?
- 8 Find the radius of the fourth orbit of hydrogen. Which orbit of $\mathrm{Be^{3+}}$ has the same radius as the first orbit of hydrogen?
- 9 Calculate the ionisation energy of $\mathrm{Li^{2+}}$ in its ground state, per ion and per mole.
- 10 Electrons in a sample of hydrogen atoms have been excited to $n=6$. How many different spectral lines can be emitted as they return to the ground state? How many of these belong to the Balmer series?
- 11 Find the ratio of the wavelength of the first Lyman line to that of the first Balmer line.
- 1 Which transition in $\mathrm{He^{+}}$ produces a line of the same wavelength as the first Balmer line of hydrogen?
- 2 In electronvolts, the ground-state energy of hydrogen is $-13.6\ \mathrm{eV}$. An electron in a hydrogen atom has energy $-1.51\ \mathrm{eV}$. Which orbit is it in, and how much energy is needed to ionise the atom from there?
- 3 A hydrogen atom in its ground state absorbs a photon of wavelength $97.2\ \mathrm{nm}$. To which level is the electron raised? How many different lines could then be emitted as the atom returns to the ground state?
- 4 In a photoelectric experiment, light of frequency $7.50\times10^{14}\ \mathrm{Hz}$ ejects electrons with a maximum kinetic energy of $1.10\ \mathrm{eV}$, and light of $1.00\times10^{15}\ \mathrm{Hz}$ gives $2.14\ \mathrm{eV}$. From these two results alone find Planck's constant and the work function of the metal.
E1. JEE Main The transition in $\mathrm{He^{+}}$ that emits light of the same wavelength as the $n=4\to n=2$ line of hydrogen is (A) $n=2\to n=1$ (B) $n=8\to n=4$ (C) $n=4\to n=2$ (D) $n=6\to n=3$
E2. NEET A metal surface has a threshold wavelength of $500\ \mathrm{nm}$. Light of wavelength $250\ \mathrm{nm}$ falls on it. The maximum kinetic energy of the photoelectrons is ($hc=1240\ \mathrm{eV\,nm}$) (A) $1.24\ \mathrm{eV}$ (B) $2.48\ \mathrm{eV}$ (C) $4.96\ \mathrm{eV}$ (D) zero
E3. Board · 3 marks State any two postulates of Bohr's model of the hydrogen atom. Calculate the frequency and wavelength of the radiation emitted when the electron in a hydrogen atom falls from $n=2$ to $n=1$ ($E_{n}=-2.18\times10^{-18}/n^{2}\ \mathrm{J}$).
E4. JEE Main Numerical answer. The ratio of the radius of the third Bohr orbit of hydrogen to the radius of the first Bohr orbit of $\mathrm{Li^{2+}}$ is $x$. Find $x$.
E5. JEE Advanced One or more options correct. In photoelectric experiments on different metals with light above the threshold frequency, (A) the graph of maximum kinetic energy against frequency has the same slope for every metal; (B) the number of electrons ejected per second is proportional to the intensity of the light; (C) the maximum kinetic energy of the electrons increases with the intensity of the light; (D) the intercept of the graph on the frequency axis is the same for every metal.
E6. Board · Case study Astronomers analysing the light of a hot star find dark lines at $656$, $486$, $434$ and $410\ \mathrm{nm}$, which they identify as absorption by hydrogen atoms in the star's outer layers. (i) Using $R_{H}=109\,677\ \mathrm{cm^{-1}}$, show that the line at $410\ \mathrm{nm}$ corresponds to the transition between $n=2$ and $n=6$. (2 marks) (ii) Name the series to which all four lines belong. (1 mark) (iii) Why do these lines appear dark rather than bright? (1 mark)
Research question. Why can a silicon solar panel never turn more than about a third of sunlight into electricity — and how are new "tandem" cells beating that limit?
Why it matters. Solar power is now the cheapest new electricity in much of the world, from Rajasthan to Chile to Australia. A solar cell works on the same principle as the photoelectric effect: a photon must deliver at least a threshold energy (the material's band gap) to free an electron. Photons with less energy are wasted; photons with more energy give up the excess as heat.
What to do.
- The band gap is $1.12\ \mathrm{eV}$ for silicon, $1.42\ \mathrm{eV}$ for gallium arsenide and about $1.60\ \mathrm{eV}$ for a typical lead-halide perovskite. Find the threshold (longest usable) wavelength for each.
- A $500\ \mathrm{nm}$ photon is absorbed by silicon. What is its energy in eV, and what fraction of it can at most be delivered as electrical energy (one band gap)?
- Bright sunlight delivers about $1000\ \mathrm{W\,m^{-2}}$. Treating it all as $600\ \mathrm{nm}$ light, estimate how many photons strike one square metre of panel per second.
- Use your answers to explain why a cell with a wide band gap wastes the infrared, while one with a narrow band gap wastes the energy of blue photons. Then find out what a tandem (perovskite-on-silicon) cell is and the latest certified efficiency record for one (for example from the US National Renewable Energy Laboratory's efficiency chart).
Tools: Calculator; Interactive 1 (set the work function as a stand-in for the band gap and watch what happens to the excess energy).
Hand in: A table of thresholds, your photon-count estimate and a short paragraph (about 150 words) explaining the efficiency limit of single-junction cells in terms of photons.
Going further. Night-vision cameras and the infrared thermometers used at airports detect long-wavelength photons. What property must their detector materials have, by the same reasoning?
🧪 Self-check quiz
Q1. Which of these has the greatest energy per photon?
Q2. Light above the threshold frequency falls on a metal. If its intensity is increased, the
Q3. All lines of the Balmer series of hydrogen correspond to transitions ending in
Q4. The energy of the electron in the second orbit of a hydrogen atom is
Q5. The radius of the second Bohr orbit of $\mathrm{He^{+}}$ is