doPhysics · Modern Physics
The Bohr Atom and Spectra
Rutherford's nucleus and why a classical atom would collapse. Bohr's postulates, the radius, speed and energy of every orbit, and the ladder of levels. The hydrogen spectrum from one formula, and the small corrections that come from momentum: recoil, reduced mass and the threshold for exciting an atom by a collision.
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Why this chapter
Why this chapter exists
A glowing tube of hydrogen gives light of only four visible colours, always the same four. Why not every colour, and why those? The answer took two steps. Rutherford found that an atom is a tiny heavy nucleus with electrons far outside it. Bohr then gave the electrons a few new rules. Out came the size of the atom, its energy levels and every line of hydrogen, all from one number, 13.6 eV. This chapter builds that model and tests it against the spectrum, the sizes of ions and the collisions of atoms.
You will learn
By the end of this chapter you will
- Describe the alpha-scattering experiment and explain why it demands a tiny, heavy, positive nucleus.
- Find the distance of closest approach of an alpha particle, and explain why a classical atom would collapse.
- State Bohr's three postulates and derive his quantisation rule from the electron's standing wave.
- Derive the radius, speed, energy, period and current of any orbit of a hydrogen-like atom, with \(K = -E\) and \(U = 2E\).
- Read an energy level diagram for excitation and ionisation energies, and explain emission and absorption spectra.
- Use the Rydberg formula for the lines of every series of hydrogen and hydrogen-like ions, and count the lines from a level.
- Correct Bohr's results for the recoil of the atom and for the reduced mass, and find the threshold for exciting an atom by a collision.
Before you begin
Before you begin
From the chapters on circular motion, you should know that a body on a circle needs a centripetal force \(mv^2/r\). From the chapters on energy and momentum, you should know conservation of energy and momentum, and what happens when two bodies stick together. From the chapter on Photons and the Photoelectric Effect, you should know that a photon has energy \(h\nu = hc/\lambda\) and momentum \(h/\lambda\). The electron's de Broglie wavelength \(h/mv\) from the chapter on Matter Waves helps in lesson 2.
Lessons
1
Inside the Atom: Rutherford's Nucleus
Alpha particles fired at a thin gold foil mostly pass straight through, yet a few turn back. The only explanation is a tiny, heavy, positive nucleus. The same picture, run by the classical rules, would make every atom collapse.
Alpha scattering and the nucleusDistance of closest approachWhy a classical atom would collapse
2
Bohr's Postulates
Bohr kept Rutherford's nucleus and added three rules. The electron circles without radiating on certain orbits, its angular momentum comes in steps of \(h/2\pi\), and it gives out one photon only when it jumps. The electron's own wave explains the second rule.
Bohrs three postulatesAn orbit holds whole waves
3
Radius, Speed and Energy of an Orbit
Two equations, the centripetal force and Bohr's rule, give every orbit of a one-electron atom. The radius grows as \(n^2/Z\), the speed falls as \(Z/n\), and the energy is \(-13.6\,Z^2/n^2\,\text{eV}\), with \(K = -E\) and \(U = 2E\). The period grows as \(n^3\), and the circling electron is a tiny current.
Orbit radius and speedKinetic potential and total energyPeriod and orbital current
4
Energy Levels, Excitation and Ionisation
The orbit energies form a ladder that crowds towards zero. Lifting the electron up the ladder costs an excitation energy; freeing it costs the ionisation energy, \(13.6\,Z^2\,\text{eV}\). An atom gives out or takes in only photons that match a gap of its ladder.
The energy level ladderExcitation and ionisation energyEmission and absorption
5
The Hydrogen Spectrum
Bohr's energies give the Rydberg formula for every line of hydrogen. The lines fall into series named by the lower level, Lyman in the ultraviolet, Balmer in the visible, Paschen and beyond in the infrared, each crowding to a limit. Atoms excited to level \(n\) give up to \(n(n - 1)/2\) lines.
The rydberg formulaThe spectral seriesCounting spectral lines
6
Recoil, Reduced Mass and Atomic Collisions
Three refinements that come from momentum. An atom recoils when it emits a photon, so the photon falls just short of the gap. The nucleus is not fixed, so the electron's mass is replaced by the reduced mass. And a particle striking an atom must bring more than the gap, because the pair cannot stop dead.
Recoil of the emitting atomReduced mass correctionThreshold for exciting an atom by collision
Read in order. The small labels are the topics of each lesson. They take your colour as you answer the checks.
0%The Workbook
Questions to solve on paper, marked by you.
Glossary
The key words of this chapter, with the meaning this chapter gives them.
- absorption spectrum
- The dark lines left in a continuous spectrum after white light passes through a cool gas. Each line is an upward jump that its atoms can make.
- alpha particle
- The nucleus of a helium atom, two protons and two neutrons, with charge \(+2e\). Some radioactive atoms emit alpha particles with a few MeV of kinetic energy.
- binding energy
- The energy needed to free a bound electron from its level and leave it at rest far away. For level \(n\) it is \(|E_n| = 13.6\,Z^2/n^2\) eV.
- Bohr radius
- The radius of the lowest orbit of hydrogen in Bohr's model, \(a_0 = 0.529\,\text{Å} = 0.529 \times 10^{-10}\,\text{m}\).
- de Broglie wavelength
- The wavelength \(\lambda = h/mv\) of the wave that goes with a particle of mass \(m\) moving at speed \(v\).
- distance of closest approach
- The distance from the centre of a nucleus at which a particle aimed straight at it stops and turns back. For an alpha particle of kinetic energy \(K\), \(d = 2Zke^2/K\); it is an upper limit on the size of the nucleus.
- emission spectrum
- The set of bright lines given out by a hot or excited gas of an element, one line for each downward jump between its levels.
- energy level
- One of the allowed energies of an atom, drawn as a horizontal line in an energy level diagram. For a hydrogen-like atom \(E_n = -13.6\,Z^2/n^2\) eV.
- excitation energy
- The energy needed to lift an atom from its ground state to a higher level, \(E_n - E_1\). For hydrogen it is \(10.2\,\text{eV}\) to \(n = 2\) and \(12.09\,\text{eV}\) to \(n = 3\).
- excited state
- Any energy level above the ground state. An atom in an excited state usually returns to a lower level within about \(10^{-8}\,\text{s}\).
- ground state
- The lowest energy level of an atom, \(n = 1\). An atom left alone settles in it.
- hydrogen-like atom
- An atom or ion with a nucleus of charge \(Ze\) and only one electron, such as hydrogen, \(\text{He}^+\) and \(\text{Li}^{2+}\). Bohr's model applies to it.
- impact parameter
- The distance by which a moving particle would miss the centre of a nucleus if it were not pushed at all. The smaller it is, the more the particle is turned.
- ionisation energy
- The energy needed to free the electron from the ground state, \(13.6\,Z^2\) eV for a hydrogen-like atom: \(13.6\,\text{eV}\) for hydrogen.
- ionisation potential
- The potential difference through which an electron must be accelerated to gain the ionisation energy, \(13.6\,\text{V}\) for hydrogen.
- nucleus
- The tiny central part of an atom that holds all its positive charge and nearly all its mass. It is about ten thousand times smaller than the atom.
- principal quantum number
- The whole number \(n = 1, 2, 3, \ldots\) that labels a Bohr orbit. It fixes the angular momentum \(nh/2\pi\), and it counts the de Broglie wavelengths round the orbit.
- reduced mass
- The mass \(\mu = mM/(m + M)\) that replaces the electron's mass \(m\) when the nucleus of mass \(M\) is allowed to move. It is slightly less than \(m\) for an ordinary atom.
- Rydberg constant
- The constant \(R = 13.6\,\text{eV}/hc = 1.097 \times 10^7\,\text{m}^{-1}\) of the Rydberg formula for the lines of hydrogen. Its inverse is \(91.2\,\text{nm}\).
- series limit
- The shortest wavelength of a spectral series, from a fall from \(n = \infty\) to its lower level, \(\lambda = n_f^2/R\) for hydrogen.
- spectral series
- All the lines of an atom that end on the same lower level. Hydrogen has the Lyman (\(n_f = 1\)), Balmer (2), Paschen (3), Brackett (4) and Pfund (5) series.
- stationary orbit
- An orbit on which, by Bohr's first postulate, the electron keeps a fixed energy and does not radiate, although it moves and accelerates.
- threshold energy
- The least kinetic energy with which a particle striking an atom at rest can excite it, \(\Delta E\,(m + M)/M\). Below it the collision must be elastic.