2. Quantum Theory and Electronic Structure

Brightness is the number of photons. An oven sends an enormous number of them, and food's water molecules absorb them by rotating faster, which is heat. One microwave photon cannot break a bond, but a trillion trillion of them can warm a cup of tea. A single ultraviolet photon can break a bond.

Wavelength is the distance from one crest to the next. Frequency is how many crests pass a point each second. Speed is how fast the crests move: speed = wavelength × frequency. All light moves at c in a vacuum, so a shorter wavelength can only mean a higher frequency.

The wavelengths of light run from smaller than an atomic nucleus to longer than a football field. A scale that jumps by a factor of 1000 at each mark is the only way to fit them all on one line. Each mark on it is 1000 times the one before.

1 eV = 1.602 × 10⁻¹⁹ J. To go from the energy of one photon to the energy of a mole of them, multiply by Avogadro's number (6.022 × 10²³). 1 eV per photon is 96.485 kJ/mol. Chemists quote bond energies in kJ/mol and physicists quote photon energies in eV.

A wave spreads its energy over its whole front, and a brighter wave has more energy in every spot. Give an electron time and it should soak up enough to escape, whatever the colour. And a brighter wave should send electrons out faster. The measurements show neither: no electrons below a threshold frequency, at once above it, and speed set by colour alone.

The least energy that frees an electron from the metal's surface. The electrons are held in by the attraction of the positive ions, and each metal holds them with a different strength. Caesium holds them loosely (1.95 eV) and lead tightly (4.25 eV). Real values depend on how clean the surface is.

Raising the collector's voltage against the electrons makes each one climb a potential hill of height eV. An electron with kinetic energy less than eV turns back. At V_s, even the fastest, with KE_max, just fails to arrive, so eV_s = KE_max.

In this experiment, light is absorbed in lumps of energy hf, one lump per electron. It is the same constant, h, that Planck needed to fit the furnace in 1900. Two quite different experiments give the same value. That agreement convinced physicists the packets are real. (Light also shows wave behaviour in other experiments: later in this unit.)

The gas sits at low pressure in a glass tube. A high voltage drives electrons through it, and they collide with the atoms. A collision gives an atom energy, and the atom gives it back as light. In a hot solid the atoms are packed together and disturb each other, which smears the light into a smooth rainbow.

A prism, or better a diffraction grating, bends each wavelength by a different amount, so the colours spread out along a line. The scale under the picture is the wavelength of each position.

Yes. No two elements have the same set of lines, so a spectrum names the elements in a flame, a lamp or a star. Helium was found in the Sun's spectrum in 1868, a quarter of a century before anyone found it on Earth.

Helium, mercury and sodium are from the NIST Handbook of Basic Atomic Spectroscopic Data (wavelengths in air). Hydrogen's are computed from the Rydberg formula, which agrees with the measured values.

Bohr did not know. He assumed it, because it fit the spectrum. The reason comes from the electron behaving like a wave, which only fits around the atom for certain energies. That is the next experiment, and it leads to orbitals.

Zero is chosen as the energy of an electron that has just escaped the atom. An electron still bound to the nucleus has less energy than that, so its energy is negative. Level 1 is the most negative (−13.6 eV) and is the most tightly bound. It takes 13.6 eV to pull the electron free from there: the ionization energy of hydrogen.

No. It is on one level or another. When it changes level it does so in one jump, and the energy goes into (or comes out as) a single photon of exactly the energy difference. That is why only certain colours come out.

No. The circles are a picture. The energy levels are real, and Bohr's formula for them is right for hydrogen. The idea of a circular track was replaced by orbitals, regions where the electron is likely to be found. Orbitals are coming up.

Bohr's model gets hydrogen right, and one-electron ions like He⁺, but fails for atoms with two or more electrons, because the electrons repel each other. For those we need orbitals, subshells and the rules for filling them: that is electron configuration, and it is what the periodic table is built on.

A bullet bounces off in a smooth spread of directions. A wave scattered by many regularly spaced atoms goes in only where the waves from neighbouring rows arrive in step, and cancels everywhere else. That gives sharp peaks at angles set by the wavelength and the spacing: n λ = D sin θ. They found a peak at the angle the formula gave for the wavelength h/p. (They found it by accident in 1925, when a nickel target they were studying was heated and became a crystal. G. P. Thomson, J. J. Thomson's son, found the same effect with thin films in 1927. They shared the 1937 Nobel Prize in Physics.)

Both descriptions are needed. Each electron arrives at the detector as a single point, like a particle. But where the electrons arrive follows the wave pattern. The wave tells you the probability of finding the electron at each place. This idea is the heart of quantum mechanics, and it is how orbitals are described later in this unit.

Wavelength = h / (mass × speed). For a 0.145 kg baseball at 40 m/s that is 1.1 × 10⁻³⁴ m, about 10¹⁹ times smaller than a nucleus, so there is nothing for it to diffract around. An electron is about 10²⁹ times lighter than a baseball, so its wavelength is large enough to fit between the atoms of a crystal.

D is the spacing between the rows of atoms on the surface of the nickel crystal, which acts as the diffraction grating. For the crystal face Davisson and Germer used it is 0.215 nm. The wave needs a wavelength about as large as D to be diffracted noticeably, which is why electrons of a few tens of volts work.

The electron goes round and round. If one trip ends out of step with where it began, the second trip overlaps the first with a shift, the third with another shift, and so on. Crests meet troughs and the wave cancels itself out. Only if the wave repeats exactly after one trip can the pattern survive, and that needs a whole number of wavelengths round the circle. A guitar string works the same way: it must fit a whole number of half-wavelengths between its ends.

Yes, for hydrogen. Combine the classical balance of forces (which gives the speed on a circular orbit), de Broglie's wavelength λ = h/mv, and the condition that a whole number of wavelengths fit. The radius comes out as r = n² × 0.0529 nm and the energy as −13.6 eV / n². The model was a stepping stone: the real picture, quantum mechanics, treats the electron as a wave spread through three-dimensional space, not a ring.

It is a combination of constants: a₀ = 4πε₀ħ² / (m e²), where m is the electron's mass, e its charge, ħ = h / 2π and ε₀ the permittivity of free space. It is the radius of the smallest standing-wave orbit.

The wave can fit round the nucleus in more patterns than a ring: with nodes, lobes and different shapes. Each allowed pattern is an orbital, labelled by a set of whole numbers, the quantum numbers. Electrons also repel each other, which splits levels of the same n into subshells. All of that is next.

No. That was Heisenberg's first way of describing it, but the deeper reason is that a particle is a wave. A wave that is spread out can have one wavelength, so one momentum, but no precise position. A wave squeezed into a small space needs many wavelengths added together, so it has no precise momentum. It is a property of the waves, however good the measurement.

It means the electron cannot have a definite path, because a path needs an exact position and speed at every moment. What we can know is where the electron is likely to be found, and that is what an orbital shows. Bohr's energy levels are right, but his circular orbits are only a picture.

Hold it in a region the size of a nucleus (5 × 10⁻¹⁵ m) and its speed would be uncertain by more than the speed of light, which is impossible. In other words it would need a huge amount of energy to stay confined. That is why electrons form a cloud around the nucleus, and not in it.

Because the uncertainty Δv = ħ / (2mΔx) is divided by the mass. For a 145 g baseball located to a millimetre it is about 4 × 10⁻³¹ m/s, which is far smaller than anything that can be measured, or that matters. The principle is always true, but it only shows up for the very small, such as electrons.

A slice through the orbital, with the nucleus in the middle. The brightness is ψ², the chance of finding the electron at each point. It is a cloud, not a path: the electron is not going round on a track. The colour is the sign (positive or negative) of the wave. For one orbital on its own the sign has no effect, but it matters when orbitals overlap to form bonds, which you will meet in the unit on bonding.

No. They are two views of the same wave. The picture shows the chance of finding the electron at each point in space (ψ² at that point), so it has the orbital's shape. The graph adds up all the directions and asks a simpler question: how likely is the electron to be at a given distance from the nucleus, anywhere on the sphere of that radius? It is P(r) = r²R(r)², and it has no shape in space.

The two can disagree. The 1s picture is brightest right at the nucleus, but the graph peaks at one Bohr radius (0.0529 nm). That is because there is a lot more room, a much bigger sphere, at a larger radius, even though the chance per point is lower. So the dip to zero at the nucleus in the graph is not an empty centre: it is just a tiny sphere.

A place where the wave is zero, so the electron is never found there. A radial node is a spherical shell. An angular node is a flat plane or a cone through the nucleus, which is why a p orbital has two lobes with a plane of zero between them. Each extra node costs energy, like the extra crest in a standing wave.

In hydrogen the energy depends only on n, so 2s and 2p have exactly the same energy: −3.40 eV. In an atom with more than one electron, the other electrons shield the nucleus and the subshells split: s is lower than p, and p lower than d. That splitting decides the order in which orbitals fill, which is electron configuration, the next unit.

Spin does not come from the wave pattern. It is a property of the electron itself, found in 1922 when a beam of silver atoms was split in two by a magnetic field (the Stern–Gerlach experiment). An electron has just two spin states, m_s = +½ or −½. That is why each orbital holds two electrons, and it leads to the Pauli exclusion principle, also in the next unit.

It is a model. Each filled subshell is drawn as a hydrogen-like orbital, with the nuclear charge replaced by the effective charge from Slater's rules, and averaged over direction. That is good enough to show the shells, how close the inner ones are, and the trends. It is not exact: a real atom needs a numerical calculation, and the exact sizes differ. Notice that the peak of the outer shell sits further out than the tabulated radius. Radii are defined in several ways, and none of them is the position of the peak.

The part of the nucleus's pull that an electron actually feels. The other electrons, mostly the inner ones, sit between it and the nucleus and cancel part of the charge. This is shielding. Zeff = Z − S, where S counts the shielding. Slater's rules give S: electrons in the same shell count a little (0.35), the shell below counts a lot (0.85), deeper ones count fully (1.00). Zeff for the outer electron rises steadily across a period, which is why atoms shrink from left to right.

Each p orbital has a lobe along one axis. In the maths, p_x depends on x/r, p_y on y/r and p_z on z/r. The density of each is the square of that, so add the three orbitals up with the same number of electrons in each: (x² + y² + z²)/r² = 1. The answer is the same in every direction, so the cloud is a perfect ball. The same happens with the five d orbitals.

That only works when every orbital in the subshell has the same number of electrons: all empty, all single (half-full, such as p³ or d⁵), or all paired (full, such as p⁶ or d¹⁰). With p⁵, one orbital has only one electron while the others have two. The sum no longer cancels, and the cloud has a dent. Rotate the 3D cloud above to see it, and try nitrogen (p³, round) against oxygen (p⁴, lopsided).

Because the big picture averages over direction, and a full subshell really is round: the lobes of all the p orbitals (or all the d orbitals) add up to a sphere. A half-full subshell is round too. Only a partly filled subshell, usually the outer valence electrons, is lopsided, and the second picture shows it. Those lobes point in particular directions, and that is what chemical bonds form along.

The two 1s electrons feel nearly the full nuclear charge. For bromine that is about +34, so they are squeezed to about 1/35 of the size they have in hydrogen. The picture uses a log scale so that the faint outer shells show up next to this tight, dense core.