SCH4U From Rutherford to the Quantum Atom: Notes and Practice

Heat a pinch of table salt in a flame and the flame turns bright yellow-orange. Strontium compounds turn a flame deep red. Every element gives off its own set of colours when it is energized, as reliable as a fingerprint. Why should that be? Why not a smooth rainbow of every colour? Answering that question forced scientists to throw out the idea that electrons can have any energy they like. In this topic you will follow the chain of experiments and ideas that took us from a tiny dense nucleus, to Bohr's energy levels, to the modern picture of electrons as clouds of probability called orbitals.

Rutherford's gold foil experiment. In the early 1900s the accepted picture was J. J. Thomson's model: a sphere of spread-out positive charge with electrons embedded in it. In Ernest Rutherford's laboratory, a beam of alpha particles (small, fast, positively charged particles) was fired at a very thin sheet of gold foil, with a fluorescent screen around the foil to show where the particles went. If Thomson were right, the positive charge would be too spread out to push hard on anything, so every alpha particle should pass straight through with only tiny deflections.

Observation	Inference
Almost all alpha particles passed straight through the foil.	An atom is mostly empty space.
A small fraction were deflected at large angles.	There is a concentrated region of positive charge that repels the positive alpha particles.
A very few bounced almost straight back.	That region is extremely small and holds nearly all of the atom's mass: the nucleus.

What the gold foil results showed and what Rutherford concluded from them.

Rutherford's nuclear model placed a tiny, dense, positive nucleus at the centre with electrons moving somewhere in the empty space around it. It explained the scattering beautifully, but it left two serious problems. First, classical physics predicts that an electron circling a nucleus is constantly accelerating and should radiate energy, spiralling into the nucleus in a tiny fraction of a second, so atoms should not be stable. Second, it said nothing about why each element emits only certain colours of light.

Light, photons and line spectra. Light behaves as a wave with wavelength $λ$ and frequency $f$ , related by the speed of light $c = 3.00 \times 1 0^{8} m/s$ . It also comes in packets of energy called photons. The energy of one photon depends only on its frequency, through Planck's constant $h = 6.63 \times 1 0^{- 34} J \cdot s$ . Short wavelength means high frequency and high energy: violet photons carry more energy than red ones. White light passed through a prism gives a continuous spectrum: every colour blends into the next. But the light from a gas of one element, energized by electricity or heat, gives an emission spectrum (a line spectrum): a few sharp, bright lines on a dark background. Shine white light through a cool sample of the same gas and you get an absorption spectrum: dark lines at exactly the same wavelengths. Only certain photon energies are ever emitted or absorbed, which means the atom itself can only change its energy by certain fixed amounts.

c = λ f E_{photon} = h f = \frac{h c}{λ}

Wave equation for light and the energy of a single photon.

Bohr's Model of the Hydrogen Atom

The electron in a hydrogen atom can only occupy certain allowed orbits, called energy levels, labelled $n = 1, 2, 3, \dots$ Each level has a fixed energy.
While the electron stays in one level it does not radiate energy, so the atom is stable.
The lowest level, $n = 1$ , is the ground state. Absorbing exactly the right amount of energy lifts the electron to a higher level, an excited state.
When the electron drops from a higher level to a lower one, the atom emits one photon whose energy equals the difference between the two levels.

E_{n} = - \frac{2.18 \times 1 0 ^{- 18} J}{n ^{2}} Δ E = E_{final} - E_{initial} E_{photon} = ∣Δ E ∣

Bohr energy levels for hydrogen. Energies are negative because the electron is bound; zero means fully removed.

Fig 1Downward jumps between hydrogen energy levels produce the bright lines of its emission spectrum.

Because the levels crowd closer together as $n$ increases, jumps down to the same final level form a family of lines called a series. Drops to $n = 1$ release the largest energies and form the Lyman series in the ultraviolet. Drops to $n = 2$ form the Balmer series, whose first four lines are visible: red (656 nm, from $n = 3$ ), blue-green (486 nm, from $n = 4$ ), and two violet lines (434 nm and 410 nm, from $n = 5$ and $n = 6$ ). Drops to $n = 3$ form the Paschen series in the infrared. Bohr's formula reproduces these wavelengths almost exactly, which is why his model was so convincing.

Where the Bohr model fails

Bohr's model predicts the spectrum of hydrogen and other one-electron species such as

He^{+}

, but it fails for any atom with two or more electrons. It cannot explain why some spectral lines are brighter than others, or why lines split into several close lines in a magnetic field. Most fundamentally, it pictures the electron on a definite circular path, and we now know an electron's position and motion cannot both be known precisely at the same time (the Heisenberg uncertainty principle). Bohr's lasting idea is that energy is quantized; his orbits had to go.

The quantum mechanical model. In 1924 Louis de Broglie proposed that electrons, like light, have wave properties. Erwin Schrödinger then wrote a wave equation for the electron in an atom (1926). Its solutions do not give paths; they give the probability of finding the electron in each region of space. A region where there is a high probability (often drawn as the space holding about 90% of the probability) of finding an electron is called an orbital. An orbital is not an orbit: it is a three-dimensional cloud shape, and each orbital can hold at most two electrons.

Quantum number	Symbol and allowed values	What it describes
Principal	$n = 1, 2, 3, \dots$	Energy level (shell) and overall size of the orbital
Secondary (angular momentum)	$l = 0$ to $n - 1$ ; letters s, p, d, f for 0, 1, 2, 3	Subshell and shape of the orbital
Magnetic	$m_{l} = - l$ to $+ l$	Orientation of the orbital in space; the number of values is the number of orbitals
Spin	$m_{s} = + \frac{1}{2}$ or $- \frac{1}{2}$	The two possible spin states of an electron

The four quantum numbers that together describe one electron in an atom.

Fig 2An s orbital is a sphere; the three p orbitals are dumbbells along the x, y and z axes; d orbitals have more complex shapes.

Spectra at work

Because every element has its own line spectrum, spectroscopy identifies elements without touching them. Astronomers read the composition of stars from the dark absorption lines in their light; helium was identified in the Sun's spectrum before it was ever found on Earth. Chemists use atomic absorption and emission spectroscopy to measure trace metals such as lead in drinking water. At McMaster University, Richard Bader developed the quantum theory of atoms in molecules, which uses the calculated electron density of a molecule to define where one atom ends and the next begins.

From Rutherford to the Quantum Atom

Key things to remember

Bohr's Model of the Hydrogen Atom

Worked examples

Where marks get lost

Flashcards 8

Practice