Atomic Structure
Inside the atom — from cathode rays and the nuclear atom to quantum numbers and the orbitals that shape all of chemistry
- How the electron, proton and neutron were discovered and what defines an atom's identity.
- Why the Rutherford nuclear model replaced Thomson's, and the puzzle it left behind.
- The quantum theory of light, \(E=h\nu\), and the photoelectric effect.
- Bohr's model, the energy formula \(E_n=-13.6\,\tfrac{Z^2}{n^2}\ \text{eV}\), and the hydrogen spectrum.
- The dual nature of matter (de Broglie) and the uncertainty principle (Heisenberg).
- The four quantum numbers, the shapes of orbitals, and the rules that build electronic configurations.
The Subatomic Particles
Dalton's atom was indivisible; experiments at the close of the nineteenth century broke it open. Studying discharge through gases at low pressure, J. J. Thomson found that cathode rays are streams of negatively charged particles — electrons — identical whatever the gas or electrode. He measured their charge-to-mass ratio \(e/m_e\); Millikan's oil-drop experiment later fixed the charge itself, and hence the tiny electron mass.
Anode (canal) rays, positive particles whose \(e/m\) was largest for hydrogen, revealed the proton. The electrically neutral neutron, of nearly the same mass as the proton, was discovered later by Chadwick. These three are the building blocks of every atom.
| Particle | Charge | Relative mass (u) | Discoverer |
|---|---|---|---|
| Electron \((e^-)\) | \(-1\) | \(0.000549\) | Thomson |
| Proton \((p^+)\) | \(+1\) | \(1.00728\) | Goldstein / Rutherford |
| Neutron \((n^0)\) | \(0\) | \(1.00867\) | Chadwick |
From Thomson to Rutherford
Thomson pictured the atom as a uniform sphere of positive charge with electrons embedded in it — the plum-pudding model. Rutherford tested it by firing \(\alpha\)-particles at thin gold foil. Most passed straight through, but a tiny fraction bounced back sharply — impossible if charge were spread out.
Rutherford concluded the atom is mostly empty space, with all positive charge and nearly all the mass concentrated in a tiny central nucleus, electrons orbiting at a distance. But classical physics doomed it: an orbiting (accelerating) charge must radiate energy and spiral into the nucleus, so the atom should collapse in an instant. It does not — a problem only the quantum theory would solve.
Rutherford's model could not explain atomic stability nor the sharp line spectra of elements. Both demanded that electron energies be quantized — a radical idea.
Atomic Number, Mass Number & Isotopes
The atomic number \(Z\) is the number of protons — it fixes the element's identity. The mass number \(A\) is the total of protons and neutrons. A nuclide is written \(^{A}_{Z}X\), so neutrons \(=A-Z\).
Hydrogen's isotopes — protium \(^{1}_{1}\text{H}\), deuterium \(^{2}_{1}\text{H}\) and tritium \(^{3}_{1}\text{H}\) — share one proton but carry zero, one and two neutrons. Because they share \(Z\), isotopes are chemically near-identical.
Light and the Quantum Theory
Light is electromagnetic radiation — oscillating electric and magnetic fields travelling at \(c=3\times10^{8}\ \text{m s}^{-1}\). Its wavelength \(\lambda\) and frequency \(\nu\) obey \(c=\nu\lambda\); the wavenumber is \(\bar\nu=1/\lambda\).
To explain black-body radiation, Planck proposed that energy is emitted or absorbed not continuously but in discrete packets — quanta. For light, each quantum (a photon) carries energy proportional to frequency.
with Planck's constant \(h=6.626\times10^{-34}\ \text{J s}\). Energy comes in whole multiples of \(h\nu\) — never fractions. This single idea launched quantum physics.
The Photoelectric Effect
When light of high enough frequency strikes a metal, electrons are ejected. Classically, brighter light should always free electrons — but experiment shows a sharp threshold frequency \(\nu_0\) below which nothing happens, however intense the light. Einstein explained it by treating light as photons: one photon ejects one electron.
Bohr's Model of the Atom
Bohr rescued the nuclear atom with three postulates: the electron moves only in certain stationary orbits without radiating; angular momentum is quantized as \(mvr=\tfrac{nh}{2\pi}\); and energy is emitted or absorbed only when the electron jumps between orbits, with \(\Delta E=h\nu\).
The radius grows as \(n^2\); the energy is negative (bound) and rises toward zero as \(n\to\infty\) (ionization). For hydrogen \((Z=1)\) the ground state is \(-13.6\ \text{eV}\) — its ionization energy.
The model works only for one-electron systems \((\ce{H},\ \ce{He+},\ \ce{Li^2+})\). It cannot handle multi-electron atoms, the splitting of lines in magnetic fields (Zeeman effect), or the dual nature of the electron — fixed later by quantum mechanics.
The Hydrogen Spectrum
Excited hydrogen emits light at sharp, discrete wavelengths grouped into series. Each line is an electron falling from a higher level \(n_2\) to a lower \(n_1\), with the wavelength given by the Rydberg formula.
| Series | \(n_1\) | Region |
|---|---|---|
| Lyman | 1 | Ultraviolet |
| Balmer | 2 | Visible |
| Paschen | 3 | Infrared |
| Brackett | 4 | Infrared |
| Pfund | 5 | Far infrared |
Dual Nature & the Uncertainty Principle
If light can behave as particles, de Broglie argued, then matter can behave as waves. Every moving particle has an associated wavelength:
Because the electron is a wave, you cannot pin down both where it is and how fast it moves. Heisenberg quantified this trade-off:
The product of the uncertainties in position and momentum has a fixed lower bound. Sharpening one blurs the other — so Bohr's neat orbits give way to probability clouds, the orbitals.
Quantum Numbers & the Shapes of Orbitals
Solving the Schrödinger equation for hydrogen yields wavefunctions \(\psi\); \(\psi^2\) is the probability of finding the electron at a point. Each allowed solution — an atomic orbital — is labelled by three quantum numbers, with a fourth for the electron's spin.
| Quantum number | Symbol | Allowed values | Tells us |
|---|---|---|---|
| Principal | \(n\) | \(1,2,3,\dots\) | shell, size & energy |
| Azimuthal | \(l\) | \(0\) to \(n-1\) | subshell & shape |
| Magnetic | \(m_l\) | \(-l\) to \(+l\) | orientation |
| Spin | \(m_s\) | \(+\tfrac12,\ -\tfrac12\) | spin direction |
The value of \(l\) names the subshell: \(l=0,1,2,3\) are \(s,p,d,f\). An \(s\) orbital is spherical; a \(p\) orbital is dumb-bell shaped along an axis. A subshell holds \(2(2l+1)\) electrons.
Building Electronic Configurations
Electrons fill orbitals by three rules that, together, reproduce the periodic table.
Orbitals fill from lowest energy upward, ordered by the \((n+l)\) rule; for equal \(n+l\), the lower \(n\) fills first.
No two electrons in an atom share all four quantum numbers — so an orbital holds at most two electrons, with opposite spins.
Within a subshell, electrons occupy separate orbitals singly with parallel spins before any orbital is doubly filled.
Putting It to Work
Problem. Find the energy of one photon of light of wavelength \(500\ \text{nm}\).
Solution. Use \(E=\tfrac{hc}{\lambda}\) with \(\lambda=500\times10^{-9}\ \text{m}\):
Problem. Calculate the energy and radius of the electron in the \(n=2\) level of hydrogen.
Solution. With \(Z=1,\ n=2\):
Problem. Find the wavelength of the line in hydrogen for the transition \(n_2=3\to n_1=2\).
Solution. Apply the Rydberg formula:
Problem. What is the wavelength of an electron \((m_e=9.1\times10^{-31}\ \text{kg})\) moving at \(2.2\times10^{6}\ \text{m s}^{-1}\)?
Solution. Use \(\lambda=\tfrac{h}{mv}\):
Problem. An electron's position is known to within \(\Delta x=1\times10^{-10}\ \text{m}\). Estimate the minimum uncertainty in its velocity.
Solution. From \(\Delta x\,\Delta(mv)\ge\tfrac{h}{4\pi}\):
Problem. Write the ground-state electronic configuration of iron \((Z=26)\) and give the number of unpaired electrons.
Solution. Filling by the Aufbau order:
By Hund's rule the \(3d^6\) set has four unpaired electrons.
Chapter Summary
Electron, proton, neutron; identity is set by \(Z\), mass by \(A=Z+N\); isotopes share \(Z\).
Rutherford's nuclear atom replaced Thomson's, but needed quantized energy to be stable.
\(E=h\nu\); the photoelectric effect proves light is particulate.
\(E_n=-13.6\,Z^2/n^2\ \text{eV}\); the Rydberg formula gives the hydrogen lines.
\(\lambda=h/mv\) and \(\Delta x\,\Delta p\ge h/4\pi\) replace orbits with orbitals.
Four quantum numbers; Aufbau, Pauli and Hund build the filling order.
Problems
Keep the constants close: \(h=6.626\times10^{-34}\ \text{J s}\), \(c=3\times10^{8}\ \text{m s}^{-1}\), \(R_H=1.097\times10^{7}\ \text{m}^{-1}\), \(m_e=9.1\times10^{-31}\ \text{kg}\). Difficulty rises down the list.
- How many protons, neutrons and electrons are in \(^{40}_{20}\text{Ca}^{2+}\)?
- Calculate the frequency and wavenumber of radiation of wavelength \(600\ \text{nm}\).
- Find the energy in joules of one mole of photons of wavelength \(300\ \text{nm}\).
- The work function of a metal is \(3.0\ \text{eV}\). Find the threshold wavelength below which photoemission occurs.
- Compute the radius and energy of the \(n=3\) orbit of \(\ce{Li^2+}\).
- Find the wavelength of the first line of the Lyman series of hydrogen.
- How much energy is needed to ionize a hydrogen atom from its \(n=2\) state?
- Calculate the de Broglie wavelength of a cricket ball of mass \(150\ \text{g}\) moving at \(30\ \text{m s}^{-1}\), and comment.
- An electron and a proton have the same kinetic energy. Which has the longer de Broglie wavelength, and why?
- Give the set of four quantum numbers for the last electron of \(\ce{Cl}\ (Z=17)\).
- Write the electronic configurations of \(\ce{Cr}\) and \(\ce{Cu}\) and explain why each departs from the naïve Aufbau order.
- For \(n=3\), list all allowed subshells, the number of orbitals in each, and the maximum number of electrons in the shell.