Subatomic particles
Particle | Discovered by | Charge | Mass |
|---|---|---|---|
Electron | J. J. Thomson (cathode rays, 1897) | −1.602 × 10⁻¹⁹ C | 9.109 × 10⁻³¹ kg |
Proton | E. Goldstein (canal rays); named by Rutherford | +1.602 × 10⁻¹⁹ C | 1.673 × 10⁻²⁷ kg |
Neutron | James Chadwick (1932) | 0 | 1.675 × 10⁻²⁷ kg |
Thomson measured the charge-to-mass ratio of the electron; R. A. Millikan's oil drop experiment measured its charge.
Atomic number Z = number of protons (= electrons in a neutral atom).
Mass number A = protons + neutrons. Number of neutrons = A − Z.
Isotopes have the same Z but different A (¹H, ²H, ³H). Isobars have the same A but different Z (¹⁴C and ¹⁴N).
Atomic models
Thomson's model pictured the atom as a sphere of positive charge with electrons embedded in it, like seeds in a watermelon.
Rutherford's α-particle scattering experiment fired α-particles at thin gold foil. Most passed straight through, a few were deflected, and about 1 in 20,000 bounced back. He concluded:
most of the atom is empty space,
the positive charge and nearly all the mass are concentrated in a tiny nucleus (radius about 10⁻¹⁵ m, compared with about 10⁻¹⁰ m for the atom),
electrons move around the nucleus.
Drawback: by classical electromagnetism, an orbiting electron should continuously radiate energy and spiral into the nucleus. Atoms are stable, so the model was incomplete. It also couldn't explain line spectra.
Electromagnetic radiation and quanta
Light is an electromagnetic wave: c = νλ, where c = 3.0 × 10⁸ m/s, ν is the frequency and λ the wavelength. The wavenumber ν̄ = 1/λ.
Planck's quantum theory: energy is emitted or absorbed not continuously but in discrete packets called quanta. The energy of one quantum is
E = hν, where h = 6.626 × 10⁻³⁴ J s (Planck's constant).
Photoelectric effect: light of high enough frequency ejects electrons from a metal surface. Below a threshold frequency ν₀, no electrons are ejected however bright the light. Einstein explained this with photons:
hν = hν₀ + ½mₑv²
The number of electrons depends on the intensity; their kinetic energy depends on the frequency.
Bohr's model of the hydrogen atom
The electron moves around the nucleus in fixed circular orbits (stationary states) without radiating energy.
Energy is absorbed or emitted only when the electron jumps between orbits: ΔE = E₂ − E₁ = hν.
Angular momentum is quantised: mvr = nh/2π.
Energy of the nth orbit: Eₙ = −2.18 × 10⁻¹⁸ (Z²/n²) J. Radius: rₙ = 52.9 n²/Z pm.
The energies are negative because the electron is bound; Eₙ approaches zero as n becomes large. Transitions between levels explain the line spectrum of hydrogen: the Lyman series (to n = 1, ultraviolet), Balmer series (to n = 2, visible), and Paschen, Brackett and Pfund series (infrared).
Limitations: Bohr's model works only for hydrogen-like (one-electron) species, can't explain the finer details of spectra or the effect of electric and magnetic fields, and contradicts the uncertainty principle.
Dual nature and uncertainty
de Broglie proposed that matter, like light, has a wave nature. A particle of momentum p has wavelength
λ = h/p = h/mv
The wavelength is significant only for tiny particles like electrons; it underlies the electron microscope.
Heisenberg's uncertainty principle: it is impossible to know both the exact position and the exact momentum of an electron at the same time: Δx · Δp ≥ h/4π. So the idea of a fixed orbit is meaningless; we can only talk of the probability of finding an electron in a region of space.
Quantum numbers and orbitals
Schrödinger's equation gives wave functions ψ for the electron; the region where the probability of finding it (ψ²) is high is an orbital. Each electron is described by four quantum numbers:
Quantum number | Symbol | Allowed values | Tells us |
|---|---|---|---|
Principal | n | 1, 2, 3, … | the shell; size and energy |
Azimuthal | l | 0 to (n − 1) | the subshell and orbital shape: l = 0 (s), 1 (p), 2 (d), 3 (f) |
Magnetic | mₗ | −l to +l | orientation; a subshell has (2l + 1) orbitals |
Spin | mₛ | +½ or −½ | the electron's spin |
A shell n contains n² orbitals and up to 2n² electrons. s orbitals are spherical; p orbitals are dumbbell-shaped (pₓ, p_y, p_z).
Filling electrons
Aufbau principle: orbitals fill in order of increasing energy, given by the (n + l) rule: lower (n + l) first; if equal, lower n first.
1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4f < 5d …
Pauli exclusion principle: no two electrons in an atom can have the same four quantum numbers, so an orbital holds at most two electrons, with opposite spins.
Hund's rule: in orbitals of equal energy (such as the three 2p orbitals), electrons occupy them singly, with parallel spins, before any pairing begins. Nitrogen is 1s² 2s² 2p³, with one electron in each 2p orbital.
Examples: O (Z = 8): 1s² 2s² 2p⁴. Na (11): [Ne] 3s¹. Fe (26): [Ar] 3d⁶ 4s².
Exceptions: half-filled and completely filled subshells are extra stable because of symmetry and exchange energy. So Cr (24) is [Ar] 3d⁵ 4s¹ and Cu (29) is [Ar] 3d¹⁰ 4s¹.
Summary
Rutherford found the nucleus; Bohr quantised orbits for hydrogen: Eₙ = −2.18 × 10⁻¹⁸ Z²/n² J.
E = hν; λ = h/mv; Δx·Δp ≥ h/4π.
Four quantum numbers n, l, mₗ, mₛ describe each electron.
Fill orbitals by aufbau (n + l), Pauli and Hund's rules; Cr and Cu are exceptions.