What is the Bohr model of the atom?

Bohr's 1913 model says an electron in hydrogen can only orbit at certain fixed radii where its angular momentum is an integer multiple of h/2pi. While in an orbit it radiates no energy; it only emits or absorbs a photon when it jumps between orbits, with photon energy equal to the energy difference between them. This exactly predicts hydrogen's line spectrum.

What is the photoelectric effect?

Light shining on a metal surface can eject electrons, but only if each photon carries at least the metal's work function energy — below that threshold frequency, no electrons are emitted no matter how bright the light. Einstein explained this in 1905 with E=hf=phi+KE_max, which won him the Nobel Prize and confirmed light behaves as discrete photons.

What is a semiconductor band gap?

Electrons in a solid can only occupy certain energy bands. In conductors the valence and conduction bands overlap so electrons flow freely; in insulators a large gap (over about 5 eV) blocks conduction; in semiconductors a small gap (around 1 eV) lets a few electrons cross at room temperature, and doping with donor or acceptor impurities adds extra energy levels that dramatically increase conductivity.

What is the difference between n-type and p-type semiconductors?

n-type semiconductors are doped with donor atoms (like phosphorus in silicon) that contribute extra free electrons as the majority charge carrier. p-type semiconductors are doped with acceptor atoms (like boron) that create extra holes — missing electrons that behave as positive charge carriers.

Forces / Modern physics

Modern physics: four ideas explained, and a nuclear lab

Three class 12 ideas that don't need a full sandbox to click, just a clear animated picture: Bohr's quantised hydrogen orbits and the line spectrum they produce (1913), Einstein's photoelectric effect — proof that light comes in discrete photons (1905), and the energy-band picture that explains why conductors, insulators and semiconductors behave so differently. A fourth, radioactivity, comes with an interactive lab: watch nuclei decay at random, date a sample, follow alpha, beta and gamma radiation, and see why nuclear reactions release millions of times more energy than chemical ones.

a quantised jump emits one photon, E=hν

1913

Bohr's model of hydrogen

Niels Bohr — quantised angular momentum

Bohr proposed that an electron in hydrogen can only orbit at radii where its angular momentum is a whole-number multiple of h/2π — mvr = nh/2π, n = 1, 2, 3… While in one of these allowed orbits the electron radiates no energy, so the atom is stable. It only emits or absorbs energy when it jumps between orbits, as a single photon whose energy equals the exact energy gap between them.

The formulas that make it work

Orbit radius: r_n = n²a₀ (a₀ = 0.529 Å, the Bohr radius).
Energy of orbit n: E_n = −13.6/n² eV — always negative (electron is bound), least negative (closest to free) as n grows.
A jump from n_i to n_f emits a photon of energy E_i−E_f; grouping by which shell the electron lands on gives the named series — Lyman (lands on n=1, UV), Balmer (n=2, visible), Paschen (n=3, infrared).

ExplainsHydrogen's exact line spectrum — every observed wavelength matches E_i−E_f to high precision, the model's biggest triumph.

LimitOnly works cleanly for one-electron atoms (H, He⁺). It treats the electron as a ball on a fixed path, which the uncertainty principle rules out — the quantum model replaced it.

a photon above the threshold frequency ejects an electron instantly

1905 (Nobel Prize 1921)

The photoelectric effect

Albert Einstein — light as discrete photons

Shine light on a metal and it can knock electrons out — but only above a minimum threshold frequency, no matter how bright the light. A dim light above threshold ejects electrons instantly; an intense light below threshold ejects none at all. Classical wave theory (energy = brightness) cannot explain this. Einstein could, by treating light as a stream of photons, each carrying a fixed energy E = hf.

Einstein's equation

Each photon transfers all its energy to one electron: hf = φ + KE_max, where φ (the work function) is the minimum energy needed to free an electron from that metal.
Below the threshold frequency f₀=φ/h, no electron is ejected — extra brightness just means more (still too-weak) photons, not more energy per photon.
Above threshold, KE_max rises linearly with frequency, not brightness — brightness only increases how many electrons are ejected per second (the photocurrent).

ExplainsThe sharp frequency threshold and the instant response — both impossible for light-as-a-continuous-wave, both natural for light-as-photons.

LimitIt doesn't erase wave behaviour — diffraction and interference still need light-as-a-wave. Light needs both pictures (wave–particle duality).

bigger gap = harder for an electron to become a free carrier

~1928–1931

Semiconductor energy bands

Bloch & Wilson — band theory of solids

Electrons in a solid can't take just any energy — they fill discrete bands. The highest filled band is the valence band; the next one up, where electrons can move freely and carry current, is the conduction band. Whether a material conducts, insulates, or does something in between comes down entirely to the size of the gap between them.

Three materials, one picture

Conductor — the bands overlap (or the conduction band is already partly filled), so electrons flow with almost no push needed.
Insulator — a large gap (E_g > ~5 eV) that thermal energy at room temperature can't bridge, so essentially no free carriers exist.
Semiconductor — a small gap (E_g ≈ 1 eV, e.g. 1.1 eV for silicon) that a few electrons cross by thermal energy alone, and that doping exploits: donor impurities (like phosphorus in silicon) add easy-to-free electrons for an n-type semiconductor; acceptor impurities (like boron) create holes — missing electrons that act as positive carriers — for a p-type one.

ExplainsWhy silicon conducts a little (and much more when doped or heated), while diamond — same crystal structure, bigger gap — never does at room temperature.

LimitIt's a single-electron approximation; real conduction also depends on carrier mobility, temperature and crystal defects that the simple band picture leaves out.

an unstable nucleus emits an alpha particle: at a random moment

1896–1932

Radioactivity and the nucleus

Becquerel, the Curies, Rutherford & Soddy — unstable nuclei

Some nuclei are unstable and change into other nuclei by emitting radiation, with no outside cause. Which nucleus goes next is pure chance, but in a large sample the number left falls in a perfectly regular way: it halves in each half-life, whether the sample is big or small, hot or cold, pure or in a compound. The energies are huge compared with chemistry because they come from the strong force inside the nucleus.

The ideas that make it work

The decay law: N = N₀e^−λt, with the half-life T_½ = 0.693/λ. The activity A = λN is measured in becquerels (decays per second).
Three kinds of radiation: alpha (a helium nucleus: A falls by 4, Z by 2), beta (an electron or positron: Z changes by 1) and gamma (a photon: nothing changes but the energy).
Binding energy: a nucleus weighs less than its parts, and the mass defect Δm is the binding energy, B = Δmc². The binding energy per nucleon peaks at iron, so fusion of light nuclei and fission of heavy ones both release energy.

ExplainsWhy a radioactive source fades by half every half-life, how carbon-14 dates ancient objects, and why a kilogram of uranium releases millions of times the energy of a kilogram of coal.

LimitThe decay law only holds for large numbers: one nucleus has no “due date”. It also does not say why a nucleus is unstable; that needs the strong and weak nuclear forces.

	Conductor	Semiconductor	Insulator
Band gap E_g	~0 eV (overlap)	~0.7–1.5 eV	> ~5 eV
Example	Copper	Silicon, germanium	Diamond, glass
Conductivity rises with	— (falls with heat)	heat, light, doping	— (stays ~0)

Nuclear lab

Four sandboxes for the explainer above. Change a value and watch the nuclei, the graphs and the worked explanation respond.

Lessons

Read the steps, then try them right in the nuclear lab below.

Class 12 — The radioactive decay law and half-life

Concept: each nucleus has the same small chance per second of decaying, so the number left falls exponentially, N = N₀e^−λt, and halves every half-life T_½ = 0.693 ÷ λ.

Open the Nuclear lab with radium-226 (half-life 1600 years). The 400 dots stand for nuclei. After one half-life about 200 are left, after two about 100, after three about 50. The count follows the dashed theory curve closely but not exactly, because the decays are random; press Reset and the pattern is different.
λ = 0.693 ÷ (1600 × 3.156 × 10⁷ s) = 1.373 × 10⁻¹¹ s⁻¹ and the mean life is 1 ÷ λ = 2308 years = 1.443 T_½.
One gram of radium-226 holds 2.664 × 10²¹ nuclei, so its activity is λN = 3.66 × 10¹⁰ Bq = 0.99 Ci: this is how the curie was defined.
Switch to carbon-14 (5730 years): λ = 3.83 × 10⁻¹² s⁻¹. Polonium-214 has a half-life of 164 µs: a short half-life means a very intense but very short-lived source.
On the lower graph ln N falls along a straight line with slope −λ.

Takeaway: N = N₀e^−λt; T_½ = 0.693 ÷ λ; the activity A = λN falls with the same half-life.

Class 12 — Radiocarbon and uranium–lead dating

Concept: a sample's radioactive atoms act as a clock: the less of the parent is left, the older it is. t = (T_½ ÷ ln 2) ln(N₀ ÷ N).

Open Radioactive dating (carbon-14). With 50% of the carbon-14 left exactly one half-life has passed: 5730 years. With 25% left: 11 460 years.
With 60% left: t = (5730 ÷ 0.693) × ln(100 ÷ 60) = 4223 years.
Try 78%: about 2050 years, the age of the Dead Sea Scrolls. About 53% gives 5300 years, the age of Ötzi the Iceman. Below about 1% (50 000 years) too little carbon-14 is left to measure.
Switch to uranium–lead dating of rocks (half-life 4.47 billion years). If there are as many lead atoms as uranium atoms (ratio 1) one half-life has passed: 4.468 billion years; a ratio of 0.5 gives t = (1 ÷ λ) ln(1.5) = 2.61 billion years.

Takeaway: use carbon-14 for organic things younger than 50 000 years and uranium–lead for rocks of millions to billions of years.

Class 12 — Alpha, beta and gamma radiation

Concept: in every decay the mass number A and the charge Z are conserved. Alpha particles are helium nuclei, beta particles are electrons or positrons and gamma rays are photons, and they penetrate matter very differently.

Open Alpha, beta, gamma with uranium-238: ²³⁸₉₂U → ²³⁴₉₀Th + ⁴₂He. A: 238 = 234 + 4; Z: 92 = 90 + 2. The energy released is Δmc² = 0.004584 u × 931.5 = 4.27 MeV, of which the alpha particle takes 4.20 MeV and the heavy thorium nucleus recoils with 0.07 MeV.
Carbon-14 decays by beta-minus emission to nitrogen-14 (Z rises from 6 to 7, A is unchanged) with Q = 0.156 MeV. Cobalt-60 gives Q = 2.82 MeV and sodium-22 emits a positron (Q = 1.82 MeV).
Switch to Penetration and shielding. Alpha particles are stopped by 0.1 mm of paper. A beta particle (1 MeV) is stopped in about 1.5 mm of aluminium (0.75 mm lets 50% through in this model).
For gamma rays in lead (μ = 0.664 per cm) 5 mm lets 71.7% through, 10 mm lets 51.5% through (the half-value thickness is 10.4 mm) and 30 mm lets 13.6% through. In aluminium 10 mm still lets 86.3% through. Gamma rays are never completely stopped, only weakened.

Takeaway: ionising power: α > β > γ; penetrating power: γ > β > α; I = I₀e^−μx for gamma rays.

Class 12 — Binding energy, fission and fusion

Concept: a nucleus is lighter than its separate nucleons; the mass defect gives the binding energy, B = Δmc². Nuclear reactions that move to a higher binding energy per nucleon release energy.

Open Binding energy & reactions. For helium-4, Δm = 2 × 1.007825 + 2 × 1.008665 − 4.002603 = 0.030377 u, so B = 28.30 MeV and B/A = 7.07 MeV. Iron-56 has the highest value, 8.79 MeV per nucleon.
Choose fission: ²³⁵U + n → ¹⁴¹Ba + ⁹²Kr + 3n. The products are lighter by 0.186033 u (0.079%), which is 173.3 MeV. One kilogram of uranium-235 would give 7.1 × 10¹³ J, about 20 million kWh and 2.4 million times as much as a kilogram of coal.
Choose D + T fusion: Q = 17.59 MeV from just 5 u of fuel, so 3.4 × 10¹⁴ J per kilogram, more than four times the fission figure. D + D gives 3.27 MeV.
Choose hydrogen burning in the Sun: 4 ¹H → ⁴He gives 26.73 MeV, and 0.7% of the mass disappears.
Alpha decay of uranium-238 gives 4.27 MeV and the beta decay of carbon-14 only 0.156 MeV. All the nuclear energies are millions of times bigger per atom than chemical ones (a few eV).

Takeaway: Q = (mass before − mass after) × 931.5 MeV; fusion of light nuclei and fission of heavy ones both climb the binding-energy curve.

Graph

The physics behind it

Formulas used in the nuclear lab

Radioactive decay Class 12

Decay law: N = N₀e^−λt; dN/dt = −λN
Half-life: T_½ = ln 2/λ = 0.693/λ; after n half-lives N = N₀/2ⁿ
Mean life: τ = 1/λ = 1.443 T_½
Activity: A = λN = A₀e^−λt; 1 Bq = 1 decay/s, 1 Ci = 3.7 × 10¹⁰ Bq
Number of atoms: N = (m/M)N_A, N_A = 6.022 × 10²³
Dating: t = (1/λ) ln(N₀/N) = (T_½/ln 2) ln(N₀/N); for a daughter D and a parent P: t = (1/λ) ln(1 + D/P)

Radiation Class 12

Alpha decay: ^A_ZX → ^A−4_Z−2Y + ⁴₂He; the alpha takes Q(A−4)/A of the energy
Beta-minus decay: ^A_ZX → ^A_Z+1Y + e⁻ + ν̄ (a neutron becomes a proton)
Beta-plus decay: ^A_ZX → ^A_Z−1Y + e⁺ + ν (a proton becomes a neutron)
Gamma emission: an excited nucleus drops to a lower level; A and Z unchanged
Conservation: mass number A, charge Z, energy and momentum are all conserved
Absorption of gamma rays: I = I₀e^−μx; half-value thickness x_½ = ln 2/μ

Nuclear energy Class 12

Mass defect: Δm = Zm_p + (A − Z)m_n − M_nucleus
Binding energy: B = Δmc²; 1 u = 931.5 MeV; per nucleon B/A (peaks at about 8.8 MeV near iron-56)
Energy of a reaction: Q = (Σm_before − Σm_after)c²; positive Q releases energy
Fission: ²³⁵U + n → fragments + 2 or 3 neutrons, about 200 MeV; a chain reaction needs more than one neutron per fission to go on
Fusion: light nuclei join, e.g. D + T → ⁴He + n, 17.6 MeV; needs about 10⁸ K
Nuclear radius: R = R₀A^1/3, R₀ = 1.2 fm, so the density is the same for all nuclei (about 2.3 × 10¹⁷ kg/m³)

Part of Forces — the first four sections are animated explainers and the nuclear lab above is an interactive sandbox; for more hands-on labs see Electricity & magnetism, Optics and the rest of Forces. See Learn for the full class 12 modern physics syllabus this maps to.