Four quantitative failures of the classical account

Take Rutherford's atom at face value. An electron on a circular Coulomb orbit of radius r has centripetal acceleration a = e²/4πε₀mₑᵣ², and Larmor prices that acceleration at P = e²a²/6πε₀c³. At the Bohr radius the bill is 4.7×10⁻⁸ W against a stored binding energy of 2.2×10⁻¹⁸ J. The loss per orbit is tiny, so the orbit stays nearly circular while it shrinks: setting dE/dt = −P with E = −e²/8πε₀r gives dr/dt = −(4/3)rₑ²c/r², and integrating from r₀ = 5.29×10⁻¹¹ m to zero yields t = r₀³/4rₑ²c = 1.6×10⁻¹¹ s — about 10⁵ turns. The prediction fails twice over: atoms visibly last, and the radiation is wrong too, since the emitted frequency is the orbital one, 6.6×10¹⁵ Hz at r₀ and rising as r⁻³/², a continuous upward chirp where hydrogen actually emits sharp lines. Every input is a measured constant, so there is nothing to adjust — 16 ps is not a model parameter but a verdict.

A classical bound charge is a periodic system, and periodic motion Fourier-decomposes into a fundamental f and integer harmonics 2f, 3f, … — so a classical atom should emit evenly spaced multiples of one frequency. Hydrogen does not. Rydberg and Ritz found that every observed wavenumber is the difference of two entries in a single list of term values, T(n) = R_H/n² with R_H = 1.0968×10⁷ m⁻¹. The rule is generative: subtract the Lyman-α wavenumber from Lyman-β and the result, 1.5233×10⁶ m⁻¹, is the observed Balmer-α line at 656.46 nm in vacuum (656.28 nm in air) — a prediction in a different series, made by arithmetic on known lines. Doubling a known line's frequency, by contrast, predicts lines that are never there. Data organised as one ledger of terms whose differences are the observables reads like a set of energy eigenvalues with transitions between them: the shape of the spectrum votes for states over orbits.

Equipartition hands every quadratic term in the energy ½k_BT regardless of temperature, so H₂ — three translations, two rotations, one vibration counting kinetic and potential parts — should hold C_V = 7/2 R from freezing point to furnace. The measurement says 5/2 R at 300 K, falling to 3/2 R below about 80 K, with vibration only beginning to stir above roughly 1000 K. Solids repeat the pattern: Dulong–Petit's 3R fits lead at room temperature, while diamond sits at 6.1 J K⁻¹ mol⁻¹, a quarter of it. The empirical rule is that a mode contributes only while k_BT is at least comparable to its level spacing — ℏω for a vibration, ℏ²/I for a rotor — and the characteristic temperatures θᵣₒₜ = ℏ²/2Ik_B ≈ 85 K and θ_vib = ℏω/k_B ≈ 6300 K do the bookkeeping for hydrogen, whose rotational step is centred near 150 K rather than at θᵣₒₜ because ortho and para H₂ must each jump ΔJ = 2, gaps of 6θᵣₒₜ and 10θᵣₒₜ. Classical mechanics cannot reproduce any of this, because equipartition's derivation never consults ω or I at all — it owns no dial by which a stiff mode could cost more to excite than a soft one. Every measured freeze-out temperature is therefore a measurement of a level spacing over k_B, and the ℏ inside that spacing is Planck's.

Boltzmann's S = k_B ln W needs W to be a pure count, but the classical phase-space volume ∫d³x d³p per particle carries dimensions of (J s)³: it counts nothing until divided by a cell volume h₀³ (with an N! for indistinguishability). Classically h₀ is arbitrary, so absolute entropies — and with them vapour-pressure constants and chemical equilibria — are undefined; only entropy differences survive. Sackur and Tetrode closed the gap in 1912 by fitting h₀ to data. For argon vapour at its 87.3 K boiling point the formula S = Nk_B[ln(V/(Nλ³)) + 5/2], with thermal wavelength λ = h/√(2πmk_BT) = 29.6 pm, gives 129.2 J K⁻¹ mol⁻¹ when h₀ = h — matching the third-law value assembled from ∫Cₚ dT/T and latent heats. A cell of (2h)³ would fall 3R ln 2 = 17.3 J K⁻¹ mol⁻¹ short. Nothing in the experiment radiates or shows a spectrum, yet the equilibrium thermodynamics of a monatomic gas that neither emits nor absorbs a photon returns Planck's constant.

None of the four failures was unanswerable alone, and contemporaries patched each locally: perhaps Maxwell's equations fail inside atoms; perhaps Ritz combinations emerge from cunningly coupled oscillators; perhaps hidden constraints immobilise the frozen modes; perhaps the entropy cell is a fitted constant with no meaning. Each patch is cheap in its own domain, and no experiment inside that domain refutes it. The inventory's force is that the patches share no parts, but the fits do. The entropy cell is h³. The freeze-out thresholds are ℏω/k_B and ℏ²/2Ik_B. The term ladder is R_H/n² with R_H = μe⁴/8ε₀²h³c, built from the same h. Even the surviving atom votes: the only action that e, mₑ and a 10⁻¹⁰ m atomic radius can form is √(r mₑₑ²/4πε₀) ≈ 10⁻³⁴ J s, the size of ℏ, so whatever halts the Larmor spiral must carry Planck's constant too. Electrodynamic, spectroscopic, thermal and thermodynamic measurements all point at 6.63×10⁻³⁴ J s. Independent patches have no reason to agree on anything; one theory with one new constant must. The rest of this unit builds that theory, and the inventory fixes its acceptance test: discrete states whose differences are the lines, level spacings per mode set by ℏ, and h³ cells in phase space.