Annus Mirabilis · Interactive critical edition in preparation
Classical mode-energy allocation
Give every resonator the same mean energy, and see whether the radiation's total energy ever settles.
Classical mode-energy allocation
Classical mode-energy allocation
Static worked exampleWorked example: with resonators up to 1 × 10¹⁴ Hz at 1500 K, the classical allocation holds 0.00644 J per cubic metre, and widening the cutoff tenfold multiplies it by 1000.
| Mean energy per resonator oscillation | 2.070974 × 10−20 J |
|---|---|
| Energy with resonators up to the cutoff | 0.006439187 J/m³ |
| Share of that energy above the probe frequency | 0.9990000 |
| Energy after widening the cutoff tenfold | 6.439187 J/m³ |
| Ratio: tenfold-widened energy ÷ current energy | 1000 |
With resonators up to 1.000 × 1014 Hz at 1500 K, the classical allocation holds 0.006439 J per cubic meter; each tenfold widening multiplies it by 1000. Every resonator oscillation carries the same mean energy kBT (source notation (R/N)T), which §1 notes is two-thirds of a free molecule's mean kinetic energy.
§2's Avogadro match and §4's later entropy argument (a separate instrument) use two disjoint limits of the same spectrum: the classical region, admitted at the 1% criterion for x ≤ 0.0198678, and the Wien region, admitted at x ≥ 4.60517. Meeting both arguments does not mean one "quantum regime" supplied both numbers.
The historical Avogadro readout, from Einstein's §2 constants
| N, from Einstein's printed §2 constants (historical) | 6.170486 × 1023 mol⁻¹ |
|---|---|
| N, as printed in §2 | 6.17 × 1023 mol⁻¹ |
| NA, defined, 2019 SI (modern) | 6.02214076 × 1023 mol⁻¹ |
| Hydrogen-atom mass from the printed N (historical) | 1.62 × 10−24 g |
| Hydrogen-atom mass, modern (labeled modern) | 1.673533 × 10−24 g |
α is a suspected historical typographical correction (10⁻⁵⁷, not the witness's 10⁻⁵⁶); R = 8.31×10⁷ erg mol⁻¹ K⁻¹ and L = 3×10¹⁰ cm s⁻¹ are declared editorial inputs, not printed values, since §2 does not appear to print either. This readout is never mixed with a modern constant in the same number.
Not modeled: The mechanism coupling matter and radiation beyond Planck's stated equilibrium condition; Cavity shape and walls; The approach to equilibrium; Any quantum hypothesis; Measured spectra (a separate instrument shows those).
Show the calculation owner: which real functions computed these numbers
src/physics/reference/radiation/classical.ts (classicalCutoffEnergyDensity, classicalTotalEnergy, meanResonatorEnergy), src/physics/reference/radiation/avogadro.ts (avogadroFromPlanckConstants), src/physics/reference/radiation/spectra.ts (regimeRelativeErrors), and src/experiments/lq02/session.ts (composing the snapshot; this component never recomputes any of these).
Predict before the numbers
Widening the resonator range from 100 THz to 1000 THz changes the energy by how much?
The result appears when you choose, say you have one in mind, or skip.
If every way a resonator can vibrate gets the same share of energy, the energy of the radiation grows without limit as higher and higher frequencies are allowed. Einstein notes that on this picture matter and light could never settle into a definite share of the energy.
Section 1 puts a gas, free electrons and resonators, electrons bound to fixed points, in a box with mirror walls. Collisions give each resonator's vibration along one line a mean energy Ē = (R/N)T, the same at every frequency; at 1500 K that is 2.07 × 10−20 J. Planck's relation between a resonator and the radiation around it, Ēν = (L3/8πν2)ρν, then fixes the density: ρν = (R/N)(8πν2/L3)T. The energy per unit volume up to a highest frequency νc is (8πkBT/3c3)νc3, which at 1500 K is 0.00644 J/m3 up to 1014 Hz, and 99.9 percent of it lies above 1013 Hz. Allow frequencies ten times higher and the energy grows a thousandfold, to 6.44 J/m3; allow them all and there is no finite total, which the instrument reports as a refusal rather than a number. Einstein concludes that the relation disagrees with experience and that on this picture there can be no definite distribution of energy between the ether and matter. Section 2 then shows that Planck's formula, which fits experience, agrees with this law at long wavelengths, and that matching the two gives N = 6.17 × 1023.
Take a resonator vibrating along one line. In a gas at temperature T each molecule's energy of motion averages (3/2)(R/N)T, which is (1/2)(R/N)T for each direction. A bound electron also has potential energy, and for a spring force the average potential energy equals the average kinetic energy, so the vibration along one line averages (R/N)T in all. With R/N = kB = 1.381 × 10−23 J/K and T = 1500 K that is 2.071 × 10−20 J. Planck showed that a resonator of frequency ν, in radiation whose energy per unit volume and unit frequency is ρν, settles at Ēν = c3ρν/(8πν2). Setting Ēν = kBT gives ρν = 8πν2kBT/c3. The energy up to νc adds this density up: the integral of 8πν2kBT/c3 from 0 to νc is (8πkBT/3c3)νc3. The factor in front is 8π × 2.071 × 10−20/(3 × (2.998 × 108)3) = 6.44 × 10−45 J s3/m3, and multiplied by (1014)3 = 1042 it gives 0.00644 J/m3. Up to 1013 Hz the cube is a thousand times smaller, 6.44 × 10−6 J/m3, so 1 − 1/1000 = 99.9 percent of the energy lies between 1013 and 1014 Hz. Each tenfold widening multiplies the energy by 103, to 6.44 J/m3 at 1015 Hz, and there is no highest frequency at which to stop, so the total is infinite. For N: at low frequency Planck's formula αν3/(eβν/T − 1) becomes (α/β)ν2T, because eβν/T − 1 is then close to βν/T. Matching (R/N)(8π/c3) = α/β gives N = (β/α)(8πR/c3); with Planck's constants that is 6.17 × 1023, and a hydrogen atom weighs 1/N gram, 1.62 × 10−24 g.
Einstein wrote Ē for the resonator's mean energy, L for the speed of light and ρν for the density, and he gave the result no name; the name it later carried in textbooks is Ehrenfest's, from 1911. Rayleigh had given a density growing as ν2T for long waves in June 1900, and Jeans corrected its coefficient in July 1905, after this paper was received. Planck had derived the resonator relation in 1900 by treating the radiation as the most disordered process conceivable, which is how Einstein cites it; Einstein uses it without Planck's energy elements. The value N = 6.17 × 1023 is Planck's own. The exponent of α as the edition reads the plate, 10−56, would give a tenth of that N; the constants that give 6.17 × 1023 need 10−57, and the provenance record leaves the printed digit to a second witness.
The explanation
Full explanation
Each resonator oscillation gets the mean energy (R/N)T whatever its frequency, so the energy up to a highest frequency grows as the cube of that frequency. Widen the range tenfold and the energy grows a thousandfold; remove the upper limit and the laboratory refuses the total, because on this model it has no finite value.
Show every step of the investigation
Set the temperature and the highest frequency, and read the energy up to it and the share lying above the probe frequency. Widen the range tenfold and compare the two energies, then remove the upper limit and read the refusal. The Avogadro readout repeats the calculation of §2 from the constants printed there.
An explanatory model, not an observation of nature. This embed starts from the laboratory’s worked defaults, not a saved run. Presentation options change the surrounding guide, never the numerical inputs.