Cross-paper connection · Molecular-dimensions companion preview
Three routes toward a molecular number.
Radiation, visible particle motion, and the viscosity of a solution ask different questions. Compare what each needs before it can supply a number.
Three methods · One comparison
What information fixes the molecular number?
Static worked example
| Method | Result | What this means |
|---|---|---|
| Radiation constantsLight paper §2 · 1905 | 6.1705 × 10²³ mol⁻¹ | Independent estimate. Planck's 1901 constants with the 1905 gas constant, which was measured without counting molecules; α scaled by 1. No confidence interval is asserted. |
| Brownian displacementBrownian paper §5 · the 1905 prediction, applied to illustrative data | 6.0143 × 10²³ mol⁻¹ Conditional 95% interval: 4.4639 × 10²³ to 7.7922 × 10²³ mol⁻¹ | Consistency check. With the 2019 SI the gas constant is NAkB by definition, so this compares the displacements with the defined value. Conditional on an independent radius and the admitted observation model. |
| Viscosity and solute diffusionDissertation · 1905, coefficient corrected 1911 | 6.0188 × 10²³ mol⁻¹ | Consistency check on illustrative inputs with the 2019 SI gas constant: a dilute-sphere inversion with coefficient 2.5. Not a historical dataset or uncertainty interval. |
| Modern reference | 6.0221 × 10²³ mol⁻¹ | Defined exactly in the 2019 SI, not measured: 6.02214076 × 10²³ mol⁻¹. |
These rows are not combined into one number. Combining them honestly would need two things this page does not have: the covariance between the routes, which all depend on the gas constant, and the two diffusion routes also on Stokes drag and the dilute-sphere model; and an allowance for model discrepancy, how far each idealized model departs from the real radiation, suspension or solution. The modern gas constant is defined using NA and kB; agreement in the illustrative diffusion rows is not independent evidence for either constant.
The radiation row reconstructs a historical calculation. The other rows use authored illustrative inputs and modern SI constants, so they are consistency checks, not independent counts or historical measurements. No Bancelin dataset is claimed here.
What diffusion cannot identify by itself
Accepted mean-square displacement: 0.86 µm² over 1 s; 100 independent coordinate increments. The Brownian owner determines the product aN: 3.0071 × 10¹⁷ m/mol.
Remove the independent radius above to reveal the family instead of a spurious unique answer. Radius, viscosity, temperature and calibration uncertainty are held exact in the displayed conditional interval. The detailed inference lab supports additional measurement models.
Read Brownian motion §5 · Open the full inference laboratory
What the viscosity correction changes
Jointly inferred solute radius: 4.2968 × 10⁻¹⁰ m. Inferred volume fraction: 0.004 1.
At fixed observations, switching the coefficient from 1 to 2.5 divides the radius by √2.5 and multiplies N by √2.5. This is not a reproduction of the entire historical revision, which also involves the choice of data. The dilute-sphere model is refused above the explicitly chosen 5% volume-fraction ceiling; that ceiling is not an accuracy guarantee.
Setting both concentration and viscosity increment to zero removes the extra information: the solute diffusion then determines only aN, not a unique radius and number.
Bookmark these accepted settings · Inspect the historical radiation calculation · Read light quanta §2
Not modeled: solvation, molecular shape, concentrated-solution interactions, measurement error and historical data fitting.
Assumptions, provenance and the code behind the numbers
These are host reference calculations, not FrankenSim/WASM execution. Hydrodynamic spheres, dilute solutions, the Stokes drag law and uniform solvent conditions are idealizations. Shared controls do not make the two sets of illustrative observations a single experiment.
Every number shown comes from the settings this laboratory last accepted, and an invalid entry leaves those numbers as they were.
The molecular number from Planck's radiation constants · The molecular number from Brownian displacements · Molecular size and number from viscosity and diffusion · How the page gathers these results
Three different pieces of physics each lead to the number of molecules in a mole: the light of hot bodies, the wandering of small spheres, and the way a dissolved substance thickens water. Each needs different things measured first, and historically their agreement helped convince physicists that molecules are real and can be counted.
Three routes are laid side by side, each computed by its own owner. From radiation, Section 2 of the light paper matches Planck's formula to the classical law at long wavelengths and gets N = (β/α)(8πR/L3) = 6.17 × 1023 from Planck's constants. From Brownian motion, Section 5 of the Brownian paper inverts λx2 = 2Dt: a mean square of 0.86 μm2 in 1 s, for spheres of 0.5 μm in water at 293.15 K, gives N = 6.01 × 1023, and only once the radius is known independently. From the dissertation, a dissolved substance's diffusivity fixes the product of its molecules' radius and N, and the extra viscosity of the solution fixes the volume the molecules fill; with a diffusivity of 500 μm2/s, 20 mol/m3 and a viscosity 1 percent above the solvent's, the two give a radius of 0.43 nm and N = 6.02 × 1023. The defined value today is 6.022 × 1023. The settings are illustrative, chosen near modern values, so their agreement here shows how the routes work rather than making a new measurement.
Radiation first. At long wavelengths Planck's formula becomes (α/β)ν2T, and the classical law of Section 1 is (R/N)(8πν2/L3)T; equating the coefficients gives N = (β/α)(8πR/L3), and with Planck's α and β, R = 8.31 × 107 erg/(mol K) and L = 3 × 1010 cm/s this is 6.17 × 1023. Brownian motion next. λx2 = 2Dt and D = RT/(6πηaN) together give N = RTt/(3πηaλx2). With RT = 8.314 × 293.15 = 2437 J/mol, t = 1 s, η = 0.001 Pa s, a = 0.5 × 10−6 m and λx2 = 0.86 × 10−12 m2, the denominator is 3 × 3.1416 × 0.001 × 0.5 × 10−6 × 0.86 × 10−12 = 4.053 × 10−21, so N = 2437/(4.053 × 10−21) = 6.01 × 1023. Last the dissertation. For solute molecules of radius a, D = RT/(6πηaN) gives aN = RT/(6πηD) = 2437/(6 × 3.1416 × 0.001 × 5 × 10−10) = 2.59 × 1014 m/mol. The viscosity law says spheres filling a fraction φ of the volume raise the viscosity by 2.5φ, the coefficient as corrected in 1911, so a specific viscosity of 0.01 means φ = 0.004. The fraction filled is also the concentration times N times each molecule's volume, φ = cN(4/3)πa3. Divide it by aN: a2 = φ/((4/3)πc × aN) = 0.004/(4.189 × 20 × 2.59 × 1014) = 1.85 × 10−19 m2, so a = 4.30 × 10−10 m and N = 2.59 × 1014/(4.30 × 10−10) = 6.02 × 1023.
Modern qualifications
Einstein's dissertation, dated 30 April 1905 and printed in 1906, applied the third route to sugar in water and found N = 2.1 × 1023. In 1911, after Bancelin had measured suspensions in Perrin's laboratory, Einstein found an error in his calculation of the viscosity: the coefficient of φ is 5/2, not 1, and the corrected N is 6.56 × 1023. Perrin's book Les Atomes of 1913 set thirteen independent determinations side by side, and their agreement did much to settle the reality of molecules. Since 2019 NA = 6.022 140 76 × 1023 mol−1 by definition.
The dissertation is a companion to the four papers, not a fifth paper in the edition. This preview makes the mathematical comparison interactive without presenting illustrative settings as historical observations.