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.
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.
Three methods · One comparison
What information fixes the molecular number?
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.
Predict before calculating
Can displacement alone distinguish a large particle from a large molecular number? With the same viscosity and diffusion observations, will correcting the viscosity coefficient increase or decrease the inferred radius?
| Method | Result | What this means |
|---|---|---|
| Radiation constants | 6.1705e23 mol^-1 | Historical reconstruction; α scaled by 1. No confidence interval is asserted. |
| Brownian displacement | 6.0143e23 mol^-1 Conditional 95% interval: 4.4639e23 to 7.7922e23 mol^-1 | Modern consistency check conditional on an independent radius and the admitted observation model. |
| Viscosity and solute diffusion | 6.0188e23 mol^-1 | Illustrative dilute-sphere inversion with coefficient 2.5. Not a historical dataset or uncertainty interval. |
| Modern reference | 6.0221e23 mol^-1 | Exactly 6.02214076 × 10²³ mol⁻¹ by definition, not measured by this laboratory. |
Do not average these rows. They have different provenance and assumptions. The modern gas constant is defined using Nₐ and kᵦ; agreement in the illustrative diffusion rows is not independent evidence for either constant.
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: 3.0071e17 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: 4.2968e-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
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. Solvation, molecular shape, concentrated-solution interactions, measurement error and historical data fitting are not modeled. Shared controls do not make the two sets of illustrative observations a single experiment.
Every displayed quantity comes from one instance-scoped accepted snapshot. Invalid drafts preserve that snapshot. This companion preview does not claim the full catalogue's launch acceptance or source review.
Historical radiation owner · Brownian inference owner · Joint viscosity–diffusion owner · Snapshot composition