Inferring the molecular number

These positions were generated with a hidden molecular number. First ask what the observations identify. Then declare the missing information, estimate the number, and test what a confidence interval does across hypothetical repeats.

Not modeled: localization error, blur, correlated or irregularly timed increments, and censoring (see the camera laboratory and kitchen mode); non-Gaussian increments; time-varying drift; polydispersity within one track set; wall effects; uncertainty in C without declared coverages; uncertainty in the gas constant itself.

Watching a small sphere wander tells you how fast it spreads. If you also know the temperature, how thick the liquid is and how big the sphere is, the spreading tells you how many molecules make up a mole, and the lab shows how sure a limited number of observations can make you.

Section 5 ends by turning its formula around: N = (t/λ_x²)(RT/(3πkP)), so a measured mean square displacement, with the temperature, the viscosity and the radius, gives the number of molecules in a gram-molecule. The instrument does this with synthetic data from a generator whose number is hidden, 3.03 × 10²³, deliberately not today's value, so the exercise tests the method and not the world. From 50 steps of 1 s in two directions, 100 squared steps, it estimates D = 8.83 × 10⁻¹³ m²/s, with a 95 percent interval from 6.81 to 11.89 × 10⁻¹³. Displacements alone fix only the product of radius and number, a·N = 1.46 × 10¹⁷ m/mol. Given the radius, 0.5 μm, the estimate is N = 2.93 × 10²³, with an interval from 2.17 to 3.80 × 10²³ that contains the hidden value. With today's defined constants the same inversion is only a consistency check, because N_A is now fixed by definition.

Start from what each step tells you. Along one axis a sphere's step Δ over a time τ has mean square 2Dτ, so D can be estimated as the average of Δ²/(2τ). The instrument records 50 steps of τ = 1 s in two directions, 100 squared steps in all, and their average divided by 2 gives D̂ = 8.83 × 10⁻¹³ m²/s. Because each step is a Gaussian draw, 100 D̂/D follows a chi-square law with 100 degrees of freedom, whose middle 95 percent runs from 74.2 to 129.6. So D lies between 100 D̂/129.6 = 6.81 × 10⁻¹³ and 100 D̂/74.2 = 11.89 × 10⁻¹³ m²/s; the generator's own D, 8.55 × 10⁻¹³, is inside. Now invert. Einstein's D = (RT/N)/(6πηa) gives a·N = RT/(6πηD̂). With R = 8.314 J/(mol K), T = 293.15 K and η = 0.001 Pa s, RT = 2437 J/mol and 6πηD̂ = 6 × 3.1416 × 0.001 × 8.83 × 10⁻¹³ = 1.664 × 10⁻¹⁴, so a·N = 2437/(1.664 × 10⁻¹⁴) = 1.465 × 10¹⁷ m/mol. A sphere twice as large with half as many molecules per mole wanders in exactly the same way, so the data cannot tell them apart. Give the radius, a = 0.5 × 10⁻⁶ m, and N = 1.465 × 10¹⁷/(0.5 × 10⁻⁶) = 2.93 × 10²³. Because N goes as 1/D, the interval turns over: the largest D gives the smallest N, so N lies between 2.17 and 3.80 × 10²³. Inverting also biases the estimate, since the average of 1/D̂ exceeds 1/D by the factor 100/98 = 1.020.

Einstein's printed form is N = (t/λ_x²)(RT/(3πkP)), with k the viscosity and P the radius, and he closed by hoping a researcher would soon decide the question. Perrin did, in 1908 and 1909, following grains of gamboge and mastic whose radius he measured separately, and found N near 7 × 10²³. The chi-square interval, the turned-over interval for N and the bias factor are modern statistics that Einstein did not give. Since 2019 N_A = 6.022 140 76 × 10²³ mol⁻¹ exactly and k_B is fixed too, so an inversion with today's constants recovers its own inputs: it checks the method and does not count molecules.