Liquid at the bottom of a narrow tube evaporates through still air. The level drops as √t, and the slope of L² against time gives the vapour's diffusivity.
Straight lines mean L² = L₀² + 2Kt. Steeper = faster evaporation.
Horizontal axis: distance from the liquid surface as a fraction of the gap. The curves are slightly bowed because of the bulk (Stefan) flow.
Eight level readings are generated from the model between the start and a 10 mm level drop (at most 60 days), each with ±0.2 mm Gaussian reading noise, then L² is fitted against time. The table compares the true D with the value recovered using the Stefan formula and with the value you would get if you ignored the bulk flow (plain Fick). Readings are synthetic, so this tests the data reduction, not the physics.
The science
Setup. A liquid surface at the bottom of a tube of length L is exposed to air that is stirred only at the open top. Vapour leaves by diffusion through the stagnant gas. At the surface the vapour is saturated: mole fraction ys = psat(T)/P from the Antoine equation log10 p = A − B/(C + T) (mmHg, °C). At the top it is the ambient value y∞ (relative humidity × ys for water, zero for the others).
Stefan flow. Because air does not dissolve in the liquid, it does not flow, so the mixture as a whole moves upward with the vapour. The molar flux is N = (c D / L) ln[(1 − y∞)/(1 − ys)] with c = P/(RT). For small ys this is Fick's law c D (ys − y∞)/L; for acetone near room temperature (ys ≈ 0.30) it is about 19 % larger. The profile is 1 − y = (1 − ys)1−ζ(1 − y∞)ζ with ζ the fraction of the gap.
The L² law. The level falls by dL/dt = N M/ρl = K/L, so L² = L₀² + 2Kt with K = (c D M/ρl) ln[(1 − y∞)/(1 − ys)]. A plot of L² against t is a straight line whose slope gives D. Because c D ∝ T0.75 and does not depend on P, pressure acts only through ys = psat/P.
Numbers. D at 25 °C in air: water 0.260, ethanol 0.132, acetone 0.105 cm²/s (typical literature values, uncertain by about 10 %), scaled as T1.75/P. Densities are held at their 25 °C values.
Simplifications. Quasi-steady diffusion (the gap changes slowly), isothermal bath (no evaporative cooling of the surface), ideal gas, no wall effects or meniscus curvature, no natural convection in the tube, constant liquid density and D with composition, and the readings in the experiment are synthetic. Above about 45 °C acetone is close to boiling and the model is no longer trustworthy. Models built with Blender, scene by Three.js.