the parabolic (Jander) rate law
/ Jander = YAHN-der /
When a reaction or an oxidation builds a product layer that the reactants must then diffuse across, the layer becomes its own brake. Early on it is thin and easy to cross, so the reaction is fast; as the layer thickens, atoms must travel farther to keep it growing, so the reaction slows down. The parabolic rate law is the mathematical shape of that self-slowing growth: the layer thickness grows not in proportion to time but as the square root of time.
The reasoning is pure diffusion. If a product layer of thickness x grows only as fast as ions can diffuse across it, then its growth rate dx/dt is inversely proportional to x (thicker layer, slower crossing). Integrating gives x^2 proportional to time, i.e. x = sqrt(k times t) — the parabolic law, the direct kinetic cousin of the sqrt(D times t) rule. Plot the square of the layer thickness (or, in Jander's version for reacting powder spheres, the square of a reacted-fraction term) against time and you get a straight line whose slope contains the diffusion coefficient. Jander adapted this to spherical powder particles reacting from the outside in, giving the widely used Jander equation for the fraction reacted versus time; it is a simplified model, accurate only while the product layer is thin compared with the particle.
A parabolic (square-root-of-time) growth is the classic signature that a solid-state process is diffusion-controlled — it is seen in the growth of a spinel reaction layer, in the protective oxide scale on a hot metal or on SiC (whose slow-growing SiO2 scale gives it excellent oxidation resistance), and in many powder reactions. The practical message: because progress slows as sqrt(t), you get rapidly diminishing returns from simply holding longer — doubling the reacted layer needs four times the time. To go faster you must raise the diffusion coefficient (fire hotter) or shorten the distance (finer powder). Jander's own honest limits: it assumes constant D, a dense adherent product, and thin layers, so it fails for coarse particles or late in the reaction, where more elaborate models (Ginstling-Brounshtein) do better.
A spinel product layer between MgO and Al2O3 thickens as sqrt(time): early growth is quick, but once the layer is thick, Mg2+ and Al3+ must diffuse ever farther across it, so plotting thickness-squared against time gives a straight line — the fingerprint of diffusion control.
Parabolic (Jander) law: a diffusion-grown product layer thickens as the square root of time (x^2 proportional to t), so growth slows as the barrier it must cross gets thicker.
Parabolic kinetics is the hallmark of diffusion control, but the Jander form is a thin-layer approximation for spherical particles. It breaks down for coarse powders and at high conversion, where the shrinking core's geometry matters and the Ginstling-Brounshtein or other models are more accurate.