Second Order Rate Constant Calculator

For a reaction that is second order in a single reactant, rate = k[A]^2, separating the variables and integrating gives 1/[A]t = 1/[A]0 + kt. Reciprocal concentration is therefore linear in time, with slope k and intercept 1/[A]0, and that linearity is the standard diagnostic for second-order behaviour. The rate constant carries units of per molar per second. The half-life, 1/(k[A]0), depends on where the reaction started, and because each successive half-life begins at half the previous concentration, the half-lives double each time: a second-order reaction decelerates far more sharply than a first-order one, whose half-life is constant. Starting from 0.500 M and reaching 0.200 M in 120 seconds gives k = (5.00 - 2.00)/120 = 0.025 per molar per second and a first half-life of 80 seconds. This integrated form applies to rate = k[A]^2, and by extension to A plus B when the two are mixed at equal initial concentrations. For A plus B at unequal concentrations, integration produces an expression containing the logarithm of the concentration ratio and the reciprocal plot is no longer linear. If one reactant is in large excess the system collapses to pseudo-first-order behaviour instead. A useful reality check on any second-order constant in solution is the diffusion-controlled ceiling, on the order of 10 to the 9th per molar per second for small molecules in water near room temperature, since two molecules cannot react faster than they encounter one another. For scale, nitrogen dioxide decomposing to nitric oxide and oxygen has k near 0.54 per molar per second at 300 degrees C.

Integrated rate laws, the second-order case among them, came out of a collaboration between a chemist and a mathematician. Augustus Vernon Harcourt and William Esson at Oxford studied the reaction of hydrogen peroxide with hydrogen iodide through the 1860s and 1870s. Harcourt supplied careful concentration-time measurements and Esson integrated the corresponding differential equations, producing expressions that could be tested by plotting a transformed concentration against time. Their work established the enduring habit of identifying a reaction order by finding which transformation of concentration yields a straight line. Second-order kinetics later became a decisive tool of physical organic chemistry. In the 1930s Edward Hughes and Christopher Ingold used the contrast between second-order kinetics, with rate proportional to both substrate and nucleophile, and first-order kinetics, with rate independent of the nucleophile, to separate the bimolecular and unimolecular substitution mechanisms now labelled SN2 and SN1. In the gas phase, Max Bodenstein's work on chain reactions rested on bimolecular rate constants, and atmospheric chemistry turned their accurate measurement into a discipline of its own, with IUPAC and NASA Jet Propulsion Laboratory panels publishing periodic evaluations of radical reaction rate constants. Marian Smoluchowski's 1917 treatment of diffusion-controlled encounters supplied the theoretical upper bound that still frames how large a solution-phase second-order constant can plausibly be.