Second Order Rate Constant Calculator
Compute second order rate constant using validated scientific equations. See step-by-step derivations, unit analysis, and reference values.
Reviewed by Manoj Kumar, Mathematics Educator
Formula
1/[A]t = 1/[A]0 + kt
The second-order integrated rate law relates the reciprocal of concentration to time linearly. 1/[A]t is the reciprocal concentration at time t, 1/[A]0 is the reciprocal initial concentration, k is the rate constant, and t is time. The half-life is t1/2 = 1/(k[A]0).
Worked Examples
Example 1: Finding Rate Constant from Concentration Data
Problem:A second-order reaction starts with [A]0 = 0.500 M. After 120 seconds, [A] = 0.200 M. Find k.
Solution:k = (1/[A]t - 1/[A]0) / t\nk = (1/0.200 - 1/0.500) / 120\nk = (5.000 - 2.000) / 120\nk = 0.025 M^-1 s^-1
Result:k = 0.025 M^-1 s^-1, Half-life = 80.0 s
Example 2: NO2 Decomposition
Problem:For 2NO2 -> 2NO + O2 at 300C, k = 0.543 M^-1 s^-1 and [NO2]0 = 0.0100 M. Find [NO2] after 10 seconds.
Solution:1/[A]t = 1/0.0100 + 0.543 * 10\n1/[A]t = 100 + 5.43 = 105.43\n[A]t = 1/105.43 = 0.009485 M
Result:[NO2] = 0.009485 M (5.15% consumed)
Frequently Asked Questions
What is a second-order reaction?
A second-order reaction has a rate that depends on the square of one reactant concentration or the product of two first-order concentrations. The integrated rate law for a second-order reaction in one reactant is 1/[A]t = 1/[A]0 + kt, where [A]t is concentration at time t, [A]0 is initial concentration, and k is the rate constant. A plot of 1/[A] versus time gives a straight line with slope k. Common examples include the dimerization of nitrogen dioxide and the reaction of iodide with persulfate.
How does the half-life of a second-order reaction differ from first-order?
For a second-order reaction, the half-life depends on the initial concentration: t1/2 = 1/(k[A]0). This means each successive half-life is longer than the previous one, as the concentration decreases. In contrast, a first-order reaction has a constant half-life of ln(2)/k regardless of concentration. This concentration-dependent half-life is a key experimental signature of second-order kinetics and explains why these reactions slow down more dramatically over time.
What are the units of a second-order rate constant?
The second-order rate constant k has units of M^-1 s^-1 (or equivalently L mol^-1 s^-1). These units ensure that the rate law rate = k[A]^2 gives a rate in M/s (mol per liter per second). For comparison, first-order rate constants have units of s^-1 and zero-order rate constants have units of M/s. The units of k can be used as a quick check of reaction order when analyzing experimental data. In the integrated rate law, k multiplied by time must have units of M^-1.
How do you identify a second-order reaction experimentally?
To identify second-order kinetics, plot 1/[A] versus time. If the plot is linear, the reaction is second order with respect to A, and the slope equals k. Alternatively, use the method of initial rates: if doubling the concentration quadruples the rate, the reaction is second order. You can also check if the half-life is inversely proportional to initial concentration. Another approach is to compare the fit of zeroth-order, first-order, and second-order integrated rate laws to the experimental data.
References
Reviewed by Manoj Kumar, Mathematics Educator · Editorial policy