Rate Constant Calculator

The Arrhenius equation, term by term

The rate constant of an elementary step follows k = A x exp(-Ea / (R x T)). A is the pre-exponential factor, sometimes called the frequency factor, and it bundles collision frequency and the orientation requirement into one number with the same units as k. Ea is the activation energy per mole of reactant, R is the molar gas constant, and T is the absolute temperature. The exponential term is the fraction of collisions whose energy clears the barrier, so it always sits between 0 and 1 and it is the reason rate constants change so steeply with temperature.

IUPAC defines the Arrhenius activation energy as an empirical parameter equal to R T squared times the derivative of ln k with respect to T, which is the slope of the Arrhenius plot multiplied by R (IUPAC Gold Book, activation energy). That definition matters because Ea is measured, not derived: it is whatever slope the data produce. For a single elementary step it sits close to the height of the energy barrier, and for a multi step reaction it is a composite of the steps and can even come out negative when the mechanism changes with temperature.

Taking logs turns the equation into a straight line, ln k = ln A - Ea/(R T), so a plot of ln k against 1/T has slope -Ea/R and intercept ln A. Every worked example below can be read off that line, and the calculator returns the exponent Ea/(R T) so you can see how large the barrier is compared with thermal energy.

Units for A, k and R

A carries the units of k, because the exponential term is dimensionless. Those units depend on the reaction order, and getting them wrong is the most common source of a factor of 1,000 in a kinetics homework answer.

R is 8.314462618 J mol^-1 K^-1, exact since the 2019 SI revision fixed the Boltzmann and Avogadro constants. NIST CODATA lists the value as 8.314 462 618... J mol^-1 K^-1 with zero uncertainty (NIST CODATA, molar gas constant). Because R is in joules, Ea must be in joules per mole inside the exponential. Enter 100 kJ/mol as 100,000 J/mol, or switch the unit menu to kcal/mol, where 1 kcal = 4.184 kJ exactly.

The temperature is the other trap. The equation needs kelvin, so 25 degrees C is 298.15 K, and a value typed in Celsius without conversion shifts the exponent by several percent. The calculator converts automatically and prints the kelvin value it used.

Worked example: k from A and Ea at 25 degrees C

Take a first order reaction with A = 1.0 x 10^13 s^-1 and Ea = 100 kJ/mol at 298.15 K. The exponent is the whole calculation, and it is large enough that the answer is tiny.

An exponent of 40.3 means the barrier is about 40 times the thermal energy R T at room temperature, and only about one collision in 3 x 10^17 clears it. That is why the rate constant is 3 x 10^-5 s^-1 rather than something near A: the exponential, not the prefactor, sets the scale. A reaction with that k has a half life of 6.4 hours, which is the sort of number that makes a room temperature reaction convenient to follow with a stopwatch or a spectrophotometer.

Worked example: two rate constants give Ea and A

Measurements usually arrive as k at two temperatures. Divide the Arrhenius equation at T2 by the same equation at T1, and A cancels, leaving ln(k2/k1) = (Ea/R)(1/T1 - 1/T2). Take k1 = 1.0 x 10^-4 s^-1 at 300.0 K and k2 = 4.0 x 10^-4 s^-1 at 320.0 K.

The check row returns the measured 4.0 x 10^-4 s^-1, which confirms the arithmetic. The same two points give a rate increase of 4.0 times across the 20 K span, which normalises to a Q10 of 2.0. When the two temperatures are close together the estimate is sensitive to measurement error, so labs fit a line through five or more points instead of trusting two.

How fast k climbs with temperature

The exponential makes the temperature sensitivity depend on the barrier. The table holds A at 1.0 x 10^13 s^-1 and varies Ea and T, and the spread is enormous.

A 50 kJ/mol barrier lets the rate constant reach 2.957 x 10^6 s^-1 at 400 K, while a 150 kJ/mol barrier is still at 2.585 x 10^-7 s^-1 at the same temperature. The gap between those two rows is nine orders of magnitude, and it comes entirely from the exponential. Q10 captures the local slope: from 298.15 K to 308.15 K with a 100 kJ/mol barrier, k rises from 3.025 x 10^-5 to 1.120 x 10^-4, a factor of 3.70. That is the origin of the rule that rates double or triple for every 10 K near room temperature, and the rule is only a rule because most organic reactions carry barriers in the 50 to 100 kJ/mol band.

Reliability engineering uses the same exponential with the barrier expressed per molecule in electronvolts. The NIST/SEMATECH e-Handbook gives the acceleration factor between two temperatures as exp[(Ea/k)(1/T1 - 1/T2)], and notes that a 0.5 eV barrier produces an acceleration factor of 133 between 25 and 125 degrees C (NIST/SEMATECH e-Handbook, section 8.1.5.1). Run 0.5 eV through the same arithmetic here and the factor comes out at 133, which is the same number.

First order half life from k

For a first order reaction the half life depends on k alone: t(1/2) = ln(2)/k = 0.693147/k. Nothing else enters, because the fraction of material that reacts in a given time does not depend on how much is there. That is why first order half lives are such a useful handle on a rate constant, and why the calculator prints one next to k.

With k = 3.025 x 10^-5 s^-1 the half life is 22,912 s, which is 6.36 hours. Raise the temperature to 400 K with the same A and Ea and k becomes 8.742 x 10^-1 s^-1, so the half life collapses to 0.79 s. A faster reaction is not a slightly faster reaction: it is a different experiment. For comparison, k = 1.5 x 10^-3 s^-1 gives a half life of 462.1 s, or 7.70 minutes, which is the range of a classic iodine clock demonstration.

Zero and second order half lives need the starting concentration, so the calculator asks for it when you pick those orders. Getting the order wrong changes the predicted half life by a factor that grows as the reaction proceeds, which is one reason a plot of concentration against time is worth the effort before any Arrhenius analysis.

Where the Arrhenius fit bends

The two parameter form is a model, and it fails in ways that are predictable enough to check.

First, A is not strictly constant. Transition state theory gives a prefactor proportional to T, and the three parameter form k = A T^n exp(-Ea/RT) fits that dependence explicitly. Over a 50 K window the difference is small, but extrapolate 200 K and a curved Arrhenius plot will hand you a biased Ea. Second, a change of mechanism bends the plot: diffusion control at high temperature or a switch of rate determining step shows up as two slopes, and a single fit through both returns a number that describes neither. Third, reactions near a catalyst surface or inside a polymer above its glass transition can slow down as temperature rises, which appears as a negative apparent Ea. Enzymes do the same past their optimum, where denaturation outruns the rate increase.

The practical rule is to fit inside the measured window and to treat any extrapolation as a guess. For measured A and Ea values on gas phase reactions, the NIST Chemical Kinetics Database collects the literature and stores the three parameter form k = A (T/298)^n exp[-(Ea/R)/T] for each entry (NIST Chemical Kinetics Database, SRD 17). Comparing your fitted A and Ea with values for the same reaction family is a quick way to spot a unit error or a stray temperature in Celsius.