Arrhenius & Eyring Calculations
Plot of ln k vs 1/T
The first line of input should be the name of the time units (e.g, hours), followed by a comma and the number of seconds per time unit (e.g., 3600 for hours). On separate lines, enter the temperature (in Celsius) and the associated rate constant, separated by commas, spaces or tabs. Entering the time units is only important if you are interested in the activation entropy; it has no bearing on the activation energy.
Press the calculate button. The routine fits the data to the Arrhenius equation as a straight line with y values of ln k and x values of 1/T:
ln k = A + ΔE* / R T
The correlation coefficient is calculated providing a measure of how linear the relationship is; the closer the correlation coefficient is to 1.0, the more linear the relationship. A is the intercept or pre-exponential factor, and ΔE*/R is the slope. The slope is multiplied by the gas constant (R = 8.314 J/K.mol) to give the negative of the activation energy, ΔE*. Typical values for the activation energy run from 20 to 150 kJ/mol. Low values are often observed for oxidation reactions, while higher values are seen for hydrolyses. Higher values of the activation energy also indicate a greater temperature dependence of the reaction rate.
The pre-exponential factor, A, is used to calculate the activation entropy, ΔS*, from the following equation:
A = e(ΔS*/R)+1 kB T / h
where kB is the Boltzmann constant (1.38 x 10-23 J/K), h is Planck's constant (6.626 x 10-34 J s) and T is the average temperature of the data.
An equation is provided that allows the reaction rate constant to be calculated at a specified temperature. From the reaction rate constant, the concentration may be calculated assuming a first-order decrease with time.
The Arrhenius calculation assumes that the pre-exponential factor, A, is constant, whereas it is actually slightly dependent upon temperature. Based on transition state theory, the Eyring equation can also be used to analyze kinetic data:
ln(k h/kB T) = ΔS*/R - ΔH*/RT
Plotting ln(k h/kB T) versus 1/T gives a line with slope of -ΔH*/R and intercept of ΔS*/R.
As a check the difference between the activation energy and enthalpy should be equal to RT.
For first-order reactions, the units of the rate constants should be in terms of inverse time. For second-order reactions, the units of the rate constants should be in terms of inverse molarity-time in order for the calculations to work out.
An example is given from The Physical Basis of Organic Chemistry by Howard Maskill (1990 edition), p. 232, using data from GR Brandon et al., Trans Faraday Soc 62, 1546 (1966) for the first-order gas-phase isomerization of bicyclo[4.2.0]oct-7-ene. The temperatures are given in Celsius and the rate constants in inverse seconds.
The calculated activation energy is 181 kJ/mol, the activation entropy is 12.4 J/K.mol, and the activation enthalpy is 176 kJ/mol.
Thanks to Professor Richard L Schowen for pointing out an error in an earlier version of this page. Of course, any mistakes in the current version are strictly my own responsibility.
Jeffrey Clymer -- Debut: April 13, 1997. Revision 8: December 30, 2015 -- Email