The Arrhenius equation

The rate of most reactions changes with temperature because the rate constant k varies with temperature. Svante Arrhenius established experimentally the relationship between rate constant k and the temperature (T) in Kelvin shown at the left below.

k = Ae^-E_a/RT
for a particular reaction
A, the preexponential factor, is constant
R is the ideal gas constant (8.314 J mol^–1 K^–1)
E_a_, the activation energy, is constant

The expression below results from taking the natural logarithm (The logarithm of a number is the power to which a fixed base must be raised to give that number. log is the logarithm to base 10. log 100 (10²) is 2. ln is the natural logarithm to base e (2.718). ln 100 is 4.61. Natural logarithms are larger than logarithms to the base 10 because the natural logarithm base is smaller.) of both sides of the Arrhenius equation.

ln k = ln A + ln (e^-E_a/RT)

ln k	=	–	E_a	×	1	+ ln A
			R		T

ln k₂	=	-	E_a	×	1	+ ln A
			R		T₂

minus

ln k₁	=	-	E_a	×	1	+ ln A
			R		T₁

equals

ln	k₂	=	-	E_a	(	1	-	1	)
	k₁			R		T₂		T₁

A form of the Arrhenius equation in which A does not appear can be derived by taking the difference between the logarithmic form at two different temperatures.

This relationship is a convenient way

to calculate E_a from k at two different temperatures
to determine k at a second temperature from k at one temperature and E_a

Important information about units in this equation:
R is 8.314 J mol^–1 K^–1(gas constant)
K is temperature in Kelvin (K = °C + 273.15)

Units of logarithms:
This has no unit because logarithms are always simple numbers.

Units of E_a and T depend on the unit of R:
Because the left-hand side of the equation is a simple number, for the two sides to be equal the units of the quantities on the right hand side must cancel when they are combined as shown.
This means that

the temperature used in the calculation must be in Kelvin so that the temperature unit in the gas constant cancels.
the E_a calculated will have units J mol^–1
!!!Energies of activation are big numbers and are usually reported in kJ mol^–1.
This means that the calculated energy of activation must be adjusted by a factor of 1000.!!!