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-Ea/RT
for a particular reaction
A, the preexponential factor, is constant
R is the ideal gas constant
Ea, the activation energy, is constant
The expression at the right 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 (7000271828182845899♠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. This suggests a graphical method for determining the activation energy for a reaction, if the rate constant at several temperatures is known
| ln k | = | – | Ea | × | 1 | + ln A |
| R | T |
| y | = | | m | | x | + c |
Because there is a linear relationship between ln
k and 1/
T, a plot of
ln k vs 1/T is a straight line with the slope –
Ea/
R.
Thus the activation energy can be calculated by multiplying the slope by –
R.
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.
| ln k2 | = | - | Ea | × | 1 | + ln A |
| R | T2 |
minus | ln k1 | = | - | Ea | × | 1 | + ln A |
| R | T1 |
equals | ln | k2 | = | - | Ea | ( | 1 | - | 1 | ) |
| k1 | R | T2 | T1 |
This relationship is a convenient way
- to calculate Ea from k at two different temperatures
- to determine k at a second temperature from k at one temperature and Ea
Recall that temperature in Kelvin MUST be used (K = °C + 273.15).
R is 8.314 J mol–1 K–1; therefore Ea must be used in joules (it is commonly given in kJ).
Calculated Ea are in joules. Ea are usually reported in kJ.