Arrhenius Equation and its Derivation | Reaction Kinetics | Chemistry

What is Arrhenius Equation?

Arrhenius equation gives us quantitative relationship, between rate constant ‘k’ temperature ‘T’ and activation energy E_a. Arrhenius equation is as follows

                                              K=Ae^-Ea/RT ---(1)

In this equation,

                K is rate constant

                A is Arrhenius constant

                E_a is activation energy

                R is general gas constant

                T is the absolute temperature

                e is the base of natural log

This equation (1) shows that

1. Rate constant (k) increases with increase in temperature.
2. Rate constant (k) has smaller value for the reaction whose energy of activation is greater.

Determination Of Energy Of Activation From Arrhenius Equation:

We know that in Arrhenius equation is

                                             K=Ae^-Ea/RT

Taking natural ‘log’ on both sides

                                             Lnk = lnA^-Ea/RT

                                             Lnk = lnA + lne^-Ea/RT

                                             Lnk = lnA-Ea/RT+lne ---(2)

Log of quantity equal to the base of log is unity.

                                             Lne = log_e e=1

So, equation (2) becomes

                                             Lnk = lnA-E_a/RT

                                             Lnk = lnA-E_a/RT+lnA ---(3)

Mathematically, natural log and common log can be related to each other as:

                                              Ln = 2.303log

                                              Lnk = 2.303logk

                                              LnA = 2.303logA

So, equation (3) becomes

                                           2.303logk=-E_a/RT+2.303logA

Dividing both sides with 2.303.

                                          Logk = -E_a/2.303RT+logA

                                          Logk = -E_a/2.303R(1/T) + logA ---(4)

This equation (4) is equation of straight line y = mx + c

       Where,                     y = log k

       Slope,                      m = -E_a­/2.303R

                                          X = 1/T

        &

                                           C = logA

Where ‘m’ is the slope of a straight line and c is an intercept of a straight line. Temperature is an independent variable while rate constant is the dependent variable and other quantities. E_a, and A are constant.

So, the graph is plotted between 1/T on the x-axis and log k on the y-axis and a straight line is obtained with a negative slope. The graph is shown in the figure below.

Tanθ = - slope of straight line (a negative quantity, because θ degree is greater than 90 degree, an tan of angle more than 90 degree is negative)

The slope of this straight line is measured by taking the tan of the angle ‘θ’ which the straight line makes with x-axis. To measure the slope, draw a line parallel to x-axis and measure the angle ‘θ’. Take tanθ which is the slope and this slope is equal to

                                Slope = -E_a/2.303R---(5)

Therefore,

                                E_a­ = -slope × 2.303 R ---(6)

The straight line of different reactions will have a different slope and different E_a values. The unit of the slope is kelvin:

From equation (v), the unit of slope can be determined as:

                              Slope = Jmol^-1/2.303JK^-1mol^-1

                              Slope = K