Arrhenius Equation: The Dependence Of Rate Constant On Temperature

We all know that most of the chemical reaction rates increase with an increase in temperature. It has been observed that for every 10oC rise in temperature, the rate of reaction gets doubled. Until 1889, there was no fixed way to relate the variation of the rate of a reaction with temperature. In 1889, the Swedish chemist, Svante Arrhenius proposed an empirical equation that related the rate constant of a chemical reaction with temperature.

Arrhenius Equation: k = \( Ae^{\frac{-E_a}{RT}} \)

Where,
k=rate constant of the reaction
A= Arrhenius Constant
Ea= Activation Energy
R= Universal Gas Constant
T=Temperature in absolute scale

The term Ea is called the activation energy. It is the minimum amount of energy that the reacting molecules must possess so that the effective collisions may result in the formation of products. It is also sometimes referred as the Activation Energy Barrier, and the reacting molecules must cross this barrier to collide effectively and form products. Arrhenius equation is used extensively in an industry by process engineers and reactor designers to model reaction kinetics.

Consider the Arrhenius Equation:

k = \( Ae^{\frac{-E_a}{RT}} \)

If we take to log on both sides of the equation, the resulting equation will be as follows:

\( ln k = ln A – \frac {E_a}{R} \frac{1}{T} \)   ……….. (1)

Now let’s compare this equation with the equation of a straight line that is y=mx+c, where the m= slope of the line and c= y-intercept.

So we have:
\( y = ln k \)
\( x = \frac {1}{T} \)
\( m = \frac {-E_a}{R} \)
\( c = lnA \)

So a plot of ln k on the y-axis and \( \frac {1}{T} \) on x-axis will give a straight line with its slop equal to –Ea/R and y-intercept equal to ln A.

The following figure shows the plot of ln k Vs ( \( \frac {1}{T} \) ).

Arrhenius Equation

Arrhenius Equation plays a very important role in the study of reaction. It gives the dependence of rate constant on temperature for a chemical reaction and provides us with a method to quantify the variation in reaction rate with temperature.

Let us now consider a reaction has activation energy Ea is occurring at a temperature T1 and has a rate constant k1. Assume that the temperature of the reaction is now altered, and it is now T2. No, we have to calculate the rate constant k2 at temperature T2.

As per Arrhenius Equation, for first condition:

\( ln K_1 = ln A – \frac {E_a}{R} \frac{1}{T_1}\) ……………. (2)

For second condition:

\( ln K_2 = ln A – \frac {E_a}{R} \frac{1}{T_2}\) ……………. (3)

Subtracting Equation (2) from equation (3) we will get,

\( ln~ k_2 – ln~ k_1 \)= \(\frac {E_a}{R} \frac{1}{T_1} – \frac {E_a}{R} \frac{1}{T_2} \)

\( \Rightarrow \) \( ln ~\frac {k_2}{k_1}\) = \(\frac {E_a}{R} ( \frac{1}{T_1} – \frac {1}{T_2} ) \)

The above equation can be used to determine the rate constant k2 at temperature T2. Equation (1) is referred as Arrhenius Equation in logarithmic form.

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Practise This Question

For a first order reaction A ⟶ P, the temperature (T) dependent rate constant (k) was found to follow the equation log k = -2000(1/T) + 6.0. The pre-exponential factor A and the activation energy Ea, respectively, are

                                                                                                                                                                                                                           (IIT-JEE, 2009)