Arrhenius Equation

The Arrhenius equation helps in providing a relationship between the rate constant of a given chemical reaction, the pre-exponential factor, A, and the absolute temperature. It also provides an insight into how the reaction rates are dependent on the absolute temperature. The Arrhenius equation is given by the formula as follows:

\[k = Ae^{\frac{-Ea}{RT}}\]

Here,

k is the rate constant of the reaction.

A refers to the pre-exponential factor which is the frequency of the properly oriented collisions between the species that are reacting.

e refers to the base of the natural logarithm, also called Euler’s number.

Ea is the activation energy of the chemical reaction (denoted by energy per mole).

R is the universal gas constant.

T refers to the absolute temperature which is related to the reaction (denoted in Kelvin).

In case the activation energy is represented in regards to the energy per the reactant molecule, then the universal gas constant is to be replaced with another constant in the Arrhenius equation known as the Boltzmann constant, kB.

In this article, we will learn about what is Arrhenius equation chemical kinetics, what is the Arrhenius equation derivation, and how to use the Arrhenius equation to determine the energy of activation.

Arrhenius Equation Graph Representation

When the nitrogen dioxide undergoes a decomposition reaction, a graph can be plotted having the rate constant (denoted by k) plotted on the Y-axis and the absolute temperature T plotted on the X-axis. You must know that the reaction rate tends to increase as you increase the temperature.

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Arrhenius Plot

When we take logarithms are on both the sides of the equation, the Arrhenius equation is written in the following manner:

\[\text{ln k = ln}(Ae^{\frac{-Ea}{RT}})\]

When we solve the equation further,

\[\text{ln k = ln(A) + ln}(e^{\frac{-Ea}{RT}})\]

\[\text{ln k = ln(A) + } (\frac{-Ea}{RT}) = ln(A) - (\frac{Ea}{RT}) = ln(A) - (\frac{Ea}{R}) (\frac{1}{T})\]

The equation here is a straight line, y = mx + c because ln(A) is constant and has a slope denoted by m = -EaR.

When we plot the log of the rate constant, ln K on the Y-axis and the absolute temperature inverse (1T) on the X-axis, the graph that results is referred to as the Arrhenius plot.

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Does Arrhenius Equation Account for the Catalysts?

Catalyst helps in lowering the activation energy which is needed by a particular reaction. Hence, the lowered activation energy which is accounted for the catalysts is substituted in the Arrhenius equation for getting the rate constant of the catalyzed reaction.

The exponential portion of Arrhenius equation, denoted by (-EaRT)results in an exponential increase in the rate constant value when the activation energy decreases. As we know that the rate of the chemical reaction is proportional directly to the rate constant of that particular reaction when there is a decrease in the activation energy, the reaction rate increases exponentially.

You must know that the rates of the reactions that are uncatalysed are affected more severely by the temperature when compared to the rates of the catalysed reactions. This is due to the reason that the activation energy lies in the numerator of the exponential term and the absolute temperature lies in the denominator of the Arrhenius equation. The activation energy of the catalysed reaction is lower comparatively, which makes the effect of the temperature on rate constant much more visible when compared to the uncatalysed reaction.

Arrhenius Equation and Pre-Exponential Factor (A)

The Arrhenius equation has a symbol A which denotes the pre-exponential factor, also called the frequency factor. This factor tends to deal with the collisions that happen between the molecules and is thought of as the frequency of the correctly oriented collisions that occur between the molecules which have sufficient energy for sparking a chemical reaction.

The pre-exponential factor is represented by the given equation:

A = ⍴Z

Wherein, Z is known as the frequency factor or the frequency of collisions and ⍴ is known as the steric factor, which tends to deal with the orientation of the molecules.

The value of A is to be determined experimentally because it tends to assume several different values for several different reactions. It also depends on the temperature wherein the reaction takes place.