Arrhenius Equation

Introduction

Juxtapose this with the scenario of cooking your favorite dish. It recommends a certain temperature and period for cooking. Now, what if you change the temperature? You are to see that the rate, by which your food gets cooked, will either become just right or becomes burnt or half-cooked. To put it in especial analogy with the intricacy of the relationship between temperature and the rate of any chemical reaction, delightfully explained by the Arrhenius equation, such occurrences do happen every day. Svante Arrhenius lent his name to this very basic equation that describes the increase of reaction rates with temperature, which became one of the cardinal principles in areas as vast as industrial chemistry and biochemistry. We will be discussing the Arrhenius equation in this post—an equation that speaks of its mathematical underpinnings and deep implications for how we understand chemical kinetics. We will consider the correct dependence of the rate constant on temperature, ratio of rate constants at different temperatures, exceptions to Arrhenius theory, mechanisms of complex reactions and more.

Main Concept: Definitions and Explanations

The Arrhenius Equation is a mathematical expression that describes the variation of temperature with the rate constant (k) of a chemical reaction. Mathematically, it has been expressed as:

$k = A e^{-\frac{E_a}{RT}}$

where ( k ) is the rate constant, ( A ) is the pre-exponential factor, $( E_a )$ is the activation energy, ( R ) is the gas constant, and ( T ) is the temperature in Kelvin. It should be noted that with increasing temperature, the exponential term $( e^{-\frac{E_a}{RT}} )$ also increases, hence increasing the rate constant ( k ). This reinforces that a higher temperature will provide additional energy to reactant molecules, which can easily overrun activation energy.

We know that on increasing the temperature, the rate of the reaction or rate constant increases.
The rate equation is given as follows:

Rate $=\mathrm{k}[\text { conc }]^{\mathrm{n}}$

Here k is the rate constant

Now, we will see the relation between k and T also known as the 'approximate dependency of k on T'.

Generally on 10^oC rise in temperature, the rate constant nearly doubles.

Temperature Coefficient: It is the ratio of two rate constants. Thus, mathematically it is given as:

$\mathrm{T}_{\text {coeff }}=\frac{\mathrm{k}_{(\mathrm{t}+10)^{\circ} \mathrm{C}}}{\mathrm{k}_{\mathrm{t}^{\circ} \mathrm{C}}}$

Thus, the temperature coefficient is showing the dependency of the rate constant(k) on temperature(T).

NOTE: The standard value of the temperature coefficient is given at t = 25^oC and (t+10) = 35^oC.

Other Faces of the Concept

1. Effect of Temperature on Rate of Reaction: Correct Dependence of K on T
It should be expected that, with increasing temperature, the rate of a chemical reaction increases because of an increase in the kinetic energy of molecules. This dependence is quantitatively articulated by the Arrhenius equation and therefore makes a balanced predictive model of reaction rates at different temperatures.

The temperature dependence of the rate of a chemical reaction can be accurately explained by the Arrhenius equation. It was first proposed by Dutch chemist, J.H. van’t Hoff but Swedish chemist, Arrhenius provided its physical justification and interpretation.

$\mathrm{k}=\mathrm{Ae}^{-\mathrm{Ea} / \mathrm{RT}}$

where A is the Arrhenius factor or the frequency factor. It is also called the pre-exponential factor. It is a constant specific to a particular reaction. R is gas constant and E_a is activation energy measured in joules/mole(J mol^–1).

2. Ratio of Two Rate Constants at Two Different Temperatures
The Arrhenius Equation can also be used to demonstrate the ratio of rate constants at two different temperatures, which are relevant for comparing how the speed of a reaction changes with changes in temperature: $\frac{k_2}{k_1} = e^{\frac{E_a}{R} (\frac{1}{T_1} - \frac{1}{T_2} )}$

3. Exeption (Arrhenius Theory)
None of the foregoing can negate the fact that, generally, many reactions do not go according to the predictions of the Arrhenius Equation. Most of these were involved with complex mechanisms of reaction or significant changes in reaction pathways due to an increased decrease in temperature.

We have the rate constant K₁ at temperature T₁ and the rate constant K₂ at temperature T₂.

We know that the Arrhenius equation is given as follows:

$
\begin{aligned}
& \log _{10} \mathrm{~K}_1=\log _{10} \mathrm{~A}-\frac{\mathrm{E}_{\mathrm{a}}}{2.303 \mathrm{RT}_1} \\
& \log _{10} \mathrm{~K}_2=\log _{10} \mathrm{~A}-\frac{\mathrm{E}_{\mathrm{a}}}{2.303 \mathrm{RT}_2}
\end{aligned}
$

On subtracting equation (i) from (ii), we get:
$
\log _{10} \mathrm{~K}_2-\log _{10} \mathrm{~K}_1=\frac{\mathrm{E}_{\mathrm{a}}}{2.303 \mathrm{RT}_1}-\frac{\mathrm{E}_{\mathrm{a}}}{2.303 \mathrm{RT}_2}
$

Thus, $\log \frac{\mathrm{K}_2}{\mathrm{~K}_1}=\frac{\mathrm{Ea}}{2.303 \mathrm{R}}\left[\frac{1}{\mathrm{~T}_1}-\frac{1}{\mathrm{~T}_2}\right]$

Although the Arrhenius equation explains the exact relationship between the rate of reaction and the temperature but there are still some exceptions in this theory. Actually, on increasing the temperature rate may decrease sometimes and may not follow the Arrhenius equation. The following examples will illustrate these exceptions.

• Bacterial decomposition
  
  From the graph, it is clear that first on increasing the temperature the rate also increases but at T_i (also known as inversion temperature) rate starts to decrease on further increasing the temperature.

Relevance and Applications

1. Complex Reaction - Mechanism of Reaction
Knowing the Arrhenius Equation allows one to infer mechanisms of complex reactions. In many-step reactions, for instance, understanding the temperature dependence of each elementary step can help in finding the overall rate of reaction.

2. Complex Reaction (When Intermediate is Incorporated)
In the presence of intermediates, the Arrhenius equation will help in estimating the rates of formation and consumption of these transient species, which are relevant for optimization in reaction conditions in industrial processes.

On the basis of mechanism, we have two types of reactions:

• Simple or elementary reaction
• Complex or multi-step reaction

Simple or Elementary reaction

• The reactions, which occur in a single step, are called simple or elementary reactions. For example:
  
  $\mathrm{NO}(\mathrm{g})+\mathrm{O}_3(\mathrm{~g}) \rightarrow \mathrm{NO}_2(\mathrm{~g})+\mathrm{O}_2(\mathrm{~g})$
  An elementary reaction is an individual molecular event that involves the breaking or making of chemical bonds. The overall reaction describes the stoichiometry of the overall process but provides no information on how the reaction occurs.

Complex Reaction

• A complex reaction takes place in a sequence of a number of elementary steps.
• The molecularity of complex reactions is not defined. The molecularity of each step can be defined but not for overall.
• The overall rate of reaction is given by the slowest step of the complex reaction.
  For example, a combination of NO₂ and CO occurs in a sequence of elementary steps.
  $\begin{array}{ll}\mathrm{NO}_2(\mathrm{~g})+\mathrm{NO}_2(\mathrm{~g}) \rightarrow \mathrm{NO}_3(\mathrm{~g})+\mathrm{NO}(\mathrm{g}) & \text { (elementary step I) } \\ \mathrm{NO}_3(\mathrm{~g})+\mathrm{CO}(\mathrm{g}) \rightarrow \mathrm{NO}_2(\mathrm{~g})+\mathrm{CO}_2(\mathrm{~g}) & \text { (elementary step II) }\end{array}$

Important Facts:

• The number of reactant molecules taking part in an elementary step or in an elementary reaction is expressed as the molecularity of that step of the molecularity of that reaction respectively.
• For elementary reactions usually, the order of reaction and molecularity are the same. Thus, it can be said that if the order of a reaction for a change is fractional it cannot be an elementary reaction.

This example will illustrate how to determine the rate law when the intermediate is involved in the rate-determining step.

$2 \mathrm{O}_3 \rightarrow 3 \mathrm{O}_2$

Mechanism
$\mathrm{O}_3 \rightleftharpoons \mathrm{O}_2+\mathrm{O}$
$\mathrm{K}_1$ and $\mathrm{K}_2$ are the forward and backward reaction constants respectively
$\mathrm{O}+\mathrm{O}_3 \rightarrow 2 \mathrm{O}_2$
$\mathrm{K}_3$ is the rate constant

In this case, step 1 is fast and step 2 is slow.

The rate law is given as follows:

rate $=\mathrm{K}_3[\mathrm{O}]^1\left[\mathrm{O}_3\right]^1$

We know from equilibrium theory that:
$
\begin{aligned}
\mathrm{K}_{\mathrm{eq}} & =\frac{\mathrm{K}_1}{\mathrm{~K}_2}=\frac{\left[\mathrm{O}_2\right][\mathrm{O}]}{\left[\mathrm{O}_3\right]} \\
{[\mathrm{O}] } & =\frac{\mathrm{K}_1\left[\mathrm{O}_3\right]}{\mathrm{K}_2\left[\mathrm{O}_2\right]}
\end{aligned}
$

Thus, rate $=\frac{\mathrm{K}_3 \cdot \mathrm{K}_1}{\mathrm{~K}_2} \frac{\left[\mathrm{O}_3\right]\left[\mathrm{O}_3\right]}{\left[\mathrm{O}_2\right]}=\frac{\mathrm{K}_3 \cdot \mathrm{K}_1}{\mathrm{~K}_2} \frac{\left[\mathrm{O}_3\right]^2}{\left[\mathrm{O}_2\right]}$
Thus, Order $=2-1=1$

3. Parallel First Order Kinetics
If parallel reaction routes prevail, then, using the Arrhenius equation, one is in a position to separate, as a function of temperature, the dependences of the different pathways involved so that better control of the desired reaction route may be obtained.

In this situation, B and C both are forming. These types of reactions are known as parallel reactions. Both these reactions are first-order reactions with rate constants K₁ and K₂ respectively and half-lives as t_(1/2)1 and t_(1/2)2.

For these parallel reactions, we need to find:

• Effective order
• Effective rate constant
• Effective t_1/2
• Effective Activation energy
• [A], [B], [C] with time (t) variation
• % of [B] and % of [C]

We know that the rate equations are given as follows:
$
\begin{aligned}
& \mathrm{r}_1=\frac{-\mathrm{dA}}{\mathrm{dt}}=\mathrm{K}_1[\mathrm{~A}] \\
& \mathrm{r}_2=\frac{-\mathrm{dA}}{\mathrm{dt}}=\mathrm{K}_2[\mathrm{~A}]
\end{aligned}
$

Thus, the overall rate of reaction is :
$
\frac{-\mathrm{dA}}{\mathrm{dt}}=\mathrm{K}_1[\mathrm{~A}]+\mathrm{K}_2[\mathrm{~A}]=\left(\mathrm{K}_1+\mathrm{K}_2\right)[\mathrm{A}]
$

Thus, rate $=\left(\mathrm{K}_1+\mathrm{K}_2\right)[\mathrm{A}]^1$
Effective Rate Constant $\left(\mathrm{K}_{\mathrm{eff}}\right)=\left(\mathrm{K}_1+\mathrm{K}_2\right)$

Effective order of reaction $=1$
Now, effective half - life $\left(\mathrm{t}_{1 / 2}\right)=\frac{0.693}{\mathrm{~K}_{\mathrm{eff}}}=\frac{0.693}{\mathrm{~K}_1+\mathrm{K}_2}$
$
\Rightarrow \frac{0.693}{\frac{0.693}{\left(t_1 / 2\right)_1}+\frac{0.693}{\left(t_1 / 2\right)_2}}
$

Thus, effective half life is given as :
$
\frac{1}{\left(t_{1 / 2}\right)_{\text {eff }}}=\frac{1}{\left(t_{1 / 2}\right)_1}+\frac{1}{\left(t_{1 / 2}\right)_2}
$

NOTE: Effective activation energy, [A], [B], [C] with time (t) variation and % of [B] and % of [C] will be discussed in later concepts.

4. Effective Activation Energy
The concept of effective activation energy follows from the Arrhenius Equation and gives information about the general height of the energy barrier in many-step reactions. This hence forms a very important parameter in the design of catalysts or optimization of reaction conditions.

We know that the Arrhenius equation is given as:
$
\begin{aligned}
& \mathrm{K}=\mathrm{A} \cdot \mathrm{e}^{-\mathrm{Ea} / \mathrm{RT}} \\
& \mathrm{K}_{\mathrm{eff}}=\mathrm{K}_1+\mathrm{K}_2
\end{aligned}
$

Thus, $\mathrm{A}_{\text {eff }} \cdot \mathrm{e}^{-\mathrm{Ea}_{\text {eff }} / \mathrm{RT}}=\mathrm{A}_1 \cdot \mathrm{e}^{-\mathrm{Ea}_1 / \mathrm{RT}}+\mathrm{A}_2 \cdot \mathrm{e}^{-\mathrm{Ea}_2 / \mathrm{RT}}$
Differentiate this equation with respect to temperature ' $\mathrm{T}^{\prime}$
Thus, we have :
$
\begin{aligned}
& \mathrm{A}_{\mathrm{eff}} \cdot \mathrm{e}^{-\mathrm{Ea}_{\mathrm{eff}} / \mathrm{RT}}\left(\frac{+\mathrm{Ea}_{\mathrm{eff}}}{\mathrm{RT}^2}\right)=\mathrm{A}_1 \cdot \mathrm{e}^{-\mathrm{Ea}_1 / \mathrm{RT}}\left(\frac{+\mathrm{Ea}_1}{\mathrm{RT}^2}\right)+\mathrm{A}_2 \cdot \mathrm{e}^{-\mathrm{Ea}_2 / \mathrm{RT}}\left(\frac{+\mathrm{Ea}_2}{\mathrm{RT}^2}\right) \\
& \mathrm{K}_{\mathrm{eff}} \mathrm{E}_{\mathrm{eff}}=\mathrm{K}_1 \mathrm{Ea}_1+\mathrm{K}_2 \mathrm{Ea}_2 \\
& \mathrm{E}_{\text {eff }}=\left(\mathrm{K}_1 \mathrm{Ea}_1+\mathrm{K}_2 \mathrm{Ea}_2\right) / \mathrm{K}_{\mathrm{eff}} \\
&
\end{aligned}
$

Recommended topic video on ( Arrhenius Equation)

Some Solved Examples

Example 1

Question: The specific rate constant (k) of a first-order reaction depends on which of the following?

1. Time
2. Concentration of reactant
3. Concentration of product
4. Temperature (correct)

Solution:

A first-order reaction's specific rate constant (k) depends on temperature. The reaction rate or rate constant increases with a rise in temperature. For example, generally, on a 10ºC rise in temperature, the rate constant nearly doubles. The temperature coefficient, which is the ratio of two rate constants, mathematically shows this dependency:

$(Tcoeff=k(t+10)^oCkt^oCT_{coeff} = \frac{k_{(t + 10)^{o}C}}{k_{t^{o}C}}Tcoeff=kt^oCk(t+10)^oC)$

Hence, the correct answer is option (4).

Example 2

Question: For a particular reaction, the temperature coefficient is 2. If the rate of the reaction at 40ºC is x, what is the rate at 80ºC?

1. 12x
2. 4x
3. 8x
4. 16x (correct)

Solution: The rate of reaction increases with an increase in temperature. For every 10ºC rise in temperature, the rate constant doubles. Therefore, starting from 40ºC to 80ºC (a 40ºC increase), the rate can be calculated as follows:

Temperature (ºC)40,50,60,70,80,

Ratex,2x,4x,8x,16x

Thus, the rate at 80ºC will be 16x. The correct answer is option (4).

Example 3

Question: Which of the following statements are in accordance with the Arrhenius equation?

1. Rate of reaction increases with an increase in temperature (correct)
2. Rate of reaction increases with a decrease in activation energy
3. Rate constant decreases exponentially with an increase in temperature
4. Rate of reaction decreases with a decrease in activation energy

Solution: According to the Arrhenius equation, the rate of a reaction increases with an increase in temperature. The equation is:

k = $A e^{-E_{a}RT}k$

where k is the rate constant, A is the Arrhenius factor or pre-exponential factor, Ea is the activation energy, Ri is the gas constant, and T is the temperature. This equation shows that the rate constant increases with an increase in temperature, which leads to an increase in the reaction rate. Hence, the correct answer is option (1).

Conclusion

The Arrhenius equation is one of the cornerstones of chemical kinetics. It describes a deep connection between temperature and reaction rate. A quantitative framework for that ability is important, not just for predicting reaction rates but also for optimizing reaction conditions and uncovering intricacies relative to mechanisms of reactions in many industrial processes. From the laboratory to everyday situations like cooking, understanding and controlling the power of chemical reactions would have been impossible without principles reflected in the Arrhenius equation.