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Arrhenius Equation-Mathematical Expression, Plot, Theory, Applications, Practice problems and FAQs
When you bring fresh vegetables or fruits from the market, your mother always washes them and arranges them in a fridge. What if she forgets to keep them in the fridge? Vegetables get rotten until the next day. Why is this so?
Does temperature have any effect on the rotting of vegetables? Clearly, the temperature in the fridge is lower than that of room temperature. Actually, increasing the temperature speeds up the chemical reaction of rotting. Does temperature have the same effect on other chemical reactions as well? The answer to this question was given by Svante August Arrhenius. He gave the mathematical relation that explains the dependence of the rate of a reaction on temperature which is known as the Arrhenius equation. Let us study more in detail about the Arrhenius equation.
Table of content
- Mathematical expression
- Arrhenius Theory
- Applications of Arrhenius equation
- Practice problems
- Frequently Asked Questions-FAQ
Mathematical expression
This relationship is in between temperature and rate constantfor a reaction:
….
Where, 
is the rate constant of a reaction at temperature, 
having activation energy
and 
is Arrhenius factor or frequency factor or pre-exponential factor.
The equation is known as the Arrhenius equation.
The rate constant of the reaction increases exponentially with the increase in temperature according to the Arrhenius equation and because the rate of a reaction is directly proportional to the rate constant. Hence, with the increase in the temperature, the rate of a reaction increases.
If we take 
on both sides of the equation 
we get;
…….
Equation is modified form of Arrhenius equation which can be used for various applications.
Arrhenius Theory
Consider an elementary reaction, 
Reaction can take place only when a molecule of 
and a molecule of 
collide. The collision leads to the formation of an unstable state which exists for a very short time. This is called a transition state. The unstable complex at the transition state is called the transition-state complex / activated complex. The minimum energy required to form the transition state is called activation energy.
Let’s consider an analogy, when enough energy is not provided to the golf ball then it is unable to reach the flag post.
But when enough energy is provided behind the shot, the golf ball reaches the flag post.
Similarly, when the energy is not enough, reactions do not proceed further and transition states can not be formed and ultimately, products can not be formed. So, to convert reactants into products, a minimum amount of energy is required which is known as Activation energy.
Let’s understand more about how a reaction proceeds through a plot between potential energy and reaction coordinates:
Activation energy is required for the conversion of reactants into activated complex and then there is a sudden decrement in energy because energy is released when the complex gets decomposed to form products.
The final enthalpyof the reaction depends upon the nature of reactants and products .
Applications of Arrhenius equation
- The Arrhenius equation can be used to calculate the rate constant for a particular reaction at two different temperatures.
Let 
be the activation energy for any given reaction and 
, 
be the rate constant at temperature 
and 
respectively.
…..
…..
Subtracting equation from equation 
This equation is known as Vant Hoff equation.
- Activation energy can be calculated from the plot of
.
We know the general equation of a straight line is 
Where, 
and
represent the co-ordinate axis, 
represents the slope of the curve and 
represents the intercept on 
axis.
Now, according to the modified form of the Arrhenius equation,
…….
If we compare the equation 
and 
, we can say;
Intercept 
and Solpe 
Therefore, the above curve between 
and 
is a straight line with a constant and negative slope
So, if the value of the slope is given, the value of activation energy can be easily calculated.
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Practice problems
Q 1. Consider the given plots obeying Arrhenius equation.
, 
is a rate constant and 
is the activation energy.
Choose the correct option.
1. Graph
is correct but Graph
is incorrect.
2. Both Graph and Graph are incorrect.
3. Graph is incorrect but the Graph is correct.
4. Both 
and 
are correct.
Answer: (D)
According to the Arrhenius equation;
InGraph as the 
increases, the value of exponential 
decreases. This leads to decrease in the value of the rate constant and it decreases exponentially. Hence, Graph is correct.
In Graph, as the value of 
increases, the value of exponential
also increases. Therefore, the value of the rate constant increases. So, the curve increases non-linearly. Hence, the Graph is also correct.
Q 2. The rate constant (
) of a reaction is measured at different temperatures (), and the data is plotted in the given figure. The activation energy of the reaction (in 
is:
(
is the gas constant)
(A) 
(B) 
(C) 
(D) 
Answer: (D)
In the plot of vs 
, the value of the slope is:
Q 3. Consider the following plots of 
for four different reactions. Which of the following orders of activation energies is correct?
(A) 
(B) 
(C) 
(D) 
Answer: (B)
It can be seen from the graph that the order of the slope would be:
In the plot of 
, the value of the slope is:
So, 
Therefore, the order of activation energy would be:
Q 4. For the reaction, 
the rate constant 
is given as:
The energy of activation (in 
is:
(A) 
(B) 
(C) 
(D) 
Answer: (A)
The rate constant is given as:
Taking 
both sides of equation;
According to Arrhenius equation;
On comparing equation 
and 
we get;
Frequently Asked Questions-FAQ
Q 1. What is the significance of the pre-exponential factor 
in the Arrhenius equation?
Answer: It tells the number of effective collisions between the molecules of reactants. 
depends on the rate constant of a reaction and it varies with the temperature.
Q 2. What is the unit of in the Arrhenius equation? Answer: According to the Arrhenius equation, 
the unit of 
will be the same as that of rate constant of the reaction. For zero-order reaction, the unit of 
will be 
for the first order, the unit will be 
.
Q 3. Why energy is released during the formation of products?
Answer: Reaction occurs in such a way that weak bonds are broken on the reactant side to form stronger bonds on the product side and bond formation always leads to a loss of energy as the reaction is moving toward a more stable system. Hence, when products are formed from reactants, the energy is released.
Q 4. What is the significance of the exponential factor 
in the Arrhenius equation?
Answer: The exponential factor refers to the fraction of collisions (also effective collisions) that have sufficient energy (activation energy) to collide and form products.
Related Topics
| Rate law | Order of a Reaction |
| Reaction Mechanism | Effect of Catalyst on Rate of a Reaction |
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