Question

For a reaction, the rate constant is expressed as $k = A{e^{ - \dfrac{{40000}}{T}}}$ . The energy of activation is:
A. $40000\;{\text{cal}}$
B. $88000\;{\text{cal}}$
C. $80000\;{\text{cal}}$
D. $8000\;{\text{cal}}$

Answer 1

Hint : The Arrhenius equation is used to describe the relation between the rate of reaction and the temperature at which many physical and chemical reactions take place. This equation has an important application and is widely used for calculation of rate constant and activation energy for a chemical reaction.

Complete Step By Step Answer:
According to Arrhenius equation, the rate constant is expressed in terms of temperature and activation energy as per following expression:
$k = A{e^{\left( { - \dfrac{{{E_a}}}{{RT}}} \right)}}\;\; - (1)$
Where, k is the rate constant, A is the pre-exponential factor, ${E_a}$ is the activation energy, R is the universal gas constant and T is temperature in kelvin.
Now, as per question the expression of rate constant is given as follows:
$k = A{e^{ - \dfrac{{40000}}{T}}}$
On comparing it with equation (1), as the base value of the expression is same so the powers will be equal and therefore, the expression will be as follows:
$\dfrac{{{E_a}}}{R} = 40000$
Substituting the value of R in calories:
$\Rightarrow \dfrac{{{E_a}}}{2} = 40000$
$\Rightarrow {E_a} = 2 \times 40000$
$\Rightarrow {E_a} = 80000{\text{ cal}}$
Hence, the value of the energy of activation for the given condition is $80000{\text{ cal}}$ . So, option (C) is the correct answer.

Note :
It is important to note that the activation energy is the minimum energy required by a reacting species to undergo a chemical reaction. On increasing the value of activation energy, the value of rate constant decreases. The unit of activation energy depends on the value of the universal gas constant considered. Here, it was considered in calories so the unit of activation energy was also expressed in calories.