Question

If for two reaction ${E_{{a_1}}} > {E_{{a_2}}}$ and $T{C_1}$ & $T{C_2}$ are temperature coefficient respectively, then which of the following is correct:-
(1) $T{C_1}$>$T{C_2}$
(2) $T{C_1}$<$T{C_2}$
(3) $T{C_1}$=$T{C_2}$
(4) None of these

Answer 1

Hint: We know that, temperature coefficient of a reaction describes the effect of temperature on the reaction rate. It is defined as the ratio of rate constants of two temperatures differ by \[10^\circ {\rm{C}}\].

Complete step by step solution:
We know that, temperature coefficient expression is,

$\dfrac{{k’}_{1}}{k_1}$ = $\dfrac{{{E_a}}}{{2.303\,R}}\left[ {\dfrac{{{T_2} - {T_1}}}{{{T_1}{T_2}}}} \right]$
               …… (1)

Here, $\dfrac{{k’}_{1}}{k_1}$is temperature coefficient, ${E_a}$ is activation energy, ${T_2}$
is final temperature, ${T_1}$ is initial temperature and R is gas constant.

For 1st reaction, activation energy is ${E_{{a_1}}}$ and temperature coefficient is $T{C_1}$. Using equation (1), temperature coefficient equation is,

$T{C_1} = \dfrac{{{E_{{a_1}}}}}{{2.303\,R}}\left[ {\dfrac{{{T_2} - {T_1}}}{{{T_1}{T_2}}}} \right]$ …… (2)

For 2nd reaction, activation energy is ${E_{{a_2}}}$ and temperature coefficient is $T{C_2}$. Using equation (1), temperature coefficient equation is,

$T{C_2} = \dfrac{{{E_{{a_2}}}}}{{2.303\,R}}\left[ {\dfrac{{{T_2} - {T_1}}}{{{T_1}{T_2}}}} \right]$ …… (3)


Now, we have to divide equation (2) and (3). We get the following equation on division.

$\dfrac{{T{C_1}}}{{T{C_2}}} = \dfrac{{{E_{{a_1}}}}}{{{E_{{a_2}}}}}$ …… (4)

In the question, it is given that ${E_{{a_1}}} > {E_{{a_2}}}$. So, from equation (4), we can say that $T{C_1} > T{C_2}$.

So, the correct answer is Option 1.

Additional Information:
Let’s learn about the rate of a chemical reaction. Rate or speed of a chemical reaction is the change in concentration of product or reactant in unit time. There are two ways of expressing rate, such as, decreasing of reactant concentration and increase of concentration of product. There are many factors on which rate of a chemical reaction depends, such as temperature, presence of catalyst, concentration of reactant etc.

Note: Rate of a chemical reaction is dependent on the temperature. When the temperature of a chemical reaction is increased by $10^\circ {\rm{C}}$, the rate constant doubles nearly. The dependence of temperature on the rate of reaction is explained by Arrhenius equation.

$k = A{e^{ - {E_a}/RT}}$

Here, k is rate constant, A is Arrhenius factor, ${E_a}$ is activation energy, R is rate constant and T is temperature.