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# 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

Last updated date: 20th Jun 2024
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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.

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.