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If a temperature increase from 20$^\circ$C to 35$^\circ$C triples the rate constant for a reaction, what is the value of the activation barrier for the reaction?

$\begin{array}{1 1} 55\;kJmol^{-1} \\ 83\;kJmol^{-1} \\ 72\;kJmol^{-1} \\ 68\;kJmol^{-1} \end{array}$

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Answer: $ 55\;kJmol^{-1}$
We have to find the activation barrier, i.e, the activation energy, give two temperatures $T_1$ and $T_2$.
Its also mentioned that the rate constant triples, so $k_2 = 3k_1$
From Arrhenius equation, $k=Ae^{\large\frac{-E_a}{RT}}$, we get:
$\Rightarrow \large\frac{k_2}{k_1} $$ = \large\frac{Ae^{\large\frac{-E_a}{RT_1}}}{Ae^{\large\frac{-E_a}{RT_2}}}$
Using the fact that $\large\frac{e^x}{e^y}$$ = e^{x-y}$ and taking the natural log on both sides and simplifying, we arrive at:
$ ln\large\frac{k_2}{k_1}$$ = \large\frac{E_a}{R}$$ (\large\frac{1}{T_1}$$ - \large\frac{1}{T_2}$$)$
Substituting the values for $k_2 = 3k_1$ and $T_1, T_2$ (in $^\circ$K), we get:
$\Rightarrow ln 3 = 1.0986 = \large\frac{E_a}{8.314\;J\;K.mol^{-1}}$$ (\large\frac{1}{293.15}$$ - \large\frac{1}{308.15}$$)$
Solving for $E_a$, we get $E_a = 55006\;Jmol^{-1} = 55\;kJmol^{-1}$
answered Jul 25, 2014 by balaji.thirumalai
edited Jul 25, 2014 by balaji.thirumalai

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