Correct Answer - Option 1 : 53.6 KJmol
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Concept:
Arrhenius equation- Temperature Dependence and Rate constant:
- The temperature dependence of the rate of a chemical reaction can be accurately explained by the Arrhenius equation.
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It has been found that for a chemical reaction with the rise in temperature by 10°C, the rate constant is nearly doubled.
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Mathematically, Arrhenius equation is given by,
\({\text{k }} = {\text{ }}{\mathbf{A}}{\text{ }}{\mathbf{e}}{\text{ }} - \frac{{{\mathbf{Ea}}}}{{{\mathbf{RT}}}}\)
where A is the Arrhenius factor or the frequency factor. It is also called the pre-exponential factor. It is a constant specific to a particular reaction, R is gas constant and Ea is activation energy measured in joules/mole (J mol–1).
Activation energy (Ea):
- It is the minimum energy that must be supplied to the reactants to enable them to cross over the energy barrier between reactants and products.
- For fast reaction; Ea is low.
- For slow reaction; Ea is high.
- An alternative form of the Arrhenius equation is given by:
\({\text{Log}}\left( {\frac{{{{\text{k}}_{\text{2}}}}}{{{{\text{k}}_{\text{1}}}}}} \right){\text{ = }}\frac{{{\text{Ea}}}}{{{\text{2}}{\text{.303}}\,{\text{R}}}}\left[ {\frac{{{{\text{T}}_{\text{2}}}{\text{ - }}{{\text{T}}_{\text{1}}}}}{{{{\text{T}}_{\text{1}}}{{\text{T}}_{\text{2}}}}}} \right]\)
Where K1 and K2 are the rate constants for a reaction at two different temperatures T1 and T2.
Calculation:
Given: K2/K1 = 2 (The rate of reaction is doubled) ; R = 8.314 JK -1 mol-1 ; T1 = 300K; T2 = 310K
As per the Arrhenius equation, \({\text{Log}}\left( {\frac{{{{\text{k}}_{\text{2}}}}}{{{{\text{k}}_{\text{1}}}}}} \right){\text{ = }}\frac{{{\text{Ea}}}}{{{\text{2}}{\text{.303}}\,{\text{R}}}}\left[ {\frac{{{{\text{T}}_{\text{2}}}{\text{ - }}{{\text{T}}_{\text{1}}}}}{{{{\text{T}}_{\text{1}}}{{\text{T}}_{\text{2}}}}}} \right]\)
\(\log \,2 = \frac{{{E_a}}}{{2.303R}}\left[ {\frac{{310 - 300}}{{300 \times 310}}} \right]\)
\({E_a} = \frac{{\log \,2 \times 2.303 \times 8.314 \times 310 \times 300}}{{10}}\)
Therefore, Ea = 53603 J = 53.6 kJ/mol
Hence, the activation energy (Ea) of the reaction whose rate doubles when its temperature changes from 300 K to 310 K is 53.6 KJ/mol