Question

How much faster would a reaction proceed at 298K than at 273K if the activation energy is $65\,kJmo{l^{ - 1}}$ ?
A. 22 times
B. 11 times
C. 33 times
D. 5.5 times

Answer 1

This question can be solved with the help of Arrhenius equation which is a formula for the temperature dependence of reaction rates.

Complete step by step solution:
This question will be solved with the help of the Arrhenius equation. In physical chemistry, the Arrhenius equation is a formula for the temperature dependence of reaction rates. This equation was proposed by Svante Arrhenius in 1889. This equation has a vast and important application in determining the rate of chemical reactions and for calculating energy of activation. Arrhenius provided a physical justification and interpretation for the formula. It is best seen as an empirical relationship. It can be used to modal the temperature variation of diffusion coefficients, population of crystals vacancies, creep rates and many other thermally induced reactions.
The Arrhenius equation gives the dependence of the rate constant of a chemical reaction on the absolute temperature as
$k = A{e^{\dfrac{{ - {E_a}}}{{RT}}}}$
Where,
$k$ is the rate constant
$T$ is the absolute temperature in kelvins
$A$ is the pre-exponential factor, which is a constant for each chemical reaction. In terms of collision theory is the frequency of correctly oriented collisions between the reacting species
$e$ is the base of the natural logarithm
${E_a}$ is the activation energy for the reaction
$R$ is the universal gas constant
If the activation energy is expressed in terms of energy per reactant molecule, the universal gas constant must be replaced with the Boltzmann constant in the Arrhenius equation.
Now, we have to find the ratio of k at different temperatures which are $298K$ and $273K$ . Comparing the two we will get,
$\Rightarrow \dfrac{{{k_{298}} = {A_{298}}{e^{\dfrac{{ - {E_a}}}{{RT}}}}}}{{{k_{273}} = {A_{273}}{e^{\dfrac{{ - {E_a}}}{{RT}}}}}}$
Substituting the values of activation energy which is $65kJmo{l^{ - 1}}$ as per given in the question, Universal gas constant which is $8.314$ and the value of temperature we get,
$\Rightarrow \dfrac{{{k_{298}}}}{{{k_{273}}}} = \dfrac{{{A_{298}}{e^{\dfrac{{ - 65000}}{{8.314 \times 298}}}}}}{{{A_{273}}{e^{\dfrac{{ - 65000}}{{8.314 \times 273}}}}}}$
$\Rightarrow \dfrac{{{e^{ - 26.24}}}}{{{e^{ - 28.64}}}}$
$\Rightarrow {e^{2.40}} = 11$
Therefore, the reaction is 11 times faster for $298K$ than at $273K$ temperature.

**Therefore, the answer is option B.

Note: **
The Arrhenius equation is a formula for the temperature dependence of reaction rates. It shows the dependence of the rate constant of a chemical reaction on the absolute temperature and is given by the equation $k = A{e^{\dfrac{{ - {E_a}}}{{RT}}}}$ .