Question

If the activation energy of a reaction is $480.9kJmo{{l}^{-1}}$ , calculate the fraction of molecules at ${{400}^{o}}C$ which have enough energy to react to form the products.

Answer 1

Hint: The fraction of the molecules present in a gas which have energies equal to or in excess of activation energy at a particular temperature. At an absolute temperature T, the fraction of molecules that have a kinetic energy than activation energy. The activation energy of a chemical reaction can be calculated by the Arrhenius equation.

Complete answer:
The energy required to form this intermediate is called activation energy.
Given Activation energy of a reaction, ${{E}_{a}}=480kJmo{{l}^{-1}}$
$R=8.314kJmo{{l}^{-1}}$ ;
Given temperature of a reaction, $T={{400}^{o}}C=(273+400)K=673K$
The temperature dependence of the rate of a chemical reaction can be accurate by the Arrhenius equation. Though this equation is applicable under a wide range of circumstances, collision theory.
According to the Arrhenius equation,
\[k={{A}^{\dfrac{-{{E}_{a}}}{RT}}}\]
fraction of the molecules which have enough energy to react to form the products is given by fraction \[={{e}^{\dfrac{-{{E}_{a}}}{RT}}}\] ----(1)
This exponential part of Arrhenius equation, expresses the fraction of reactant molecules that possess enough kinetic energy to react, as governed by Maxwell-Boltzmann distribution. Depending on the magnitude of activation energy and temperature, this fraction can range from zero, where no molecules have enough energy to react, to unity, where all molecules have energy to react.
Substituting the activation energy, temperature and R values in the equation (1),
fraction molecule required=\[{{e}^{\dfrac{-480.9X{{10}^{3}}}{8.314X673}}}={{e}^{-85.95}}=4.75X{{10}^{-38}}\] .

Note:
fraction of molecular collisions that have sufficient energy to react. It is believed that catalyst provides an alternate pathway or reaction mechanism by reducing the activation energy between reactants and products and hence lowering the potential barrier. From the Arrhenius equation, that lowers the value of activation energy faster will be the rate of reaction.