Question

The activation energy for a reaction is changed from 184 kJ/mol to 59.0 kJ/mol at 600. K by the introduction of a catalyst. If the uncatalyzed reaction takes about 3900 years to occur, about how long will the catalyzed reaction take?

Answer 1

Answer:

The catalyzed reaction will take time of $5.11\times 10^{-8} years$.

Explanation:

According to the Arrhenius equation,

$K=A\times e^{\frac{-Ea}{RT}}$

The expression used with catalyst and without catalyst is,

$\frac{K_2}{K_1}=\frac{A\times e^{\frac{-Ea_2}{RT}}}{A\times e^{\frac{-Ea_1}{RT}}}$

$\frac{K_2}{K_1}=e^{\frac{Ea_1-Ea_2}{RT}}$

where,

$K_2$ = rate of reaction with catalyst

$K_1$ = rate of reaction without catalyst

$Ea_2$ = activation energy with catalyst  = 59.0 kJ/mol = 59000 J/mol

$Ea_1$ = activation energy without catalyst  = 184 kJ/mol = 184000 J/mol

R = gas constant

T = temperature = $600K$

Now put all the given values in this formula, we get:

$\frac{K_2}{K_1}=e^{\frac{184,000 kJ-59000 kJ}{R\times 300}}=7.632\times 10^{10}$

The reaction enhances by $7.632\times 10^{10}$  when catalyst is present.

Time taken by reaction without catalyzed = 3900 years

Time taken by reaction with catalyzed = x

$x=\frac{3900 year}{7.632\times 10^{10}}=5.11\times 10^{-8} years$

The catalyzed reaction will take time of $5.11\times 10^{-8} years$.