Question

a) Derive an integrated rate equation for rate constant of a first order reaction.
b) Draw a graph of potential energy V/S reaction co-ordinates showing the effect of catalyst on activation energy $\left(E_a\right)$ of a reaction.

Answer 1

a) Consider a general first order reaction
R $\to$ P
The differential rate equation for given reaction can be written as
Rate$=-\frac{d\left[R\right]}{dt}=K[R{]}^1$
Rearrange above equation.
$\frac{d\left[R\right]}{\left[R\right]}=-K\times dt$
Integrating on both sides of the given equation
$\int \frac{d\left[R\right]}{\left[R\right]}=-k\int dt$
ln[R]$=-Kt+I$ ..$\left(1\right)$
Where I is Integration constant
At $t=0$ the concentration of reactant [R]$=[R{]}_0$ where $[R{]}_0$ is initial concentration of reactant
Substituting in equation $\left(1\right)$ we get
ln $[R{]}_0=(-K\times 0)+I$
ln $[R{]}_0=I$ $\left(2\right)$
Substitute I value in equation $\left(1\right)$
ln $\left[R\right]=-Kt+ln[R{]}_0$
$Kt=ln[R{]}_0-lnR$
$Kt=ln\frac{[R{]}_o}{\left[R\right]}$
or $K=\frac{1}{t}ln\frac{[R{]}_0}{\left[R\right]}$.
or $K=\frac{2.303}{t}log\frac{[R{]}_0}{\left[R\right]}$.