Question

Derive an expression for the rate constant of a First order reaction.

Answer 1

Any reaction is called a first order reaction if a change in concentration of just one reactant determines the rate of reaction. For a reaction as follows

A$\to$ Product or products

${\left[A\right]}_o=$Initial concentration

${\left[A\right]}_t=$ concentration at time t.

since reaction in first order-

Rate$=-\frac{d\left[A\right]}{dt}=k\left[A\right]-\frac{d\left[A\right]}{dt}=-k.dt$

On integrating, we get

${\int }_{{\left[A\right]}_o}^{\left[A\right]}\frac{d\left[A\right]}{\left[A\right]}=-k{\int }_0^{t}dt\ln \left[A\right]-\ln {\left[A\right]}_o=-k[t-0]\ln \frac{\left[A\right]}{{\left[A\right]}_o}=-kt\ln \frac{{\left[A\right]}_o}{\left[A\right]}=kt\log \frac{2.303}{t}\frac{{\left[A\right]}_o}{\left[A\right]}=k$