Derive an integrated rate equation for the rate constant of a first-order reaction.

Explanation

Consider a general first order reaction.
`R to ` Products.
The differential rate equation is
rate`=(d[R])/(dt)=d[R]`
Where K, is the rate constant of the first order reaction.
On Rearranging `(d[R])/([R])= -k dt`
Integrating on both the sides, we get.
`ln[R]= -Kt +I " "...(i)`
Where I is the constant of Integration.
When `t = 0, [R] = [R]_(0)` and hence using equation (i)
`I= ln[R]_(0)" "...(ii)`
Rearranging this equation.
`ln[R]_(0)-ln[R] = Kt`
or `K=(1)/(t)"ln" ([R]_(0))/([R])" "...(iii)`
or `K= (2.303)/(t)"log"([R]_(0))/([R])" "...(iv)`