Derive an intergrated rate for the first order reaction.

Explanation

Consider a first order reaction, Reactants `rarr` Products
Rate of a first order reaction is given by
`"Rate "=(-d[R])/(dt)=k[R]^(1)`
`-(d[R])/([R])=kdt`
Integrating on both sides
`-int(d[R])/([R])=k int dt`
`-ln [R]=kt+I rarr (1)`
Where I is the integration constant.
When `t=0` the concentration of the reaction `[R]=[R]_(0)`, where `[R]_(0)` is the initial concentration of the reactant.
`-ln[A]_(0)=I`
Substituting the value of (2) in equation (1), we ge
`-ln[R]=kt-ln[R]_(0)`
`n[R]_(0)-ln[R]=k`
`ln([R]_(0))/([R])=kt`
`k=(1)/(t)ln([R])/([R])rarr(3)`
`k=(2.303)/(t)log""([R]_(0))/([R])rarr(4)`
This is an integrated rate equation for the first order reaction.