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

Explanation

Consider a first order reaction
`RrarrP`
A first order reaction is one on which the rate is directly proprtional to the first power of the reactant concentration
Rate `alpha[R]^(2)`
Rate`=K[R]^(1)"                 ….(1)"`
  `Krarr" rate constant"`
`"But Rate "=(-d[R])/(dt)`
`(-d[R])/(dt)=k[R]"              …(2)"`
`"Rearrange equ 2 "(d[R])/([R])=-Kdt"             …(3)"`
`"Integrate equ 3"`
`int(1)/([R])d[R]=-K int dt`
`ln[R]=-kt+I"     …4"`
`"When "t=0, [R]=[R]_(0)`
`[R]_(0)=" Initial concentration of reactant R"`
`ln [R]_(0)=-K(o)+I[Irarr" integration constant"]`
`I=ln[R]_(0)`
`"Substituting the given the value of I in eqn… 4"`
`ln[R]=-Kt+ln[R]_(0)`
`KT=ln[R]_(0)-ln[R]`
`Kt=(ln[R]_(0))/([R])`
`Kt=2.303ln([R]_(0))/([R])`
`K=(2.303)/(t)log10""([R]_(0))/([R])`
`therefore The is th expression for the rate constant for first order reaction