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

Explanation

consider a first order reaction , R `to` P
`"Rate" = -(d[R])/(dt) = k[R]^(1) or (d[R])/([R]) = - kdt `
Integrating both sides , In [R] = - kt +1 ........... (1)
where .I . is integration constant.
At t=0 , [R] =`[R]_(0)` where `[R]_(0)` is initial concentration of the reactant .
Substituting the value of I in equation (1) In `[R ] = -kt + ln[R]_(0)`......... (2)
Rearranging this equation , `ln([R])/([R]_0)= -kt `
` or k = (1)/(t) ln ([R]_0)/([R]) or k = (2.303)/(t) log ""([R]_0)/([R])` ...........(3)