Derive an intergrated rate for the first order reaction.

Explanation

The rate of the reaction is directly proportional to the `1^(@)` power of the concentration of reactant R.
`R rarr P`
rate `= (-d[R])/(dt) = k[R]`
`(d[R])/(dt) = -K[R]` or `(d[R])/([R]) = -K dt`
Integration both sides
`int. (d[R])/([R]) = -intK dt`
`log = R = -Kt + I` ....(1)
Where I is integration constant.
To find I when `t = 0, [R] = [R_(0)]`
`log[R_(0)] - K xx 0 + I`
`I = log[R_(0)]`
Substituting in (1) we get
`log R_(0) = log R = Kt`
`ln.([R_(0)])/([R]) = Kt`
or `K = (1)/(t)ln.([R_(0)])/([R])`