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

Explanation

(a) Consider a general first order reaction, `R-P`
The differential rate equation for the above reaction can be written as,
Rate `=(-d[R])/(R) = k[R]^(1)`
`= (d[R])/(R) = -k xx dt`
Integrating on both sides of the above equation, we get
`int(d[R])/(R) = -k xx int dt`
In `[R] = - kt + I" ".........(1)`
Where `I to` constant of integration
At time t = 0, the concentration of the reactant `R= [R]_0` where `[R]_0` is the initial concentration of the reactant.
Substituting in equation 1, we get
`ln [R_0] = -(k xx 0) + I`
`ln [R]_(0) = I" "..........(2)`
Substituting the value of I from equation 2 in equation 1, we get
`ln[R] = -kt + ln [R]_(0)`
`kt = ln [R]_(0) - ln[R]" ".......(3)`
`kt = ln[R]_(0) -ln[R] " "....(3)`
`kt = ln[R]_(0) - ln[R]`
`kt = (ln[R]_(0))/([R])`
or, `k = (2.303)/t xx "log" ([R]_(0))/([R])`
Equation 4 is called as the integrated form of rate of rate constant for a first order reaction.
(b)