Derive an intergrated rate for the first order reaction.

Explanation

Consider a first order reaction,
`R to P`
A first order reaction is one in which the rate is directly proportional to first power of the reactant concentration. Therefore, according to rate law,
Rate `alpha [R].`
Rate `=k[R]." "...(1)`
Where `k` is rate constant or velocity constant
But, Rate `= (d [R])/( dt)`
`- (d [R])/([R])= k[R] " "...(2)`
Rearrange the equation (2), we get `-(d[R])/( [R])=-kdt " "...(3)`
`int (1)/( [R]) d[R] = -k int dt`
`"In" [R] = -kt +1 " "...(4)`
When `t=0, [R]=[R]`, is the initial concentration of reactant R.
`"In" [R]_^(@) = -k xx 0+1`
Where I is called integration constant
`I="In" [R]_0`
Substituting the value of I in equation (4) we get,
`"In" [R] = -kt + "In" [R]_0`
`kt = "In" [R]_0 - "In" [R]`
`kt =2.303 "In" ([R]_0)/([R])`
`k= (2.303)/(t) log_(10)"" ([R]_0)/([R])`