Derive the integrated rate law equation for `1 ^(st)` order reaction and write its one use.

Explanation

Integrated rate law method is uded to determine the order of reaction.
Let us consider the reaction ` A to ` Product,
If the reaction follows first order kinetics, then rate of recation is given by :
`- (d [A])/( dt ) = k [A] " "...(i)`
On rearrangining the equation (i), we have `- (d [A])/( [A} ) = kdt `
On integrating the above equation we get `- int (d [A] )/( [A] ) = int dt [ because k ` is constatn ]
`- ln [A] = kt + C,` (where C = integration constant ...(ii)
When ` t =0,` then `[A] = [A _(0) ]`
Thus , using equation (ii), we have, `- ln [A _(0) ] = C " "...(iii)`
Putting the value of constant in equation (ii), we have `- ln [A] = kt -ln [A _(0) ]`
`implies ln [A _(0) ] - ln [A] = kt `
`or ln ""([A _(0) ] )/( [A]) = kt `
`or k = (1)/(t) ln ""([A _(0) ] )/( [A]) `
`k = (2. 303)/( t ) log "" ( [A _(0)])/( [A] ) `
Equation (iv) is know as integrated rate equation for first order reaction . Thus , any reaction which follows first order kinetics, must obey equation (iv).
It is used to calculate the rate constant of first order reaction . It is also used to determine the order of a reaction.