Derive an integrated rate equation for a first order reaction.

Let us, consider a unimolecular first order reaction represented by the general equation, A → Product At t = 0 a 0 At t = t a - x x The initial concentration of A is a mole L - 1 and its concentration after time t is ( a – x ) mole L - 1 . This means during the time interval t , x mole L – 1 of A has reacted. The rate of reaction at any time t is given by the following first order kinetics. dx dt = k a - x where k is the rate constant of the reaction. dx a - x = kdt This is differential rate equation and can be solved by integration. ∫ dx a - x = k ∫ dt - ln a - x = k . t + C . . . . . ( 1 ) where C is integration constant. The constant C can be evaluated by applying the initial condition of the reaction i.e. when t = 0 , x = 0 . Putting these in equation (1), we get C = – ln a Putting the value of C in equation (1), we get – ln ( a – x ) = k . t – lna k = 1 t ln a a - x = 2 . 303 t log a a - x . . . . . ( 2 ) Equation (2) integrated rate equation for a first order reaction.