J. lndiau Cbem. Soc., Vol. 74, Marcb 1997, pp. 190-191 Kinetics of Two-step Irreversible Consecutive First Order Reactions JEBAKUMAR JEEVANANDAM", R. GOPALAN•" and R. "Department of Chemistry, Madras Christian College, Tambaram, Madras-600 OS9 "Department of Tagore Arts College, Pondicherry-60S 008 MQIJuscript receilled 12 July /995, accepted 27 October 1995 Analytical expressions for the concentrations and the rates of formation of Intermediate and product are known for consecutive reactions. These expressions break down when the rate constants for the first and the second steps are equal. Using the approximate method and the I'H{)Spital rule, equations have been derived for the concentrations and the rates ol formation of the Intermediate and the product and also the time when the rate of product formation is a maximum, when the rate constants of the two steps are equal. A two-step irreversible consecutive reaction 1 can be repre- sented as (1) If the initial concentration of A is (A] 0 and its concentra- tion at time t is [A], then the rate equation for the disap- pearance of A is -d(A]/dt = kt (A] (2) Integration of this simple first order equation, subject to the condition that [A)= (A] 0 when t = 0, gives [A] = [A]o exp (-ktt) (3) The net rate of formation of X is d[X]Idt = kt[A]-k2[X] (4) Inserting equation (3) in equation (4) gives ·d[X]/dt = kt(A]o exp (-ktt) ,.... kz[X] (5) Integrating the above expression gives kt [X] = [A]o (kz _ kt) { exp (-ktt) - exp (k2t)} (6) The concentration of [X] goes through a maximum. The maximum can be mathematically represented as d[X]Idt = 0 Combining equations (6) and (7), we get ln{kt/kz) tmax = (kl kz) (7) (8) tmax corresponds to the time when the concentration of X, the intermediate, is a maximum, which is also the moment at which the rate of formation of the final product Z is maximum. If the intermediate X and the final product Z are initially absent, at any time t, by stoichiometric balance, [A] + [X] + [Z] = [A]o .so that {Z] = [A]o - [A) - [X] (9) (10) Inserting equations (3) and (6) in equation (10) leads to 190 [A] . [Zl = (k2-:t)[kz{1-exp(-ktt)}-kt{1-exp(:-kzt)}] (11) These equations apply to radioactivity 2 where two succes- sive disintegrations are involved and also to chemical reac- tions. An example for irreversible first order consecutive reaction is the thermal isomerisation3 of 1,1-dicyclo- propylethylene to produce which further isomerises to give bicyclo[3.3.0)pent-1-ene. Results and Diseussion Equations (6), (8) and (11) contain terms such as (k 2 --k,) and/or ln(k 1 /k:V. When k 1 = k 2 , these equations give rise to indeterminate forms 4 of the type 0/0. To treat this special case, when k 1 = k 2 , expressions for tmax, [X], {Z], d[X]/dt and d[Z]/dt have been derived using the approximate method' and the de )'Hospital ruJe6. Derivation for tmax : The expression for tmu when kt = k,. is an indeterminate quantity of the form 0/0. Using the approximate method an expression for tmax is derived when k1 .= k,.. i.e the difference (k 1 -t,.) is taken as very small. The value of In (1 + x) can be found by summing the series 7 , . x } x 3 x 4 In (1 + x) = 1-2+3-4 ... ... (12) when x 2 < 1. When x is very small, the terms containing higher powers may be neglected and the series truncated as In (1 + x) .= x (13) The term 1n(k 1 /k,.) in equation (8) can be written as and approximated as I {1 (kt - kz)} _ (kt - k2) n+ kz- k2 (14) The approximation is valid as (k1 -k:V is very small. There- fore, ln (kt/k2) tmax = · (kt:-kz)