An organic compound undergoing number of parallel reaction can give product P 1 , P 2 , P 3 , ……… . P n . The rate constant of these parallel reactions are k , 2 K , 3 k …… . nk and activation

Question

An organic compound undergoing number of parallel reaction can give product P 1 ​ , P 2 ​ , P 3 ​ , ……… . P n ​ . The rate constant of these parallel reactions are k , 2 K , 3 k …… . nk and activation energy for each parallel reaction is E , 2 E , 3 E ……… n E respectively. The overall activation energy of reaction is given by x E ( 2 n + 1 ) ​ . The value of x is ____ .

TestHub · Accepted Answer

Let rate cont. K 0 ​ = K + 2 K + 3 K + …… n E ​ K 0 ​ = A e − RT E 0 ​ ​ d T d K 0 ​ ​ = A e − RT E 0 ​ ​ R T 2 E 0 ​ ​ d T d K 0 ​ ​ = − K 0 ​ R T 2 E 0 ​ ​ ​ ​ ​ ​ ​ K 0 ​ 1 ​ d T d K 0 ​ ​ = R T 2 − E 0 ​ ​ K 0 ​ 1 ​ [ dT d ​ ( K + 2 K + 3 K + ……… . nK ) ] = RT 2 E 0 ​ ​ K 0 ​ 1 ​ [ RT 2 K . E ​ + RT 2 2 K .2 E ​ + RT 2 3 K .3 E ​ + RT 2 nk . nE ​ ] = RT 2 E 0 ​ ​ RT 2 K . E ​ [ 1 2 + 2 2 + 3 2 + …… . n 2 ] = RT 2 E 0 ​ ​ K ( 1 + 2 + 3 + … . n ) K . E ​ 6 ( n ) ( n + 1 ) ( 2 n + 1 ) ​ = E 0 ​ n ( n + 1 ) ( 2 n + 1 ) = E 0 ​ 2 n ( n + 1 ) ​ 6 E 0 ​ = 3 ( 2 n + 1 ) ​ E

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