An organic compound undergo 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 en

Question

An organic compound undergo 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 1 ​ , 2 E 1 ​ , 3 E 2 ​ , …… .. n E respectively. The overall activation energy of reaction is given by X E ( 2 n + 1 ) ​ the value of " X " is ...

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Let the rate constant be: k 0 ​ = k + 2 k + 3 k + … + nk 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 + 2 k + … + nk ) ​ ] = ( R T 2 E 0 ​ ​ ) k 0 ​ 1 ​ [ R T 2 k ⋅ E ​ + R T 2 2 k ⋅ 2 E ​ + R T 2 3 k ⋅ 3 E ​ + … + R T 2 nk ⋅ n E ​ ] = R T 2 E 0 ​ ​ R T 2 ⋅ k 0 ​ k ⋅ E ​ [ 1 2 + 2 2 + 3 2 + … + n 2 ] = R T 2 E 0 ​ ​ R T 2 ⋅ k 0 ​ k ⋅ E ​ 6 ( n ) ( n + 1 ) ( 2 n + 1 ) ​ = R T 2 E 0 ​ ​ k ( 1 + 2 + 3 + … + n ) k ⋅ E ​ ⋅ 6 ( n ) ( n + 1 ) ( 2 n + 1 ) ​ = E 0 ​ 6 ⋅ 2 n ( n + 1 ) ​ E n ( n + 1 ) ( 2 n + 1 ) ​ = E 0 ​ E 0 ​ = 3 ( 2 n + 1 ) E ​

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