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  • Consider a mechanism in which a single reactant produces several products by the  following parallel first order reactions: <img alt="A \stackrel{k_{1}}{\longrightarrow} B, A \stac

Consider a mechanism in which a single reactant produces several products by the  following parallel first order reactions:
A \stackrel{k_{1}}{\longrightarrow} B, A \stackrel{k_{2}}{\longrightarrow} C ,\stackrel{k_{3}}{\mathrm A\longrightarrow} D
which one of the following is the rate expression for [A].

Option: 1

\left(k_{1}+k_{2}+k_{3}\right)[\mathrm{A}]


Option: 2

k_{3}\right)[\mathrm{A}]


Option: 3

k_{2}\right)[\mathrm{A}]


Option: 4

k_{1}\right)[\mathrm{A}]


Answers (1)

best_answer

Parallel First Order Kinetics -

In this situation, B and C both are forming. These types of reactions are known as parallel reactions. Both these reactions are first order reactions with rate constants K1 and K2 respectively and half-lives as t(1/2)1 and t(1/2)2.

For these parallel reactions, we need to find:

  • Effective order
  • Effective rate constant
  • Effective t1/2
  • Effective Activation energy
  • [A], [B], [C] with time (t) variation
  • % of [B] and % of [C]

We know that the rate equations are given as follows:

\\\mathrm{r_{1}\:=\: \frac{-dA}{dt}\:=\: K_{1}[A] }\\\\\mathrm{r_{2}\: =\: \frac{-dA}{dt}\: =\: K_{2}[A]}

\\\mathrm{Thus,\: overall\: rate\: of\: reaction\: is:}\\\\\mathrm{\frac{-dA}{dt}\: =\: K_{1}[A]\: +\: K_{2}[A]\: =\: (K_{1}\: +\: K_{2})[A]}

\mathrm{Thus,\: rate\: =\: (K_{1}\: +\: K_{2})[A]^{1}}

\mathrm{\mathbf{Effective\: Rate\: Constant(K_{eff})}\: =\: (K_{1}\: +\: K_{2})}

\mathrm{\mathbf{Effective\: order\: of\: reaction}\: =\: 1}

\mathrm{Now, effective\: half-life(t_{1/2})\:=\: \frac{0.693}{K_{eff}}\: =\: \frac{0.693}{K_{1}\: +\: K_{2}}}

                                                           \mathrm{\Rightarrow\: \frac{0.693}{\frac{0.693}{(t_{1/2)_{1}}}+\frac{0.693}{(t_{1/2})_{2}}}}

\\\mathrm{Thus,\: effective\: half\: life\: is\: given\: as:}\\\\\mathrm{\frac{1}{(t_{1/2})_{eff}}\: =\: \frac{1}{(t_{1/2})_{1}}\: +\:\frac{1}{(t_{1/2})_{2}} }

NOTE: Effective activation energy, [A], [B], [C] with time (t) variation and % of [B] and % of [C] will be discussed in later concepts.

 

\begin{aligned} \frac{-d[\mathrm{A}]}{d t} &=k_{1}[\mathrm{A}]+k_{2}[\mathrm{A}]+k_{3}[\mathrm{A}] \\ &=\left(k_{1}+k_{2}+k_{3}\right)[\mathrm{A}] \end{aligned}

Therefore, option(1) is correct

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manish

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