# Kinetics of Parallel Reactions

## Parallel Reactions

Reactions in which initial species react to give multiple products simultaneously.Let's consider a reactant 'A' reacts to form product 'B' and 'C' simultaneously is given below-

Where k

_{1}and k

_{2}are the rate contents.

If the initial concentration of 'A' is'[A]

_{o}' and that of 'B' and 'C' be zero. After time 't', the concentration of A, B and C be [A], [B,] and [C] respectively.

**If we assume that both the reactions are first order**, then both the rates R

_{1}and R

_{2}can be given as-

Rate R

_{1}= −d[A]/dt = d[B]/dt = k

_{1}[A]

^{1}---Eq.-1

Rate R

_{1}= −d[A]/dt = d[C]/dt = k

_{2}[A]

^{1}---Eq.-2

Now total rate of reaction-

−d[A]/dt = R

_{1}+ R

_{2}---Eq.-3

or, −d[A]/dt = k

_{1}[A] + k

_{2}[A] ---Eq.-4

or, −d[A]/dt− = (k

_{1}+ k

_{2})[A] ---Eq.-5

or, d[A]/dt = −(k

_{1}+ k

_{2})[A] ---Eq.-6

on integrating the above equation with respect to 't', we get the following equation-

[A] = [A]

_{o}e

^{−(k1 + k2)t}---Eq.-6

Similarly,

d[B]/dt = k

_{1}[A] = k

_{1}[A]

_{o}e

^{−(k1 + k2)t}---Eq.-7

[B] = − k

_{1}[A]

_{o}/(k

_{1}+ k

_{2})(e

^{−(k1 + k2)t}) + C ---Eq.-8

When 't' = 0, [B] = 0

Putting the value of 't' and [B] in Eq.-8 we get-

C = k

_{1}[A]

_{o}/(k

_{1}+ k

_{2}) ---Eq.-9

Now putting the value of 'C' in Eq.-8 we get-

[B] = k

_{1}[A]

_{o}/(k

_{1}+ k

_{2})(1 − e

^{−(k1 + k2)t}) ---Eq.-10

Likewise [B]-

[C] = k

_{2}[A]

_{o}/(k

_{1}+ k

_{2})(1 − e

^{−(k1 + k2)t}) ---Eq.-11

**Now the ratio of [B] to [C]**-

[B]/[C] = k

_{1}/k

_{2}---Eq.-12

Percentage Yield of [B] = k

_{1}/(k

_{1}+ k

_{2}) x 100 ---Eq.-13

Percentage Yield of [C] = k

_{2}/(k

_{1}+ k

_{2}) x 100 ---Eq.-14

**If we assume that one is 1st order and other is 2nd order reactions, then-**

d[B] = k

_{1}[A]

^{1}

d[C] = k

_{2}[A]

^{2}

So, − d[A]/dt = k

_{1}[A] + k

_{2}[A]

^{2}

or, [A] = k

_{1}/e

^{k1t(k1 + k2[A]o) − k2[A]o}

**Case-1**

When k

_{2}[A]

_{o}<<< k

_{1}

then, [A] = [A]

_{o}e

^{ − k1t}

**Case-2**

When k

_{2}[A]

_{o}>>> k

_{1}

then, 1/[A] = 1/[A]

_{o}e

^{ + k2t }

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