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 k₁ and k₂ 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 R₁ and R₂ can be given as-
Rate R₁ = −d[A]/dt = d[B]/dt = k₁[A]¹     ---Eq.-1
Rate R₁ = −d[A]/dt = d[C]/dt = k₂[A]¹     ---Eq.-2
Now total rate of reaction-
−d[A]/dt = R₁ + R₂     ---Eq.-3
or, −d[A]/dt = k₁[A] + k₂[A]     ---Eq.-4
or, −d[A]/dt− = (k₁ + k₂)[A]     ---Eq.-5
or, d[A]/dt = −(k₁ + k₂)[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 = k₁[A] = k₁[A]_oe^{−(k₁ + k₂)t}      ---Eq.-7
[B] = − k₁[A]_o/(k₁ + k₂)(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 = k₁[A]_o/(k₁ + k₂)      ---Eq.-9
Now putting the value of 'C' in Eq.-8 we get-

[B] = k₁[A]_o/(k₁ + k₂)(1 − e^{−(k₁ + k₂)t})      ---Eq.-10

Likewise [B]-

[C] = k₂[A]_o/(k₁ + k₂)(1 − e^{−(k₁ + k₂)t})      ---Eq.-11

Now the ratio of [B] to [C]-
[B]/[C] = k₁/k₂      ---Eq.-12
Percentage Yield of [B] = k₁/(k₁ + k₂) x 100      ---Eq.-13
Percentage Yield of [C] = k₂/(k₁ + k₂) x 100      ---Eq.-14


If we assume that one is 1st order and other is 2nd order reactions, then-
d[B] = k₁[A]¹
d[C] = k₂[A]²
So, − d[A]/dt = k₁[A] + k₂[A]²
or, [A] = k₁/e^{k₁t(k₁ + k₂[A]_o) − k₂[A]_o}
Case-1
When k₂[A]_o <<< k₁
then, [A] = [A]_oe ^{− k₁t}
Case-2
When k₂[A]_o >>> k₁
then, 1/[A] = 1/[A]_oe ^{+ k₂t}

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