For the first order parallel reactions : reaction
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the value of the ratio $(\frac{[ x ] _{t}}{[ y ] _{t} + [ z ] _{t}})$ , at any given instant t , is :

Question

For the first order parallel reactions : reaction , the value of the ratio ( [ y ] t ​ + [ z ] t ​ [ x ] t ​ ​ ) , at any given instant t , is :

TestHub · Accepted Answer

For parallel first-order reactions, the concentration of reactant $X$ at time $t$ is $[X]_{t} = [X]_{0} e^{- (k_{1} + k_{2}) t}$ . By conservation of mass, the total concentration of products formed is $[Y]_{t} + [Z]_{t} = [X]_{0} - [X]_{t} = [X]_{0} (1 - e^{- (k_{1} + k_{2}) t})$ . Setting up the required ratio gives:

$\frac{[ X ] _{t}}{[ Y ] _{t} + [ Z ] _{t}} = \frac{[ X ] _{0} e ^{- (k_{1} + k_{2}) t}}{[ X ] _{0} ( 1 - e ^{- (k_{1} + k_{2}) t} )} = \frac{e ^{- (k_{1} + k_{2}) t}}{1 - e ^{- (k_{1} + k_{2}) t}}$

Multiplying the numerator and denominator by $e^{(k_{1} + k_{2}) t}$ simplifies the expression to $\frac{1}{e ^{(k_{1} + k_{2}) t} - 1}$ (Option A).

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