Concept Explanation: Pseudo-First Order Kinetics

This problem involves a reaction with multiple reactants and a complex rate law. The key concept is simplifying the kinetics under specific concentration conditions to form a pseudo-first order reaction.

Step-by-Step Solution:

Step 1: Analyze the Given Rate Law

The rate law is given as: $rate=k{C}_A{C}_B{C}_C^2$

Step 2: Apply the Given Conditions

The conditions state that the initial concentrations of reactants B and C are much greater than the concentration of reactant A ($b>>a$ and $c>>a$). This means that as reactant A is consumed, the concentrations of B and C remain effectively constant throughout the reaction.

Step 3: Derive the Pseudo-First Order Rate Constant (k')

Since $C_B$ and $C_C^2$ are constant, they can be combined with the original rate constant $k$ to form a new, effective rate constant $k^{\prime }$.

The rate law simplifies to: $rate=k\cdot \left(C_B\right)\cdot \left(C_C^2\right)\cdot C_A=k^{\prime }{C}_A$

Therefore, by comparing the two expressions, we find: $k^{\prime }=k\cdot C_B\cdot C_C^2$

Since we are dealing with initial concentrations, we substitute $C_B$ with $b$ and $C_C$ with $c$: $k^{\prime }=k\cdot b\cdot c^2$

Final Answer: The correct option is $k^{\prime }=kb{c}^2$.

Related Topics & Formulae

1. Rate Law: An equation that shows the relationship between the reaction rate and the concentrations of reactants. For a general reaction $aA+bB\to products$, the rate law is $Rate=k{\left[A\right]}^m{\left[B\right]}^n$, where $m$ and $n$ are the orders of the reaction with respect to A and B.

2. Reaction Order: The power to which a reactant's concentration is raised in the rate law. The overall order is the sum of all individual orders (e.g., $m+n$).

3. Pseudo-First Order Reaction: A higher-order reaction that is made to behave like a first-order reaction by keeping the concentrations of all but one reactant in large excess. The rate law simplifies to $Rate=k^{\prime }\left[A\right]$, where $k^{\prime }=k\cdot \left(\text{constant concentrations of other reactants}\right)$.

4. Integrated Rate Law for First-Order: For a reaction $A\to Products$ with rate law $Rate=k\left[A\right]$, the concentration of A at time $t$ is given by $ln\left[A\right]=ln[A{]}_0-kt$.