A following mechanism has been proposed for a reaction
`2A +B rarr D rarr E`
`A+ B rarrC+D `(slow)
`A+C rarr E` (fast)
The rate law expression for the reaction by RDS methd is:

Options

A. `r=k[A]^(2)[B]`
B. `r=k[A][B]`
C. `r=k[A]^(2)`
D. `r=k[A][C]`

Answer

To derive the rate law expression for the given reaction mechanism using the Rate Determining Step (RDS) method, we can follow these steps:

### Step 1: Identify the Rate Determining Step (RDS)
The first step in the mechanism is given as:
\[ A + B \rightarrow C + D \] (slow)

Since this step is the slowest, it is the Rate Determining Step (RDS). The rate of the overall reaction will be determined by this step.

### Step 2: Write the Rate Law for the RDS
The rate law for a reaction can be expressed in terms of the concentration of the reactants involved in the RDS. For the RDS:
\[ A + B \rightarrow C + D \]

The rate law can be written as:
\[ \text{Rate} = k [A]^m [B]^n \]
where \( m \) and \( n \) are the stoichiometric coefficients of A and B in the RDS.

### Step 3: Determine the Stoichiometric Coefficients
In the RDS, the stoichiometric coefficients for A and B are both 1. Therefore, we can substitute these values into the rate law:
\[ \text{Rate} = k [A]^1 [B]^1 \]
This simplifies to:
\[ \text{Rate} = k [A] [B] \]

### Step 4: Final Rate Law Expression
Thus, the final rate law expression for the overall reaction based on the RDS is:
\[ \text{Rate} = k [A][B] \]

### Summary
The rate law expression for the reaction based on the Rate Determining Step method is:
\[ \text{Rate} = k [A][B] \]

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