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The problem involves determining the rate law for a given reaction mechanism. The reaction mechanism is provided as:

1. \( 2A + B \leftrightarrow C \)
   - Rate constants: \( k_1 = 9.25 \times 10^{-2} \) (forward), \( k_1' = 2.32 \times 10^{-1} \) (reverse)

2. \( C \rightarrow D + E \)
   - Rate constant: \( k_2 = 8.55 \times 10^{-7} \)

3. \( D + A \rightarrow F \)
   - Rate constant: \( k_3 = 4.41 \times 10^{-4} \)

### Step-by-Step Approach:

#### Step 1: Identify the rate-determining step
The rate-determining step (RDS) is the slowest step, controlling the overall rate of the reaction. Typically, the step with the smallest rate constant is the RDS. Here, \( k_2 = 8.55 \times 10^{-7} \) is significantly smaller than the other rate constants, indicating that the reaction \( C \rightarrow D + E \) is the slowest step and the rate-determining step.

#### Step 2: Write the rate law for the slowest step
The rate law for the rate-determining step (RDS) is generally based on the reactants of that step. In this case, the slowest step is \( C \rightarrow D + E \), so the rate law would initially be:

\[
\text{Rate} = k_2 [C]
\]

#### Step 3: Express \( [C] \) in terms of \( A \) and \( B \)
Since \( C \) is an intermediate, we use the equilibrium expression from the first step \( 2A + B \leftrightarrow C \) to express \( [C] \) in terms of the concentrations of \( A \) and \( B \). At equilibrium, the forward rate equals the reverse rate:

\[
k_1 [A]^2 [B] = k_1' [C]
\]

Solving for \( [C] \):

\[
[C] = \frac{k_1 [A]^2 [B]}{k_1'}
\]

#### Step 4: Substitute \( [C] \) into the rate law
Now, substitute this expression for \( [C] \) into the rate law for the RDS:

\[
\text{Rate} = k_2 \left( \frac{k_1 [A]^2 [B]}{k_1'} \right)
\]

Simplifying:

\[
\text{Rate} = \frac{k_2 k_1}{k_1'} [A]^2 [B]
\]

Thus, the rate law for the overall reaction is:

\[
\boxed{\text{Rate} = k[A]^2 [B]}
\]

This corresponds to the third option in the provided choices: \( \text{Rate} = k[A]^2[B] \).

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Would you like further details or have any questions?

Here are five related questions:
1. How does the rate-determining step affect the overall reaction rate?
2. What is the significance of intermediate species in reaction mechanisms?
3. How do you determine equilibrium constants in reversible reactions?
4. What factors influence whether a reaction is fast or slow?
5. How do temperature changes affect reaction rates and constants?

**Tip:** Always look for the slowest step (smallest rate constant) to identify the rate-determining step in multi-step reactions.

For the theoretical reaction 2A + B → E + F, the reaction mechanism was determined to be: A + B ⇌ C (k1 = 9.25 × 10^-2, k1' = 2.32 × 10^-1), C → D + E (k2 = 8.55 × 10^-7), D + A → F (k3 = 4.41 × 10^-4). What is the rate law for the overall reaction?

Learn how to determine the rate law for a multi-step reaction mechanism involving 2A + B → E + F. This problem covers key concepts in chemical kinetics, including identifying the rate-determining step and expressing intermediates using equilibrium constants. Ideal for undergraduate chemistry students.

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