5.9.A.1 Rate Law Expression: Approximations for Non-Rate-Limiting First Elementary Reactions:

1. Elementary Reactions and Rate-Limiting Step:

i. Elementary Reaction Characteristics:
a. Single-Step: One step, no intermediates.
b.Rate Law: Formulated directly from stoichiometry, e.g., 

$\text{rate} = k[A]^a[B]^b$

.
c. Molecularity: Deals with number of reactants (unimolecular, bimolecular, etc.).
d. No Intermediates: Reactants are directly converted into products.
e. Arrhenius Behavior: Rate constant depends upon temperature,

$k = A e^{-E_a/RT}$

.

ii. Determining the Rate-Limiting Step:
a. Slowest Step: The step that possesses maximum activation energy.
b. Experimental Data: Rate law signifies the rate-determining step.
c. Energy Barrier: The highest energy on the reaction coordinate diagram.
d. Rate Law Analysis: Relates rate law to an elementary step.
e. Steady-State Approximation: Addresses intermediates at steady concentration.

In short, the rate-determining step is the slowest, highest-energy step of a reaction mechanism.

2. Pre-Equilibrium Approximation:

The pre-equilibrium approximation is used in reaction mechanisms where an intermediate step reaches equilibrium before going on to the next step. Under this approximation, it is assumed that the intermediate concentration remains constant over time since the reaction sets an equilibrium state quickly before going on with the subsequent steps.

i. When It’s Applied:

a. Intermediate Formation: A mechanism in which formation of an intermediate early on, and this intermediate is rapidly in equilibrium with its reactants.

b. Fast Initial Step: In mechanisms with a fast first step which forms an equilibrium prior to the slow rate-limiting step.

c. Simplification of Complex Mechanisms: The pre-equilibrium approximation simplifies complex reaction mechanisms, particularly when concentrations of intermediates cannot be modeled or measured directly.

ii. How It Works:

– Equilibrium Hypothesis: Assume that the first portion of the reaction (with the intermediate) occurs very quickly in equilibrium.
– Constant Concentration: The intermediate concentration is assumed to be constant during the reaction.
– Rate Law: An equilibrium constant from the first step is used in writing the intermediate concentration in terms of reactant concentrations in order to provide an overall rate law simplified.

iii. Example:

In a mechanism like:

1. A+B⇌I (fast equilibrium)
2. I→C (slow step, rate-determining)

The pre-equilibrium approximation is that the intermediate (I) is in equilibrium with (A) and (B), and therefore its concentration can be written in terms of the equilibrium constant (K). This enables us to condense the two steps into one rate law for the slow step.

3. Rate Law Expression:  To derive the rate law from a reaction mechanism, we analyze the individual steps and their stoichiometry. The crucial aspect is to understand how intermediates and the rate constants of each step influence the overall rate law.

i.  General Process:
a. Write out the mechanism: Decompose the overall reaction into a series of elementary steps.
b. Identify the rate-limiting step: The slowest step determines the overall reaction rate.
c. Express the rate law for the rate-limiting step: The rate law for this step is based on the concentrations of the reactants involved.
d. Account for intermediates: If the rate-limiting step includes an intermediate, express its concentration using the equilibrium constant from previous steps (if applying the pre-equilibrium approximation) or through steady-state approximations.

ii. Steps for Deriving the Rate Law:

a. Write the Mechanism:

Consider a simple reaction mechanism:

1. $A + B \rightleftharpoons I$

   A+B⇌I (fast equilibrium)

2. I→C (slow step, rate-determining)

b. Identify the Rate-Limiting Step:

In this case, the second step is the slow one, making it the rate-determining step. We will formulate the rate law based on this step.

c. Rate Law for Rate-Determining Step:
For the rate-determining step, the rate law is derived from the concentration of the reactants involved.

For step 2, the rate law is: rate=k2​[I]

where (k2) represents the rate constant for the second step, and ([I]) denotes the concentration of the intermediate.

d. Express Intermediate Concentration:
The intermediate (I) is in equilibrium with (A) and (B) from step 1. Assuming fast equilibrium, we can use the equilibrium constant (K1) for the first step:

$$K_1=\frac{[I]}{[A][B]}$$

$$[I]=K_1[A][B]$$

e . Substitute the Intermediate into the Rate Law:
Insert the expression for
[I] into the rate law for the rate-determining step:

rate = k₂[I] = k₂K₁[A][B]

f. Simplify the Rate Law:
Let k = k₂K₁, which represents the overall rate constant:

rate = k[A][B]

Therefore, the overall rate law for the reaction is:

rate = k[A][B]

This indicates that it is a bimolecular reaction since the rate is influenced by the concentrations of both A and B.

Role of Intermediates:
In this scenario, the intermediate I does not show up in the final rate law because we applied the pre-equilibrium approximation to express it in terms of the concentrations of A and B. If the intermediate had been stable or involved in several steps, a steady-state approximation might have been necessary instead.

4. Reaction Mechanisms:

i.Pre-Equilibrium Approximation in Rate Law Simplification:

The pre-equilibrium approximation is a technique which assumes that an intermediate quickly reaches equilibrium in the first step of a reaction mechanism in order to simplify the rate law by expressing intermediates’ concentrations in terms of reactants and equilibrium constants.

ii. How It Helps:
– Avoids Intermediate Tracking: Instead of determining the intermediates, their concentrations are expressed through the equilibrium constant.
– Simplifies Rate Law: Reduces involved multi-step reactions into simple-to-compute forms. For example:

$$A + B \rightleftharpoons I \quad \text{(fast)} \quad \text{and} \quad I \rightarrow C \quad \text{(slow)}$$ $$\text{rate}=k[A][B]$$

iii. Other Approximations:
– Steady-State Approximation: Assumes the intermediate concentration is uniform across the reaction and reduces the rate law to reactants and rate constants.

iv. Summary:
Approximations like pre-equilibrium and steady-state simplify the rate law calculation to a lesser degree by reducing trace intermediates tracking and focusing the slow, rate-determining step.