Problem 176 - A Complete Resource Book in Chemistry for JEE Main

Question

The following mechanism proposed for a reaction \(2 \mathrm{~A}+\mathrm{B} \rightarrow \mathrm{D}+\mathrm{E}\) is as \(\mathrm{A}+\mathrm{B} \rightleftharpoons \mathrm{C}\) (fast) \(\mathrm{A}+\mathrm{C} \rightarrow \mathrm{E}+\mathrm{D}(\mathrm{slow})\) The order of reaction is

Step-by-Step Solution

Answer

The order of reaction is three.

Step 1: Identify Rate-Determining Step

The given mechanism consists of two elementary reactions. The second step, \( \mathrm{A} + \mathrm{C} \rightarrow \mathrm{E} + \mathrm{D} \), is labeled as the slow step. Therefore, this is the rate-determining step that will govern the rate of the overall reaction.

Step 2: Write the Rate Law for the Rate-Determining Step

For the slow step \( \mathrm{A} + \mathrm{C} \rightarrow \mathrm{E} + \mathrm{D} \), the rate is dependent on the concentrations of \( \mathrm{A} \) and \( \mathrm{C} \). Therefore, the rate law for this step can be written as \( \text{Rate} = k[\mathrm{A}][\mathrm{C}] \).

Step 3: Express Intermediate Concentration in Terms of Reactants

The intermediate \( \mathrm{C} \) is formed in the fast equilibrium step \( \mathrm{A} + \mathrm{B} \rightleftharpoons \mathrm{C} \). According to the equilibrium, \( K = \frac{[\mathrm{C}]}{[\mathrm{A}][\mathrm{B}]} \). Rearranging gives \( [\mathrm{C}] = K[\mathrm{A}][\mathrm{B}] \).

Step 4: Substitute Intermediate's Concentration

Substitute \( [\mathrm{C}] = K[\mathrm{A}][\mathrm{B}] \) back into the rate law from the rate-determining step: \( \text{Rate} = k[\mathrm{A}][K[\mathrm{A}][\mathrm{B}]] = kK[\mathrm{A}]^2[\mathrm{B}] \).

Step 5: Determine Reaction Order

The derived rate law is \( \text{Rate} = kK[\mathrm{A}]^2[\mathrm{B}] \). Therefore, the overall order of the reaction is the sum of the powers of the concentration terms: \( 2 + 1 = 3 \). Thus, the order of the reaction is three.

Key Concepts

Rate-Determining Step

In a sequence of reactions, the rate-determining step (RDS) is the slowest step. Imagine it as a bottleneck in a factory line, where production is hampered until that step catches up. This step dictates the speed of the entire reaction, similar to how the slowest runner determines the pace of a relay team.
The rate-determining step governs the rate law of the overall chemical reaction. In this particular exercise, the second step \( \mathrm{A} + \mathrm{C} \rightarrow \mathrm{E} + \mathrm{D} \) is the slow step and therefore the rate-determining one.
To find out which step is slowest, scientists usually rely on experimental data or made an educated guess based on what they know about the chemical system. By identifying the RDS, one can derive the rate law, which will then tell us how fast the reaction goes under certain conditions.

Elementary Reactions

Elementary reactions are simple processes that involve a direct transformation of reactants to products. These occur in a single step and do not require any further breakdown into sub-steps. Think of them like small parts of a whole puzzle, each elementary step contributing to the complete picture of a reaction mechanism.
Our exercise involves two elementary reactions:

• \( \mathrm{A} + \mathrm{B} \rightleftharpoons \mathrm{C} \) - a fast equilibrium step where reactants interconvert rapidly.
• \( \mathrm{A} + \mathrm{C} \rightarrow \mathrm{E} + \mathrm{D} \) - a slower step that limits the overall pace of the reaction.

Understanding each step is crucial because summing up these elementary reactions gives the overall balanced equation of the complex reaction. They provide the full sequence of what occurs at a microscopic level during the reaction.

Reaction Order

The reaction order provides insight into how the concentration of reactants affects the reaction rate. It is determined by adding up the powers of concentration terms in the rate law expression.
In this case, our derived rate law is: \[ \text{Rate} = kK[\mathrm{A}]^2[\mathrm{B}] \]
Where:

• \([\mathrm{A}]^2\) implies that the reaction is second-order with respect to \([\mathrm{A}]\).
• \([\mathrm{B}]\) implies it is first-order with respect to \([\mathrm{B}]\).

This results in an overall reaction order of three: \(2 (\text{from } [\mathrm{A}]^2) + 1 (\text{from } [\mathrm{B}]) = 3\).
Reaction order is key in predicting how changes in concentration affect the reaction rate, helping chemists to control and optimize conditions for desired reactions.