Problem 81 - A Complete Resource Book in Chemistry for JEE Main

Question

The rate of a certain hypothetical reaction \(\mathrm{A}+\mathrm{B}+\mathrm{C} \longrightarrow\) Products is given by \(\mathrm{r}=-\frac{\mathrm{d}[\mathrm{A}]}{\mathrm{dt}}=k[\mathrm{~A}]^{1 / 2}[\mathrm{~B}]^{1 / 3}[\mathrm{C}]^{\mathrm{L} / 4}\) The order of the reaction is (a) \(13 / 12\) (b) \(13 / 14\) (c) \(12 / 13\) (d) \(13 / 11\)

Step-by-Step Solution

Answer

The reaction order is (a) \(13/12\).

Step 1: Understand the Rate of Reaction Equation

The given rate equation is \( r = - \frac{d[A]}{dt} = k[A]^{1/2}[B]^{1/3}[C]^{L/4} \). This represents how the rate of the reaction depends on the concentration of reactants A, B, and C, with corresponding powers.

Step 2: Determine Reaction Order for Each Reactant

The order with respect to \( A \) is \( 1/2 \), for \( B \) it is \( 1/3 \), and for \( C \), it is \( L/4 \). A reaction order for each reactant is the power to which its concentration is raised in the rate law.

Step 3: Total Reaction Order Calculation

To find the total order of the reaction, sum the exponents of all reactants in the rate equation: \( 1/2 + 1/3 + L/4 \).

Step 4: Find Common Denominator

First, find a common denominator for the fractions: \( 1/2 = 6/12 \), \( 1/3 = 4/12 \), \( L/4 = 3L/12 \).

Step 5: Sum the Orders

The total order is \( 6/12 + 4/12 + 3L/12 = (10 + 3L)/12 \).

Step 6: Match with Given Options

For the order to match any given option, solve \((10 + 3L)/12 = 13/12\) (since the total order should equate to one of the given choices and be a single fraction involving L). The simplicity of format suggests \(3L = 3\), thus \(L = 1\).

Step 7: Verify Answer

With \(L = 1\), calculate the total order as \((10 + 3 \times 1)/12 = 13/12\). Therefore, the order of the reaction is indeed \(13/12\).

Key Concepts

Rate Equation Analysis

The understanding of rate equations is crucial to determining how fast a chemical reaction proceeds. In the provided exercise, the rate equation is expressed as \( r = - \frac{d[A]}{dt} = k[A]^{1/2}[B]^{1/3}[C]^{L/4} \). This equation shows how the rate of reaction is dependent on the concentration of the reactants \( A \), \( B \), and \( C \). It's essential to notice the exponents of each reactant's concentration. These exponents illustrate the individual reaction orders with respect to each reactant.
The constant \( k \) represents the rate constant, which is specific to the reaction and the particular conditions under which it occurs. It's worth noting that the rate constant is influenced by factors such as temperature and the presence of a catalyst. By analyzing each component of this rate equation, you can uncover the dynamics of the reaction and predict how changes in concentration might impact the reaction rate.

Reaction Kinetics

Reaction kinetics is the science of exploring the rate at which a chemical reaction occurs. It involves studying the factors that affect the speed of reactions, such as reactant concentrations, temperature, and the physical state of the reactants.
In the context of the given exercise, we explore the reaction kinetics through the rate equation. Each exponent on the reactant concentration tells us the degree to which the reactant influences the rate of reaction.

• The higher the exponent, the more impact a change in that reactant's concentration will have on the reaction rate.
• If an exponent is zero for any reactant, it implies that the rate does not depend on the concentration of that reactant.
• In our case, the fractional exponents indicate non-integer reaction orders, which means the reaction mechanism might be complex or involve multiple steps.

Understanding reaction kinetics allows chemists to develop models that predict the behavior of reactions under various conditions.

Fractional Reaction Orders

Fractional reaction orders often surprise those new to reaction kinetics, but they are common in complex reactions. In our exercise, we see fractional orders of \( 1/2 \), \( 1/3 \), and \( L/4 \). These fractions indicate that the relationship between concentration and rate isn't straightforward.
Such fractions suggest that the reaction might involve intermediate species or transitions that aren't visible in the overall equation. It can imply a stepwise mechanism, where the observed rate law is a result of several elementary steps.
Here's what fractional orders imply:

• They can point to reactions where the reactant undergoes transformations before affecting the rate.
• Often, they are seen in reactions that involve catalysts or inhibitory effects.
• The calculated overall order, like \( 13/12 \) in our problem, offers insights into the complexity and intermediates of the reaction.

Understanding these concepts is key to mastering reaction mechanisms and the application of kinetic theory in solving real-world chemical problems.