Problem 4 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

For the reaction \(2 \mathrm{~A}+3 \mathrm{~B}+\frac{3}{2} \mathrm{C} \rightarrow 3 \mathrm{P}\), which statement is correct ? [Main Sep. \(\mathbf{0 3}, \mathbf{2 0 2 0}\) (II)](a) \(\frac{\mathrm{dn}_{\mathrm{A}}}{\mathrm{dt}}=\frac{3}{2} \frac{\mathrm{dn}_{\mathrm{B}}}{\mathrm{dt}}=\frac{3}{4} \frac{\mathrm{dn}_{\mathrm{C}}}{\mathrm{dt}}\) (b) \(\frac{\mathrm{dn}_{\mathrm{A}}}{\mathrm{dt}}=\frac{\mathrm{dn}_{\mathrm{B}}}{\mathrm{dt}}=\frac{\mathrm{d} n_{\mathrm{C}}}{\mathrm{dt}}\) (c) \(\frac{\mathrm{dn}_{\mathrm{A}}}{\mathrm{dt}}=\frac{2}{3} \frac{\mathrm{dn}_{\mathrm{B}}}{\mathrm{dt}}=\frac{4}{3} \frac{\mathrm{dn}_{\mathrm{C}}}{\mathrm{dt}}\) (d) \(\frac{\mathrm{dn}_{\mathrm{A}}}{\mathrm{dt}}=\frac{2}{3} \frac{\mathrm{dn}_{\mathrm{B}}}{\mathrm{dt}}=\frac{3}{4} \frac{\mathrm{dn}_{\mathrm{C}}}{\mathrm{dt}}\)

Step-by-Step Solution

Answer

The correct answer is (d).

Step 1: Understanding Reaction Rate Expressions

For a balanced chemical reaction: \(2\text{A} + 3\text{B} + \frac{3}{2}\text{C} \rightarrow 3\text{P}\), each reactant and product is related by a stoichiometric ratio. Generally, the rate of change of concentration of each species is related as: \(\frac{1}{\text{stoichiometry}}\frac{dn}{dt}\).

Step 2: Express Rates for Reaction Components

Based on the reaction's stoichiometric coefficients, the rate expressions for components are: \(\frac{1}{2}\frac{dn_{A}}{dt} = \frac{1}{3}\frac{dn_{B}}{dt} = \frac{2}{3}\frac{dn_{C}}{dt} = \frac{1}{3}\frac{dn_{P}}{dt} \). Here, \(\frac{dn_{A}}{dt}\), \(\frac{dn_{B}}{dt}\), and \(\frac{dn_{C}}{dt}\) are rates of consumption of A, B, and C, respectively.

Step 3: Simplify and Compare Options

Simplifying the relationships, we find: \(\frac{dn_{A}}{dt} = \frac{2}{3}\frac{dn_{B}}{dt} = \frac{3}{4}\frac{dn_{C}}{dt}\). Now, compare these with the given options to find the correct choice.

Step 4: Identify the Correct Answer

Comparing the simplified expression \(\frac{dn_{A}}{dt} = \frac{2}{3}\frac{dn_{B}}{dt} = \frac{3}{4}\frac{dn_{C}}{dt}\) with the listed options shows that option (d) matches our derived expression.

Key Concepts

Reaction Rate Expressions

In chemical kinetics, reaction rate expressions describe how the reaction proceeds by relating the change in concentration of reactants and products over time. These expressions are vital for understanding the dynamics of a reaction. The rate at which a reactant is consumed or a product is formed can be depicted using differential rate expressions. For instance, for a general reaction involving a reactant A, the rate expression can be written as \( \frac{d[A]}{dt} \), where \([A]\) denotes the concentration of A over time. In our specific reaction \( 2 \text{A} + 3 \text{B} + \frac{3}{2} \text{C} \rightarrow 3\text{P} \), the rate expressions must account for the stoichiometry of each reactant and product. By using the stoichiometric coefficients, we can write rate expressions that describe the relative rates at which each substance reacts or is formed. This ensures that the relationship between them is maintained, as observed in how A, B, and C are being consumed. It is a balanced approach to capture every component involved in the reaction's kinetics.

Stoichiometric Coefficients

Stoichiometric coefficients are the numbers in front of molecules in a chemical reaction that indicate the proportion in which reactants are used and products are formed. These coefficients are crucial because they maintain the balance of atoms during a chemical reaction, ensuring mass conservation. In the reaction \( 2 \text{A} + 3 \text{B} + \frac{3}{2} \text{C} \rightarrow 3\text{P} \), the coefficients (2, 3, and \( \frac{3}{2} \)) determine how much of each reactant is needed to produce a certain amount of product. Stoichiometric coefficients allow us to set up the rate expressions as seen in the equation: \( \frac{1}{2} \frac{dn_{A}}{dt} = \frac{1}{3} \frac{dn_{B}}{dt} = \frac{2}{3} \frac{dn_{C}}{dt} = \frac{1}{3} \frac{dn_{P}}{dt} \). This relationship helps us understand the precise reaction dynamics, linking the consumption rates of A, B, and C to the creation of P. By scaling the rate of change of each substance by its stoichiometric coefficient, we achieve a balanced perspective of how the overall reaction progresses.

Rate of Reaction

The rate of reaction refers to how quickly or slowly a reaction proceeds, which can be crucial for both laboratory settings and industrial applications. The rate is essentially how the concentration of reactants decreases or how products increase over time. It is expressed as \( \, \frac{1}{\text{stoichiometric coefficient}} \cdot \frac{d[n]}{dt} \, \) for any component in a given reaction.In the example \( 2\text{A} + 3\text{B} + \frac{3}{2}\text{C} \rightarrow 3\text{P} \), the rate of reaction is deduced from these coefficient adjustments. As a rule of thumb, a larger coefficient for a reactant means that reactant is consumed faster. The relative rates we calculated, such as \( \frac{dn_{A}}{dt} = \frac{2}{3} \frac{dn_{B}}{dt} = \frac{3}{4} \frac{dn_{C}}{dt} \), show how quickly each reactant is used up in relation to each other. Understanding the rate of reaction allows chemists to control the conditions under which reactions occur and optimize for desired outcomes. This aspect of chemical kinetics ensures that chemical processes are efficient and safe, especially when scaling reactions to larger volumes.