Problem 77 - A Complete Resource Book in Chemistry for JEE Main

Question

For the reaction \(2 \mathrm{~A}+\mathrm{B} \longrightarrow 3 \mathrm{C}+\mathrm{D}\) which of the following does not express the reaction rate? (a) \(\frac{\mathrm{d}[\mathrm{D}]}{\mathrm{dt}}\) (b) \(-\frac{\mathrm{d}[\mathrm{A}]}{2 \mathrm{dt}}\) (c) \(\frac{\mathrm{d}[\mathrm{C}]}{3 \mathrm{dt}}\) (d) \(-\frac{\mathrm{d}[\mathrm{B}]}{\mathrm{dt}}\)

Step-by-Step Solution

Answer

Option (c) is incorrect.

Step 1: Understand the Reaction

Given the chemical reaction: \(2 \mathrm{A} + \mathrm{B} \rightarrow 3 \mathrm{C} + \mathrm{D}\). We need to express the rate of reaction based on the changes in the concentration of the reactants and products.

Step 2: Define the Reaction Rate Formulas

The rate of reaction is defined by the change in concentration of reactants or products per unit time. For the given reaction, the rate can be expressed as:- \(\text{Rate} = -\frac{1}{2} \frac{\mathrm{d}[\mathrm{A}]}{\mathrm{dt}}\)- \(\text{Rate} = -\frac{\mathrm{d}[\mathrm{B}]}{\mathrm{dt}}\)- \(\text{Rate} = \frac{1}{3} \frac{\mathrm{d}[\mathrm{C}]}{\mathrm{dt}}\)- \(\text{Rate} = \frac{\mathrm{d}[\mathrm{D}]}{\mathrm{dt}}\)

Step 3: Analyze the Options

Let's examine each option compared to the standard reaction rate expressions:- Option (a) \(\frac{\mathrm{d}[\mathrm{D}]}{\mathrm{dt}}\) matches one of the rate expressions.- Option (b) \(-\frac{\mathrm{d}[\mathrm{A}]}{2\mathrm{d}t}\) matches the rate expression for \(\mathrm{A}\).- Option (c) \(\frac{\mathrm{d}[\mathrm{C}]}{3\mathrm{d}t}\) is incorrectly expressed; it should be \(\frac{1}{3} \frac{\mathrm{d}[\mathrm{C}]}{\mathrm{dt}}\).- Option (d) \(-\frac{\mathrm{d}[\mathrm{B}]}{\mathrm{dt}}\) matches the rate expression for \(\mathrm{B}\).

Step 4: Identify Incorrect Expression

From Step 3, the incorrect expression is option (c) \(\frac{\mathrm{d}[\mathrm{C}]}{3\mathrm{d}t}\), as it does not follow the correct form \(\frac{1}{3} \frac{\mathrm{d}[\mathrm{C}]}{\mathrm{dt}}\).

Key Concepts

Chemical Kinetics

Chemical kinetics is a branch of chemistry that studies the speed or rate at which chemical reactions occur. This is vital for understanding how fast reactions happen and how different conditions, such as temperature and concentration, affect these rates. It involves analyzing the steps or mechanisms behind reactions.
In the case of our given reaction, \(2 \mathrm{A} + \mathrm{B} \rightarrow 3 \mathrm{C} + \mathrm{D}\), chemical kinetics helps us understand how quickly reactants are converted into products over time. By studying these rates, chemists can optimize reactions to be more efficient in industrial and laboratory settings. Thus, chemical kinetics becomes fundamental for developing new products and improving processes, as it helps determine the optimal conditions needed for reactions.

Rate Expression

A rate expression shows the mathematical relationship between reaction rate and the concentration of reactants or products. It is an essential tool in chemical kinetics as it quantifies the rate of a reaction.
For the reaction \(2 \mathrm{A} + \mathrm{B} \rightarrow 3 \mathrm{C} + \mathrm{D}\), we can express its rate through various equations, which relate to changes in concentration over time, namely:

• \(\text{Rate} = -\frac{1}{2} \frac{\mathrm{d}[\mathrm{A}]}{\mathrm{dt}}\)
• \(\text{Rate} = -\frac{\mathrm{d}[\mathrm{B}]}{\mathrm{dt}}\)
• \(\text{Rate} = \frac{1}{3} \frac{\mathrm{d}[\mathrm{C}]}{\mathrm{dt}}\)
• \(\text{Rate} = \frac{\mathrm{d}[\mathrm{D}]}{\mathrm{dt}}\)

These expressions are derived by observing how quickly the concentration of one reactant or product changes relative to time, which can then be related to other components of the reaction through stoichiometry.

Stoichiometry

Stoichiometry involves the quantitative relationships between reactants and products in a chemical reaction. It is critical in the calculation of reaction rates because it tells us how the concentration of one substance relates to another.In the chemical equation \(2 \mathrm{A} + \mathrm{B} \rightarrow 3 \mathrm{C} + \mathrm{D}\), stoichiometry is the basis for expressing the reaction rates properly. Each component of the reaction is represented by a coefficient, indicating their relative amounts.
For example, if 2 moles of \(\mathrm{A}\) are used up for every mole of \(\mathrm{B}\), it means the rate of disappearance of \(\mathrm{A}\) can be halved in comparison to \(\mathrm{B}\)'s disappearance to obtain the common reaction rate. This is why the fraction \(\frac{1}{2}\) appears in its rate expression.

Concentration Change

Concentration change refers to how the concentration of reactants or products varies as a reaction proceeds. It's closely monitored in relation to time, giving us the rate at which a reaction advances.
In a reaction like \(2 \mathrm{A} + \mathrm{B} \rightarrow 3 \mathrm{C} + \mathrm{D}\), tracking these concentration changes is crucial. For instance, if \(\mathrm{C}\) increases by 3 units as \(\mathrm{A}\) decreases by 2 units, this direct relationship is derived from the stoichiometric ratios of the reaction. The rate expressions use coefficients to adjust these changes so that all components of the reaction align properly.

• Reactants typically decrease in concentration, noted by a negative rate expression.
• Products typically increase, noted by a positive rate expression.

This technique aids in determining not just the speed, but also the efficiency of chemical processes.