Problem 37 - A Complete Resource Book in Chemistry for JEE Main

Question

Which of the following relation is correct for a first order reaction? \((k=\) rate constant; \(\mathrm{r}=\) rate of reaction; \(\mathrm{C}=\) conc, of reactant) (a) \(k=\mathrm{r} \times \mathrm{C}^{2}\) (b) \(k=\mathbf{r} \mathbf{x}\) (c) \(k=\mathrm{C} / r\) (d) \(k=\mathrm{r} / \mathrm{C}\)

Step-by-Step Solution

Answer

The correct relation is (d): \( k = \frac{r}{C} \).

Step 1: Understanding First Order Reactions

In a first-order reaction, the rate of reaction is directly proportional to the concentration of one reactant. This means we can express the rate of reaction, \( r \), as \( r = k \cdot C \), where \( C \) is the concentration of the reactant and \( k \) is the rate constant.

Step 2: Expressing the Rate Constant

Given the relationship \( r = k \cdot C \), we can rearrange this equation to solve for the rate constant \( k \). This gives us \( k = \frac{r}{C} \).

Step 3: Identifying the Correct Relation

Now, compare this derived expression \( k = \frac{r}{C} \) with the given options:- Option (a): \( k = r \cdot C^2 \)- Option (b): \( k = r \cdot x \)- Option (c): \( k = \frac{C}{r} \)- Option (d): \( k = \frac{r}{C} \)The correct expression from our derivation is option (d): \( k = \frac{r}{C} \).

Key Concepts

Understanding the Rate Constant

The rate constant, often denoted by the symbol \( k \), is a fundamental component in the study of chemical kinetics. It is a unique value that determines the speed of a reaction for a given set of conditions. In a first-order reaction, the rate constant simplifies the relationship between the rate of the reaction and the concentration of the reactant.
By definition, a first-order reaction is one where the rate is dependent on the power of one of the concentration terms. As such, this rate is proportional to the concentration of a single reactant. To determine \( k \), we rearrange the relationship for a first-order reaction:

• Rate of reaction, \( r = k \cdot C \)
• Rearranging gives: \( k = \frac{r}{C} \)

This equation signifies how \( k \) relates to both the rate and the concentration, making it a crucial element in predicting how a chemical reaction will proceed under predefined conditions.

Defining the Rate of Reaction

The rate of reaction \( r \) is a measure of how fast a chemical process occurs. It quantifies the change in concentration of a reactant or product per unit time. For a first-order reaction, the rate is directly determined by the concentration of a singular reactant. In simplified terms, if you imagine a solution of chemicals reacting together, the rate of reaction tells you how quickly this transformation happens. The formula connecting the rate to the concentration and the rate constant is:

• \( r = k \cdot C \)

Understanding this relationship helps in numerous applications:

• Predicting how a change in reactant concentration affects reaction speed.
• Estimating how long a reaction may take under certain conditions.

Thus, the rate of reaction gives valuable insight into the dynamics of the chemistry involved.

Importance of Concentration of Reactant

In the context of first-order reactions, the concentration of reactant \( C \) is pivotal. It reflects how many particles of the reactant are present in a given volume, which ultimately influences how often these particles collide and react. For reactions that follow first-order kinetics, there's a straightforward linear dependency of the reaction rate on reactant concentration. This is captured in the fundamental equation:

• \( r = k \cdot C \)

Where:

• \( r \) is the rate of reaction.
• \( k \) is the rate constant.
• \( C \) is the concentration of the reactant.

Here's why understanding reactant concentration is important:

• As the concentration increases, so does the rate of reaction.
• It allows chemists to control and modify reaction conditions to achieve desired results faster.
• Tracking changes in concentration helps in determining kinetic parameters experimentally.

In summary, concentration measurements are crucial in the study and application of chemical reactions, providing the basis for calculating other critical kinetic parameters.