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) \(k=\mathrm{r} \mathrm{x}\) (c) \(k=\mathrm{C} / r\) (d) \(k=\mathrm{r} / \mathrm{C}\)

Step-by-Step Solution

Answer

The correct relation for a first-order reaction is (d) \(k = r / C\).

Step 1: Understand the order of reaction

A first-order reaction implies that the rate of reaction is directly proportional to the concentration of one reactant. The general form for a first-order reaction rate is given by \( r = k \times C \), where \( r \) is the rate of the reaction, \( k \) is the rate constant, and \( C \) is the concentration of the reactant.

Step 2: Analyze each option

Let's consider each option: - Option (a): \( k = r \times C^2 \) suggests a second-order reaction involving two molecules of the reactant, which isn't applicable to a first-order reaction. - Option (b): \( k = r C \) is missing an operation between \( r \) and \( C \), making it nonsensical. - Option (c): \( k = C / r \) would imply the rate constant is inversely proportional to the rate of reaction, which isn't correct. - Option (d): \( k = r / C \) matches the defined rate law \( r = k \times C \). Dividing both sides by \( C \) gives \( k = r / C \).

Step 3: Select the correct option

Based on the rate law for first-order reactions, the relationship \( k = \frac{r}{C} \) is correct because it reflects the definition of the rate constant in terms of reaction rate and concentration.

Key Concepts

Rate Law

The rate law is a mathematical expression that links the reaction rate with the concentration of the reactants. For a first-order reaction, the rate law is expressed as:\[ r = k \times C \]Here,

• \( r \) represents the rate of reaction, which is the speed at which the reactants are transformed into products.
• \( k \) is the rate constant, a specific coefficient that relates to the speed of reaction at a given temperature.
• \( C \) signifies the concentration of the reactant.

This formulation indicates that a change in concentration will directly affect the rate of reaction. For example, if the concentration doubles, the rate of the reaction will also double, illustrating a direct proportionality.

Rate Constant

The rate constant, \( k \), is a crucial factor in the study of chemical kinetics. It determines the speed at which a reaction proceeds and is unique for each reaction at a given temperature. In the context of first-order reactions, it is derived using the formula:\[ k = \frac{r}{C} \]Some important points about rate constants:

• The unit of \( k \) varies with the order of reaction. For a first-order reaction, it is usually expressed in s\(^{-1}\).
• The value of \( k \) is independent of the concentration but dependent on temperature and the presence of a catalyst.
• A higher value of \( k \) implies that the reaction happens at a faster rate.

Understanding the rate constant helps chemists predict how a reaction might behave under different conditions.

Reaction Order

The reaction order is an important concept in chemical kinetics. It indicates how the concentration of reactants affects the rate of reaction. For first-order reactions:

• The reaction rate depends linearly on the concentration of one reactant. This means when the reactant concentration changes, the rate changes in a linearly corresponding fashion.
• The overall order is determined solely by the exponent of the concentration term in the rate law equation. Here, it is "1" for a first-order reaction as seen in the equation \( r = k \times C \).

Recognizing the reaction order assists in predicting the kinetics of a reaction, allowing scientists to tailor conditions for desired outcomes. Moreover, different reaction orders necessitate distinct mathematical treatments to describe the rate accurately.

Chemical Kinetics

Chemical kinetics is the branch of chemistry that examines reaction rates and mechanisms. It explains how different variables influence the speed of chemical reactions:

• Temperature: An increase usually raises the rate constant \( k \), thereby speeding up the reaction.
• Concentration: As depicted in a first-order reaction, doubling the concentration doubles the reaction rate.
• Catalysts: They lower the activation energy, thus increasing the rate without being consumed in the process.

A solid grasp of chemical kinetics allows chemists to control reactions, optimizing processes in both industrial and laboratory settings. Since reactions do not all proceed at the same rate, understanding these parameters is essential to sustaining the efficiency and safety of chemical processes.