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) \(k=r x\) (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 \( k = \frac{\mathrm{r}}{\mathrm{C}} \) (Option D).

Step 1: Understand First Order Reaction

A first-order reaction is one where the rate of reaction is proportional to the first power of the concentration of one reactant. This means that the rate equation is expressed as \( \mathrm{r} = k \times \mathrm{C} \), where \( \mathrm{r} \) is the rate of reaction, \( k \) is the rate constant, and \( \mathrm{C} \) is the concentration of the reactant.

Step 2: Analyze Option A

Option (a) states that \( k = \mathrm{r} \times \mathrm{C}^2 \). In a first-order reaction, our relationship based on the definition should be \( \mathrm{r} = k \times \mathrm{C} \) rather than a \( \mathrm{C}^2 \) term. Therefore, (a) is incorrect.

Step 3: Analyze Option B

Option (b) states \( k = r x \), which is ambiguous and does not relate rate, rate constant, and concentration in a clear mathematical form relevant to a first-order reaction. Thus, this option is incorrect as it is not a defined relationship.

Step 4: Analyze Option C

Option (c) gives \( k = \frac{\mathrm{C}}{r} \). This does not directly relate to how rate constant \( k \) and concentration \( \mathrm{C} \) are related in a first-order reaction, where they must follow \( r = k \times C \). Therefore, this option is incorrect.

Step 5: Analyze Option D

Option (d) states \( k = \frac{\mathrm{r}}{\mathrm{C}} \). Rearranging the equation of a first-order reaction, you start from \( \mathrm{r} = k \times \mathrm{C} \) and solve for \( k \), resulting in \( k = \frac{\mathrm{r}}{\mathrm{C}} \), which is indeed the correct relationship for a first-order reaction.

Key Concepts

Rate Constant

In the context of chemical reactions, the rate constant, denoted as **k**, is a crucial parameter that helps us understand the speed of a reaction. The rate constant is specific to a particular reaction at a given temperature.
The value of **k** gives us insight into how quickly reactants are converted to products.
For a first-order reaction, the relationship between the rate constant and other factors is given by:

• The rate of reaction cannot be directly inferred from the rate constant alone without knowing the concentration of the reactant.
• The larger the rate constant, the faster the reaction proceeds, provided similar conditions prevail.

The units of the rate constant depend on the overall order of the reaction. For a first-order reaction, the units are reciprocal time, such as s^-1. This is because a first-order reaction's rate is directly proportional to the concentration of one reactant.

Rate of Reaction

The rate of reaction is a measure of how fast or slow a chemical reaction occurs. In a first-order reaction, the rate can be defined in relation to the concentration of the reactant and the rate constant using the equation: \( \mathrm{r} = k \times \mathrm{C} \).
This implies that:

• The rate is directly proportional to the concentration of the reactant.
• If the concentration of a reactant doubles, the rate of reaction also doubles, assuming the reaction remains first-order.

The rate of reaction is typically measured in molar concentration per unit time, such as moles per liter per second (mol/L·s). It is essential to understand this relationship to predict how changes in concentration will affect the speed of the reaction.

Concentration of Reactant

The concentration of reactant, often symbolized as **C**, plays a significant role in determining the rate of chemical reactions. In a first-order reaction, the reaction rate is directly dependent on the concentration of one specific reactant.
Understanding this dependency allows us to predict:

• How changes in the amount of reactant affect the rate of the reaction.
• Why reactions may slow down as the reactant is consumed.

In a first-order reaction, as the reactant concentration decreases due to consumption, the reaction rate also diminishes. This connection is fundamental because it helps explain why the rate of reaction changes over time and eventually slows as the reactant nears depletion. In laboratory settings, this principle is used to control and manipulate the speed of reactions by adjusting concentrations.