Problem 18 - A Complete Resource Book in Chemistry for JEE Main

Question

Among which of the following factor the specific reaction rate of a first- order reaction depends on (a) temperature (b) concentration of reactant (c) pressure (d) volume

Step-by-Step Solution

Answer

The specific reaction rate of a first-order reaction depends on temperature.

Step 1: Understanding First-Order Reactions

A first-order reaction is characterized by a rate that depends linearly on the concentration of a single reactant. The rate equation for a first-order reaction is given by:\[ r = k[A] \]where \( r \) is the rate, \( k \) is the rate constant, and \([A]\) is the concentration of the reactant.

Step 2: Examining the Rate Constant

The rate constant \( k \) is a specific factor that dictates how fast a reaction proceeds. For a first-order reaction, it is independent of the reactant's concentration, pressure, or volume. However, the rate constant \( k \) is dependent on temperature, often increasing with temperature due to the Arrhenius equation:\[ k = A e^{-E_a/RT} \]where \( A \) is the pre-exponential factor, \( E_a \) is the activation energy, \( R \) is the universal gas constant, and \( T \) is the temperature.

Step 3: Conclusion on Dependence Factors

Based on the analysis of the rate constant, the specific reaction rate \( k \) of a first-order reaction is dependent on temperature. It is not affected by the concentration of the reactant, pressure, or volume, as these do not alter \( k \).

Key Concepts

Reaction Rate

The reaction rate describes how quickly a chemical reaction occurs. In the context of first-order reactions, the rate is proportional to the concentration of a single reactant. This means that as the concentration of the reactant decreases, the reaction rate also decreases. An important point to remember is that unlike higher-order reactions, a first-order reaction does not rely on the concentration of multiple reactants.

• The rate equation for a first-order reaction is expressed as: \[ r = k[A] \]where \( r \) is the reaction rate, \( k \) is the rate constant, and \([A]\) is the concentration of the reactant.
• Since the reaction rate depends linearly on the concentration, plotting the concentration versus time results in an exponential decay curve reflecting the decreasing rate as the reaction proceeds.

Understanding these concepts helps clarify why first-order reactions are crucial for both theoretical studies and practical applications, such as radioactive decay and drug metabolism.

Temperature Dependence

Temperature plays a pivotal role in chemical reactions, including first-order reactions. As temperature increases, the molecules involved in the reaction move more rapidly and collide more frequently with each other. This increase in molecular collisions enhances the likelihood that reactants will successfully convert to products, effectively increasing the reaction rate.

The relationship between temperature and reaction rate is quantitatively expressed by the Arrhenius equation:\[ k = A e^{-E_a/RT} \]

• In this equation, \( k \) is the rate constant, \( A \) is the pre-exponential factor (frequency of collisions), \( E_a \) is the activation energy (energy required to initiate the reaction), \( R \) is the universal gas constant, and \( T \) is the temperature in Kelvin.
• The equation shows that the rate constant \( k \) increases exponentially with an increase in temperature, making reactions faster at higher temperatures.

By analyzing the Arrhenius equation, it becomes evident why temperature is a critical factor in determining the speed of a reaction.

Rate Constant

The rate constant, represented by \( k \), is a crucial component in understanding chemical kinetics. It serves as a proportionality factor that links the reaction rate to the concentration of reactants in the rate law equation.

• For first-order reactions, the rate constant is independent of the concentration of reactants, unlike higher-order reactions where the rate may depend on the concentrations of multiple substances.
• The rate constant is primarily determined by temperature, as evidenced by its presence in the Arrhenius equation. Any changes in temperature can significantly affect the rate constant \( k \), thereby altering the speed of the reaction.

Understanding the nature of the rate constant helps chemists predict how a reaction will behave under different conditions, providing crucial insights for experimental and industrial processes.