Question

A substance reacts according to first-order kinetics. The rate constant for the reaction is \(1 \times\) \(10^{-2} \mathrm{sec}^{1} .\) Its initial concentration is IM. Its initial rate is (a) \(2 \times 10^{2} \mathrm{Ms}^{-1}\) (b) \(1 \times 10^{2} \mathrm{Ms}^{-1}\) (c) \(1 \times 10^{-2} \mathrm{Ms}^{-1}\) (d) \(2 \times 10^{-2} \mathrm{Ms}^{-1}\)

Short answer

(c) \(1 \times 10^{-2} \ \mathrm{Ms}^{-1} \)

Step-by-step solution

Understand First-Order Kinetics

First-order kinetics means that the rate of reaction is directly proportional to the concentration of the one reactant that is involved. The rate law for a first-order reaction can be expressed as \( r = k[A] \), where \( r \) is the rate, \( k \) is the rate constant, and \( [A] \) is the concentration of the reactant.

Apply the Rate Law

For this problem, we know that the rate constant \( k \) is \( 1 \times 10^{-2} \ \mathrm{sec}^{-1} \) and the initial concentration \( [A]_0 \) is \( 1 \ \mathrm{M} \). Substituting these values into the rate law, \( r = k[A] = (1 \times 10^{-2} \ \mathrm{sec}^{-1})(1 \text{ M}) \).

Calculate the Initial Rate

Perform the multiplication to find the initial rate: \( r = 1 \times 10^{-2} \ \mathrm{M} \cdot \mathrm{sec}^{-1} \). Therefore, the initial rate of the reaction is \( 1 \times 10^{-2} \ \mathrm{Ms}^{-1} \).

Compare with Given Options

The initial rate calculated is \( 1 \times 10^{-2} \ \mathrm{Ms}^{-1} \). Looking at the provided options, option (c) matches our calculated value of \( 1 \times 10^{-2} \ \mathrm{Ms}^{-1} \).

Key concepts

Rate Constant in First-Order Kinetics

The rate constant, often denoted as \( k \), is a crucial component in understanding reaction kinetics, especially in first-order reactions. First-order reactions have a direct dependency between the reaction rate and the concentration of the reactant. The rate constant serves as a proportionality factor within this relationship.
In the context of first-order kinetics, the rate law is represented as \( r = k[A] \). This formula indicates that the rate \( r \) is directly proportional to the concentration \( [A] \) of the reactant. The rate constant remains constant for a given reaction at a specific temperature.
It is expressed in units of \( ext{time}^{-1} \) (e.g., \( ext{sec}^{-1} \)) in first-order reactions. This unique unit indicates how the concentration of the reactant impacts the rate over time. By knowing the rate constant, you can predict how fast a reactant will convert into products, given its concentration.

Understanding Initial Rate

The initial rate of a reaction is the rate at which reactants transform into products at the very beginning, usually immediately after the reactants are mixed. Calculating the initial rate is valuable in situations where a swift estimation of reaction speed at the onset is needed.
To calculate the initial rate for a first-order reaction, one uses the equation \( r = k[A] \), using the known initial concentration of the reactant \( [A]_0 \) and the rate constant \( k \). This gives a clear picture of how quickly the reaction begins. For example, if \( k = 1 \times 10^{-2} \ ext{sec}^{-1} \) and \( [A]_0 = 1 \ ext{M} \), the initial rate \( r \) is simply \( k[A]_0 = 1 \times 10^{-2} \ ext{M} \cdot \text{sec}^{-1} \).
The initial rate is paramount for understanding how quickly the reaction progresses right after the start, and it can inform further investigative or practical approaches regarding the reaction.

Role of Concentration in Reaction Rate

Concentration plays a pivotal role in influencing the rate of reactions, particularly for first-order reactions. In a first-order reaction, the initial concentration of the reactant directly dictates the initial rate of conversion to products.
The general rate expression for a first-order reaction, \( r = k[A] \), makes it evident that as the concentration of reactant \( [A] \) changes, the rate of the reaction changes correspondingly. A higher concentration means more reactant molecules are available to react, leading to an increase in the rate of reaction.
This relationship underscores the importance of keeping track of the concentration during experiments, as it can significantly impact the reaction's speed and outcome. Furthermore, since concentration is directly measurable, it provides a practical means to control and predict the reaction kinetics effectively.