Question

The rate constant of first-order reaction is \(10^{-2} \mathrm{~min}^{-1}\). The half-life period of reaction is (a) \(693 \mathrm{~min}\) (b) \(69.3 \mathrm{~min}\) (c) \(6.93 \mathrm{~min}\) (d) \(0.693 \mathrm{~min}\)

Short answer

The half-life is 69.3 min, which is option (b).

Step-by-step solution

Understanding Half-Life Equation for First-Order Reactions

The formula to find the half-life \( t_{1/2} \) of a first-order reaction is \( t_{1/2} = \frac{0.693}{k} \), where \( k \) is the rate constant.

Substitute the Given Rate Constant

Given that the rate constant \( k = 10^{-2} \mathrm{~min}^{-1} \). Substitute \( k \) into the half-life equation: \( t_{1/2} = \frac{0.693}{10^{-2}} \).

Calculate the Half-Life Period

Perform the division: \( t_{1/2} = \frac{0.693}{0.01} = 69.3 \mathrm{~min} \).

Identify the Correct Answer from Options

Among the provided options, the half-life period we calculated, \( 69.3 \mathrm{~min} \), matches option (b).

Key concepts

Rate Constant

In first-order reactions, the rate constant, denoted as \( k \), is a crucial factor determining the speed at which a reaction proceeds. This constant is unique to each reaction and remains unchanged, provided the temperature remains constant. To put it simply, the rate constant is part of the mathematical relationship that describes how quickly reactants turn into products in a given reaction.

For a reaction with the rate constant \( k = 10^{-2} \mathrm{~min}^{-1} \), this value tells us that the reaction is relatively slow, as the numerical value is small. Typically, an increase in the numerical value of the rate constant indicates a faster reaction.

• The units of the rate constant vary based on the order of reaction. For first-order reactions, the units are \( \mathrm{time}^{-1} \), such as \( \mathrm{min}^{-1} \).
• First-order reactions depend only on the concentration of one reactant, which simplifies calculations.
• The rate constant is integral when determining important reaction metrics, such as the half-life, as seen in our example.

Half-Life

The half-life of a reaction, denoted as \( t_{1/2} \), is the time it takes for half of the reactant to be consumed. For first-order reactions, there's a handy equation to calculate this:

\[ t_{1/2} = \frac{0.693}{k} \]
This relationship highlights that the half-life is inversely proportional to the rate constant \( k \). This means that a smaller rate constant results in a longer half-life, and similarly, a larger rate constant shortens the half-life.

• The factor 0.693 arises from the natural logarithm of 2, which is a constant used in exponential decay processes.
• One remarkable feature of first-order reactions is that the half-life remains constant throughout the reaction. This characteristics greatly simplifies planning and predicting when a reaction will reach a certain stage.
• For our exercise, plugging in the rate constant into the half-life formula gives us \( t_{1/2} = 69.3 \mathrm{~min} \), a clear indication of the reaction's behavior over time.

Kinetics

Kinetics is the study of the speed or rate of a chemical reaction and the factors affecting it. In the context of first-order reactions, kinetics helps us examine reactions where the rate is directly proportional to the concentration of one reactant.

Understanding kinetics allows chemists to predict how changes in conditions, such as temperature or reactant concentration, can influence the speed of a reaction. This is particularly useful in industrial and laboratory settings, where maximizing efficiency and yield is crucial.

• First-order kinetics implies that doubling the reactant concentration doubles the reaction rate, offering a straightforward relationship.
• Kinetics can provide insights into the reaction mechanism, thereby helping researchers develop better catalysts or reaction conditions.
• Kinetics is also vital in pharmacology, as it helps predict how drugs metabolize within the body by following first-order reaction patterns.

By diving into the kinetics of a reaction, we gain a deeper understanding of the dynamic processes driving chemical transformations, thus enabling controlled manipulation of reaction conditions to achieve desired outcomes.