Question

For a particular first order reaction, it takes 48 minutes for the concentration of the reactant to decrease to \(25 \%\) of its initial value. What is the value for rate constant (in \(\mathrm{s}^{-1}\) ) for the reaction? a. \(2.4 \times 10^{-4} \mathrm{~s}^{-1}\) b. \(1.8 \times 10^{-3} \mathrm{~s}^{-1}\) c. \(3.18 \times 10^{-4} \mathrm{~s}^{-1}\) d. \(4.8 \times 10^{-4} \mathrm{~s}^{-1}\)

Short answer

The rate constant is \( 4.8 \times 10^{-4} \mathrm{~s}^{-1} \).

Step-by-step solution

Understand the Formula

For a first order reaction, the rate constant \( k \) can be determined using the formula for the rate law: \[ k = \frac{1}{t} \ln \left(\frac{[A]_0}{[A]}\right) \]where \( t \) is the time, \([A]_0\) is the initial concentration, and \([A]\) is the concentration at time \( t \).

Assign Known Values

In this problem, the concentration of the reactant decreases to \( 25\% \) of its initial value in 48 minutes. Thus, \([A] = 0.25 [A]_0\), and \( t = 48 \) minutes.

Convert Time to Seconds

Since the rate constant \( k \) needs to be in \( \mathrm{s}^{-1} \), convert 48 minutes to seconds:\[48 \text{ minutes} = 48 \times 60 = 2880 \text{ seconds} \].

Substitute Values into the Formula

Substitute the known values into the rate constant formula:\[k = \frac{1}{2880} \ln \left(\frac{[A]_0}{0.25[A]_0}\right)\]Since \( \ln \left(\frac{[A]_0}{0.25 [A]_0}\right) = \ln(4) \), simplify to:\[ k = \frac{1}{2880} \times \ln (4) \]

Calculate the Rate Constant

Calculate \( \ln(4) = 1.3863 \). Now calculate the rate constant:\[ k = \frac{1.3863}{2880} = 4.81 \times 10^{-4} \mathrm{~s}^{-1} \]

Choose the Correct Answer

Compare the calculated rate constant with the given options:d. \( 4.8 \times 10^{-4} \mathrm{~s}^{-1} \) is the closest match.

Key concepts

Rate Constant

The rate constant, denoted as \( k \), is a crucial parameter in the study of reaction kinetics. It quantifies the speed at which reactants are converted to products in a chemical reaction. Essentially, it tells us how fast a reaction is occurring under specific conditions. For first-order reactions, the rate constant is especially significant because it remains constant, even if the concentration of the reactants changes.
In mathematical terms, the rate constant for a first-order reaction can be calculated using the equation:

• \( k = \frac{1}{t} \ln \left(\frac{[A]_0}{[A]}\right) \)

Here, \( t \) represents the time elapsed, \([A]_0\) is the initial concentration of the reactant, and \([A]\) is the concentration at time \( t \).
This relationship highlights the logarithmic dependence of the reactant concentration on time, indicating that the rate of reaction decreases over time as the reactants are consumed. Therefore, the rate constant is indispensable for predicting the course of a reaction and for comparing the kinetics of different reactions.

Reaction Kinetics

Reaction kinetics is the branch of chemistry that deals with the rates of chemical reactions. It focuses on understanding how different factors such as concentration, temperature, and the presence of catalysts influence the speed of a reaction.
For first-order reactions, such as the one given in the exercise, the rate of reaction depends solely on the concentration of one reactant. This makes it easier to study and model compared to reactions involving multiple reactants. The rate equation for a first-order reaction is:

• Rate = \( k[A] \)

Where \( k \) is the rate constant and \([A]\) is the concentration of the reactant. This equation shows that the rate of reaction at any given moment is directly proportional to the concentration of the reactant.
Because the concentration of the reactant reduces as the reaction progresses, the reaction order in kinetics helps explain the dynamic changes occurring during the transformation from reactants to products.

Half-Life Calculation

The half-life of a reaction, denoted as \( t_{1/2} \), is the time required for the concentration of a reactant to fall to half of its initial value. For first-order reactions, the half-life is particularly unique and insightful because it remains constant throughout the reaction.
The half-life for a first-order reaction can be determined using the formula:

• \( t_{1/2} = \frac{0.693}{k} \)

In this equation, \( 0.693 \) is the natural logarithm of 2, which comes into play because the reactant concentration halves. This consistent half-life is a characteristic feature for first-order kinetics, distinguishing it from second or zero-order reactions.
Understanding half-life is crucial in various fields such as pharmacology, where it helps in determining how long a drug stays active in the body, and in environmental science, where it helps in understanding the degradation of pollutants.