Question

This reaction was monitored as a function of time: \(A \longrightarrow B+C\) A plot of \(\ln [\mathrm{A}]\) versus time yields a straight line with slope \(-0.0045 / \mathrm{s}\) a. What is the value of the rate constant \((k)\) for this reaction at this temperature? b. Write the rate law for the reaction. c. What is the half-life? d. If the initial concentration of \(\mathrm{A}\) is \(0.250 \mathrm{M},\) what is the concentration after 225 s?

Short answer

a. The rate constant \(k\) is 0.0045 / \mathrm{s}. b. The rate law is \(\text{rate} = k [\mathrm{A}]\). c. The half-life is \(t_{1/2} = \frac{\ln(2)}{k} = \frac{0.693}{0.0045 / \mathrm{s}} \). d. After 225 seconds, the concentration of \(A\) is \(0.250 \mathrm{M} \cdot e^{-0.0045 / \mathrm{s} \cdot 225 \mathrm{s}}\).

Step-by-step solution

Determining the Rate Constant

The slope of the plot of \(\ln [\mathrm{A}]\) versus time is equal to the negative of the rate constant \(k\) for a first-order reaction. Since the slope is given as \(-0.0045 / \mathrm{s}\), the value of the rate constant \(k\) is \(0.0045 / \mathrm{s}\).

Writing the Rate Law

For a first-order reaction, the rate law can be expressed as \(\text{rate} = k [\mathrm{A}]\). The rate at which \(A\) is consumed is directly proportional to its concentration at any point in time.

Calculating the Half-Life

The half-life of a first-order reaction is given by the equation \(t_{1/2} = \frac{\ln(2)}{k}\). Substituting the determined rate constant \(k = 0.0045 / \mathrm{s}\) into this formula will give the half-life.

Computing the Concentration after Time

For a first-order reaction, the concentration of \(A\) over time is given by \([\mathrm{A}] = [\mathrm{A}_0]e^{-kt}\), where \([\mathrm{A}_0]\) is the initial concentration. Using the initial concentration \(0.250 \mathrm{M}\) and time \(225 \mathrm{s}\), along with the rate constant \(k = 0.0045 / \mathrm{s}\), we can calculate \([\mathrm{A}]\) after 225 seconds.

Key concepts

Rate Constant

Understanding the rate constant is essential in the study of chemical reaction kinetics. It's a measure of the speed at which a reaction proceeds. Specifically, in the context of a first-order reaction such as
\(A \longrightarrow B + C\),
the rate constant \(k\) is determined from the slope of a plot of \(\ln [A]\) versus time. Since the given slope is \(-0.0045 / \mathrm{s}\), the rate constant is \(0.0045 / \mathrm{s}\). This value indicates how rapidly reactant \(A\) is converted to products \(B\) and \(C\) per second under the given conditions.

Rate Law

The rate law expresses the relationship between the reaction rate and the concentrations of reactants. For a first-order reaction like the one we're discussing, the rate law is given by:
\(\text{rate} = k [A]\).
This mathematical model states that the rate at which reactant \(A\) is consumed is directly proportional to its instantaneous concentration. As \(A\) decreases, the rate at which it reacts diminishes accordingly. It's a simple yet powerful way of quantifying how reactants change over time.

Chemical Reaction Half-Life

Half-life, symbolized by \(t_{1/2}\), is the time required for half of the reactant to be consumed in a chemical reaction. For first-order reactions, the half-life is independent of the initial concentration and solely depends on the rate constant \(k\). The equation
\(t_{1/2} = \frac{\ln(2)}{k}\)
allows us to calculate the half-life. Using the rate constant \(0.0045 / \mathrm{s}\), we find a specific numerical value representing the time in which the concentration of reactant \(A\) decreases by half. This concept is vital for predicting how long a reaction will take to reach a certain point.

Reaction Concentration Over Time

In first-order kinetics, the concentration of a reactant decreases exponentially over time. The key equation representing this behavior is:
\([A] = [A_0]e^{-kt}\).
Here, \([A_0]\) is the initial concentration, \(k\) is the rate constant, and \(t\) is the time elapsed. This formula allows us to predict the remaining concentration of reactant \(A\) at any given time. By plugging in the values for \([A_0]\), \(k\), and \(t\), we can determine the concentration of \(A\) after 225 seconds, illustrating how it diminishes as the reaction progresses towards completion.