Question

A reaction of the form $$ \mathrm{aA} \longrightarrow $$ gives a plot of \(\ln [\mathrm{A}]\) versus time (in seconds), which is a straight line with a slope of \(-7.35 \times 10^{-3} .\) Assuming \([\mathrm{A}]_{0}=\) \(0.0100 M,\) calculate the time (in seconds) required for the reaction to reach 22.9\(\%\) completion.

Short answer

For a first-order reaction, the integrated rate equation is \(\ln [\mathrm{A}] = -kt + \ln [\mathrm{A}]_0\). Given the initial concentration \([\mathrm{A}]_0 = 0.0100 M\) and the rate constant \(k = 7.35 \times 10^{-3} s^{-1}\), we can find the concentration of A at 22.9% completion, \([\mathrm{A}] = 0.00771 M\), and then solve for the time \(t\). The equation becomes \(\ln (0.00771) = -(7.35 \times 10^{-3} s^{-1})t + \ln (0.0100)\), which gives us \(t \approx 429.5s\). Thus, the time required for the reaction to reach 22.9% completion is approximately 429.5 seconds.

Step-by-step solution

Write down the integrated rate equation for a first-order reaction

For a first-order reaction, the integrated rate equation is given by: \[ \ln [\mathrm{A}] = -kt + \ln [\mathrm{A}]_0 \] where \([\mathrm{A}]\) is the concentration of A at time \(t\), \([\mathrm{A}]_0\) is the initial concentration of A, \(k\) is the rate constant, and \(t\) is the time in seconds.

Calculate the rate constant (k) using the given slope

We are given that the slope of the plot of \(\ln [\mathrm{A}]\) versus time is \(-7.35 \times 10^{-3}\). Since the slope of this plot is equal to the negative of the rate constant (-k), we can find the value of k as: \[ k = 7.35 \times 10^{-3} s^{-1} \]

Write down the equation for the percentage completion of the reaction

The percentage completion of the reaction is given by the fraction of A that has reacted. To find the time when the reaction reaches 22.9% completion, we can write the equation for the percentage completion as: \[ \frac{[\mathrm{A}]_0 - [\mathrm{A}]}{[\mathrm{A}]_0} = 0.229 \]

Solve the equation for the concentration of A at 22.9% completion

We know that the initial concentration \([\mathrm{A}]_0\) is \(0.0100 M\). We can now use the equation from Step 3 to find the concentration of A at 22.9% completion: \[ [\mathrm{A}] = [\mathrm{A}]_0(1 - 0.229) = 0.0100 M (1 - 0.229) = 0.00771 M \]

Substitute the known values into the integrated rate equation and solve for the time (t)

We can now substitute the known values of \([\mathrm{A}]\), \([\mathrm{A}]_0\), and \(k\) into the integrated rate equation from Step 1: \[ \ln [\mathrm{A}] = -kt + \ln [\mathrm{A}]_0 \] \[ \ln (0.00771) = -(7.35 \times 10^{-3} s^{-1})t + \ln (0.0100) \] Now, rearrange the equation and solve for \(t\): \[ t = \frac{\ln (0.00771) - \ln (0.0100)}{-7.35 \times 10^{-3} s^{-1}} \approx 429.5s \] Therefore, the time required for the reaction to reach 22.9% completion is approximately 429.5 seconds.

Key concepts

Integrated rate equation

The integrated rate equation is a fundamental concept in chemical kinetics, especially for first-order reactions. It provides a mathematical relationship that allows us to determine the concentration of a reactant over time. For a first-order reaction, the equation is given by: \[\ln [\mathrm{A}] = -kt + \ln [\mathrm{A}]_0\]Here,

• \([\mathrm{A}]\) is the concentration of the reactant at time \(t\).
• \([\mathrm{A}]_0\) is the initial concentration of the reactant.
• \(k\) is the rate constant, which is specific to the reaction and the conditions like temperature.
• \(t\) represents time.

This equation is derived using the rate law for first-order reactions and allows us to predict how the concentration changes. It's useful in calculating the time taken to reach a certain percentage of reaction completion, as seen in the given problem. Understanding this equation helps in analyzing reaction kinetics and comparing reaction rates under different conditions.

Rate constant

The rate constant, denoted as \(k\), is a crucial factor in the study of reaction rates. For first-order reactions, the rate constant determines how quickly a reaction proceeds. The value of \(k\) is affected by variables such as temperature and the presence of a catalyst. In the context of the given exercise, \(k\) is calculated using the slope of the graph of \(\ln[\mathrm{A}]\) versus time.

• The slope is equal to \(-k\), so the rate constant is the negative of the slope. For example, with a slope of \(-7.35 \times 10^{-3}\), we have:

\[k = 7.35 \times 10^{-3} \text{ s}^{-1}\]Knowing \(k\) helps predict the speed of the reaction and, combined with the integrated rate equation, allows us to find the concentration of reactants over time. It is a specific characteristic of a reaction under given conditions.

Reaction completion time

Determining the time required for a reaction to reach a certain level of completion is an important application of the integrated rate equation. In the exercise, the task is to find the time needed for the reaction to achieve 22.9% completion. First, we identify the remaining concentration at 22.9% completion using the equation for percentage completion:\[\frac{[\mathrm{A}]_0 - [\mathrm{A}]}{[\mathrm{A}]_0} = 0.229\]This helps determine the concentration of \([\mathrm{A}]\) at this completion stage. Knowing \([\mathrm{A}]_0 = 0.0100 \text{ M}\), we calculate \([\mathrm{A}]\) as follows:\[[\mathrm{A}] = 0.0100 \text{ M} \times (1 - 0.229) = 0.00771 \text{ M}\]Next, substituting \([\mathrm{A}] = 0.00771 \text{ M}\), the initial concentration, and the rate constant into the integrated rate equation, we solve for \(t\):\[t = \frac{\ln (0.00771) - \ln (0.0100)}{-7.35 \times 10^{-3} \text{ s}^{-1}} \approx 429.5 \text{ seconds}\]Thus, the time to reach 22.9% completion is about 429.5 seconds. This calculation showcases how kinetics can predict when a reaction milestone is achieved.

Concentration

Concentration in chemical kinetics refers to the amount of a substance in a given volume of solution, typically expressed in moles per liter (M). It plays a significant role in understanding and calculating reaction dynamics. In the given problem, concentration is vital in determining the reaction's progress over time.

• \([\mathrm{A}]\): Represents the concentration of reactant at a specific time \(t\).
• \([\mathrm{A}]_0\): The initial concentration or starting point before the reaction proceeds.

At every step, concentration informs us about how much of the reactant remains or has reacted. Calculating the concentration at different times enables us to measure the rate of the reaction, using equations like those in the first-order kinetics model. With the integrated rate equation, the link between concentration change and the passage of time becomes apparent, helping predict future states of the reaction based on current data.