Question

A first-order reaction is \(75.0 \%\) complete in \(320 . \mathrm{s}\). a. What are the first and second half-lives for this reaction? b. How long does it take for \(90.0 \%\) completion?

Short answer

a. The first and second half-lives for this reaction are both \(199.7 \, s\). b. It takes approximately \(646.6\, s\) for the reaction to reach \(90.0\%\) completion.

Step-by-step solution

Write the given information

It is given that the reaction is 75.0% complete in 320 seconds. That means 25.0% of the reactant remains at that time.

Determine the concentration of reactant at time t

As the reaction is 75.0% complete, the remaining percentage of reactant is 100 - 75 = 25.0%. Thus, the concentration of the reactant at time t is: \[A_t = 0.25A_0 \]

Calculate the rate constant (k)

We can use the first-order reaction formula given earlier: \[ k = \frac{1}{t} \ln{ \frac{A_0}{A_t} }\] Substitute the given values, A₀ and Aₜ: \[ k = \frac{1}{320s} \ln{ \frac{A_0}{0.25A_0} }\] \[ k = \frac{1}{320s} \ln{4} \] Now, calculate the value of k: \[ k ≈ 0.00347 \, s^{-1} \]

Calculate the first half-life

For a first-order reaction, the half-life (t₁/₂) is given by: \[t_{1/2} = \frac{0.693}{k}\] Substitute the value of k: \[t_{1/2} = \frac{0.693}{0.00347\, s^{-1}}\] Now, calculate the value of t₁/₂: \[t_{1/2} ≈ 199.7\, s\]

Calculate the second half-life

For first-order reactions, the half-life is constant. Therefore, the second half-life will be equal to the first half-life: \[t_{2nd\, 1/2} = t_{1st\, 1/2} = 199.7\, s\]

Calculate the time for 90% completion

We are asked to find the time it takes for the reaction to reach 90.0% completion. That means 10.0% of the reactant remains: \[A_t = 0.1A_0\] Use the first-order reaction formula: \[ t = \frac{1}{k} \ln{ \frac{A_0}{A_t} }\] Substitute the given values, A₀ and Aₜ: \[ t = \frac{1}{0.00347\, s^{-1}} \ln{ \frac{A_0}{0.1A_0} }\] \[ t = \frac{1}{0.00347\, s^{-1}} \ln{10} \] Now, calculate the value of t: \[ t ≈ 646.6\, s \] Therefore, the reaction will reach 90.0% completion in approximately 646.6 seconds. #Summary# a. The first and second half-lives for this reaction are both 199.7 seconds. b. It takes approximately 646.6 seconds for the reaction to reach 90.0% completion.

Key concepts

Understanding Half-Life in First-Order Reactions

The concept of half-life is a key factor when studying first-order reactions. Simply put, the half-life is the time it takes for half of the reactant concentration to decrease to half its original value. For first-order reactions, this duration remains constant irrespective of the starting concentration. This is quite different from other types of reactions. It gives first-order kinetics a distinctive advantage in predicting how a reaction proceeds over time.
In the provided problem, the reaction's half-life is calculated using the formula \(t_{1/2} = \frac{0.693}{k}\), where \(k\) is the rate constant. From the solutions, the half-life is found to be approximately 199.7 seconds. This means every 199.7 seconds, the concentration of reactants will halve until the reactants are almost completely converted to products.
The regularity of this half-life in first-order reactions helps chemists plan and control chemical processes efficiently, as they can predict how long it will take for a reaction to achieve a certain degree of completion.

Rate Constant and Its Importance

The rate constant, denoted as \(k\), is a vital parameter in understanding first-order reactions. It provides insight into the speed of the reaction; a larger rate constant means a faster reaction occurs over time. To calculate \(k\) in a first-order reaction, the equation \(k = \frac{1}{t} \ln{ \frac{A_0}{A_t} }\) is utilized. Here, \(A_0\) is the initial concentration of reactants, \(A_t\) is the concentration at time \(t\), and \( \ln\) denotes the natural logarithm.
In the example problem, after substituting given values, \(k\) is about 0.00347 s\(^{-1}\).* This numerical value not only quantifies the reaction's pace but also aids in further calculations like determining half-life or time for a specific completion percentage. Understanding the rate constant is crucial, as it allows students and professionals to compare different reactions and adjust conditions to optimize processes in research and industry.

Deciphering Reaction Completion

Reaction completion is a term used to describe how much of the reactant has been transformed into the product. It's frequently expressed as a percentage to indicate progress. In the discussed problem, 75% and 90% completion are analyzed, representing the extent to which reactants have changed.
To calculate the time required to reach a certain completion level, use the formula associated with first-order reactions: \(t = \frac{1}{k} \ln{ \frac{A_0}{A_t} }\). For 90% completion, this meant only 10% of the reactant remained, leading to a solution indicating the reaction's near-total transformation.
Understanding reaction completion helps chemists gauge the efficiency and progress of their reactions. By knowing how effectively a reaction proceeds, it has practical applications in fields such as pharmaceuticals, where precise amounts of reactant transformation are essential for effective medication dosages.

Analyzing Concentration of Reactants

The concentration of reactants during a reaction provides fundamental insight into its progression. First-order reactions specifically depend largely on the concentration, yet the rate affects percentage reduction, and not the absolute amount. This concentration drops systematically with each passing half-life.
In the provided exercise, starting with 100% concentration, after 320 seconds, only 25% of the original reactant remains, indicating a 75% completion due to half-life dynamics. The concentration at any time \(t\) can be represented as \(A_t = A_0 \times e^{-kt}\), with exponential decay characteristic of first-order kinetics.
By continuously measuring the concentration of reactants, chemists can fine-tune processes, combining theoretical calculations with practical maintenance to ensure desired reaction speeds and results. Concentration measures become guides, indicating when and how to apply catalysts, heat, or pressures to optimize reaction pathways.