Question

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

Short answer

a. The rate constant (k) is 0.0045 /s. b. The rate law is Rate = k[A]. c. The half-life can be calculated using the formula t_{1/2} = ln(2)/k. d. The concentration after 225 s can be found using the first-order kinetics equation.

Step-by-step solution

Determine the rate constant (k)

For a first-order reaction, the rate constant (k) can be determined from the slope of a plot of \(\ln [A]\) versus time. In this case, the slope of the line is given as \(-0.0045 / \mathrm{s}\), which is actually the value of the rate constant (k).

Write the rate law

The rate law for a first-order reaction can be written as: \[\text{Rate} = k[A]\] where \(k\) is the rate constant and \([A]\) is the concentration of reactant A.

Calculate the half-life

The half-life (\(t_{1/2}\)) of a first-order reaction is given by the formula: \[t_{1/2} = \frac{\ln 2}{k}\] Given the rate constant \(k = -0.0045 / \mathrm{s}\), we can substitute the value into the formula to find the half-life.

Determine the concentration after 225 s

For a first-order reaction, the relationship between concentration and time is given by: \[\ln \left(\frac{[A]}{[A]_0}\right) = -kt\] where \([A]\) is the concentration at time \(t\), \([A]_0\) is the initial concentration, and \(k\) is the rate constant. Using the initial concentration \(0.250 \mathrm{M}\), the rate constant \(-0.0045 / \mathrm{s}\), and the time 225 s, we can solve for the concentration \([A]\) at 225 s.

Key concepts

Chemical Kinetics

Chemical kinetics is the branch of chemistry that deals with the rates of chemical reactions and the mechanisms by which they proceed. It involves studying how different variables such as concentration, temperature, and presence of catalysts affect the speed of a reaction.

Understanding kinetics helps chemists to control reactions to optimize yield and efficiency. For instance, knowing how quickly a reactant is consumed can inform the timing of reactants addition in industrial processes. In the educational context, kinetic principles enable students to predict the behavior of reactions over time, which is essential in both lab and theoretical chemistry.

Half-life of Reaction

The half-life of a reaction, commonly denoted as \(t_{1/2}\), is the time required for half of the reactant to be converted into product in a chemical reaction. For first-order reactions, the half-life is particularly unique because it is independent of the initial concentration of the reactant.

This constant half-life feature helps simplify many calculations involved in chemical kinetics and is critical for applications such as drug dosing in pharmacology, where half-life dictates the frequency and dosage of medication administration.

Reaction Rate Law

The reaction rate law, or simply the rate law, is an equation that relates the rate of a reaction to the concentrations of the reactants and the specific rate constant. For a first-order reaction, the rate law is expressed as \(\text{Rate} = k[A]\), where \(k\) is the rate constant and \([A]\) is the concentration of the reactant.

The rate law is determined experimentally and cannot be deduced from the stoichiometry of the reaction alone. This equation is pivotal in predicting how changing the concentration of reactants will impact the speed of a reaction. For higher order reactions, the rate law can become more complex, including terms for the concentration of each reactant raised to a power often corresponding to the molecularity of the reaction.

Concentration vs Time Relationship

For first-order reactions, there is a direct logarithmic relationship between the concentration of a reactant and time. The formula \(\ln \left(\frac{[A]}{[A]_0}\right) = -kt\) encapsulates this relationship, where \([A]\) denotes the concentration of reactant at time \(t\), \([A]_0\) is the initial concentration, and \(k\) is the rate constant.

The negative sign in the equation indicates that the concentration of the reactant decreases over time. By using this relationship, one can determine the remaining concentration of a reactant after a certain period or estimate how long it will take for the reactant to reach a particular concentration. This becomes particularly useful in predicting the progress of a reaction and in designing industrial and pharmaceutical processes.