Question

A certain reaction has the following general form: $$\text { aA } \longrightarrow \mathrm{bB}$$ At a particular temperature and \([\mathrm{A}]_{0}=2.00 \times 10^{-2} \mathrm{M}, \)concentration versus time data were collected for this reaction, and a plot of \(\ln [\mathrm{A}]\) versus time resulted in a straight line with a slope value of \(-2.97 \times 10^{-2} \mathrm{min}^{-1}\)a. Determine the rate law, the integrated rate law, and the value of the rate constant for this reaction. b. Calculate the half-life for this reaction. c. How much time is required for the concentration of A to decrease to \(2.50 \times 10^{-3} \mathrm{M} ?\)

Short answer

The reaction follows a first-order rate law with a rate constant of \(k = 2.97 \times 10^{-2}\, \mathrm{min}^{-1}\). The half-life is approximately 23.35 minutes, and it takes about 78.77 minutes for the concentration of A to decrease to \(2.50 \times 10^{-3}\, \mathrm{M}\).

Step-by-step solution

Determine the rate law and integrated rate law

We are given that the reaction has a general form: \(\text{aA} \longrightarrow \text{bB}\) It is also given that a plot of \(\ln [\mathrm{A}]\) versus time results in a straight line with a slope value of \(-2.97 \times 10^{-2}\,\mathrm{min}^{-1}\). This information tells us that the reaction follows a first-order rate law, because only in first-order reactions, the plot of \(\ln [\mathrm{A}]\) versus time is a straight line. Now that we know that the reaction follows a first-order rate law, we can write the rate law and the integrated rate law as follows: Rate law: \(R = k[\mathrm{A}]\) Integrated rate law: \(\ln [\mathrm{A}] = - kt + \ln [\mathrm{A}]_{0}\)

Find the value of the rate constant k

We now need to find the value of the rate constant, k. We know that the slope of the straight line in the plot of \(\ln [\mathrm{A}]\) versus time is \(-2.97 \times 10^{-2}\,\mathrm{min}^{-1}\) which is equal to the value of k: \(-k = -2.97 \times 10^{-2}\,\mathrm{min}^{-1}\) Therefore, the rate constant k is: \(k = 2.97 \times 10^{-2}\,\mathrm{min}^{-1}\)

Calculate the half-life of the reaction

For a first-order reaction, the half-life formula is given by: \(t_{1/2} = \frac{\ln 2}{k}\) Substituting the value of k, we get: \(t_{1/2} = \frac{\ln 2}{2.97 \times 10^{-2}\,\mathrm{min}^{-1}}\) \(t_{1/2} \approx 23.35\,\mathrm{min}\)

Calculate the time required for the concentration of A to decrease to 2.50 x 10^(-3) M

We need to find the time it takes for the concentration of A to decrease to \(2.50 \times 10^{-3}\,\mathrm{M}\). Using the integrated rate law, we can solve for t: \(\ln [\mathrm{A}] = - kt + \ln [\mathrm{A}]_{0}\) \(\ln (2.50 \times 10^{-3}) = - (2.97 \times 10^{-2}\,\mathrm{min}^{-1}) t + \ln(2.00 \times 10^{-2})\) Solving for t, we get: \(t \approx 78.77\,\mathrm{min}\) In conclusion, the rate law of the reaction is \(R = k[\mathrm{A}]\), with a rate constant of \(k = 2.97 \times 10^{-2}\,\mathrm{min}^{-1}\). The half-life of the reaction is approximately 23.35 minutes, and it takes about 78.77 minutes for the concentration of A to decrease to \(2.50 \times 10^{-3}\,\mathrm{M}\).

Key concepts

Rate Law

Understanding the rate law is vital in the study of reaction kinetics. This mathematical equation relates the rate of a chemical reaction to the concentration of the reactants. For a reaction where the concentration of the reactant A affects the rate, the rate law can be expressed as \( R = k[\mathrm{A}]^n \), where \( R \) is the reaction rate, \( k \) is the rate constant, \( [\mathrm{A}] \) represents the concentration of A, and \( n \) is the order of the reaction with respect to A. A first-order reaction, where \( n=1 \) as seen in the provided exercise, indicates that the rate is directly proportional to the concentration of the reactant A. You can use experimental data to determine the order of a reaction by plotting concentration versus time and analyzing the resulting graph.

Integrated Rate Law

The integrated rate law is the equation that links concentration with time for a chemical reaction. It's developed by integrating the rate law over time. For first-order reactions, the integrated rate law is given as \( \ln [\mathrm{A}] = -kt + \ln [\mathrm{A}]_{0} \), where \( [\mathrm{A}] \) is the concentration of the reactant A at time \( t \), \( [\mathrm{A}]_{0} \) is the initial concentration of A, \( k \) is the rate constant, and \( t \) is the time elapsed. This equation describes how the concentration of the reactant decreases over time and is key for predicting the concentration at any given time during a first-order reaction.

Rate Constant

The rate constant \( k \) is a proportionality factor in the rate law that provides critical information about the reaction speed. In the context of first-order reactions, it has units of inverse time such as \( \mathrm{min}^{-1} \) or \( \mathrm{s}^{-1} \). The rate constant can be determined from experimental data, typically by measuring the slope of the line in a time-based plot, just as we saw with the given slope of \( -2.97 \times 10^{-2}\,\mathrm{min}^{-1} \) in the exercise. This value is crucial as it also permits us to calculate other important kinematic features such as the half-life of a reaction.

Reaction Half-life

The half-life of a reaction, frequently denoted as \( t_{1/2} \), is the time required for the concentration of a reactant to drop to half its initial value. In first-order reactions, the half-life is constant regardless of the initial concentration and can be calculated using the formula \( t_{1/2} = \frac{\ln 2}{k} \). This aspect of first-order reactions showcases their unique feature: the half-life does not change over the course of the reaction, in stark contrast to zero or second-order reactions where half-life varies with concentration.

Chemical Kinetics

Chemical kinetics is the field of chemistry that deals with the speeds, or rates, at which chemical reactions occur. This branch of chemistry is not only concerned with the rate at which reactions proceed but also the mechanism by which they occur. Kinetics involves studying how different variables such as temperature, concentration, and catalysts affect the rate of a reaction. By analyzing the rate laws, integrated rate equations, rate constants, and half-life data, chemists can decipher the nuances of reaction pathways and optimize conditions for desired reaction speeds, which have applications ranging from industrial synthesis to pharmaceuticals.