Problem 39 - Chemistry: The Central Science

Question

(a) Define the following symbols that are encountered in rate equations for the generic reaction \(\mathrm{A} \rightarrow \mathrm{B}:[\mathrm{A}]_{0}, t_{1 / 2}[\mathrm{~A}]_{\mathrm{b}} k\). (b) What quantity, when graphed versus time, will yield a straight line for a first-order reaction? (c) How can you calculate the rate constant for a first- order reaction from the graph you made in part (b)?

Step-by-Step Solution

Answer

(a) The symbols are defined as follows: 1. \([\mathrm{A}]_{0}\): initial concentration of reactant A. 2. \(t_{1/2}\): half-life of the reaction. 3. \([\mathrm{A}]_{\mathrm{b}}\): concentration of reactant A at any given time. 4. \(k\): rate constant of the reaction. (b) For a first-order reaction, the quantity to be graphed for a straight line is the natural logarithm of the concentration of reactant A (\(\ln [\mathrm{A}]\)). (c) To calculate the rate constant \(k\) from the graph, find the slope of the straight line and take its negative: \(k = - \frac{\ln[\mathrm{A}]_{2} - \ln[\mathrm{A}]_{1}}{t_{2} - t_{1}}\).

Step 1: (a) Definition of symbols

Let's define the given symbols: 1. \([\mathrm{A}]_{0}\): This represents the initial concentration of reactant A at time t=0. 2. \(t_{1/2}\): This is the half-life of the reaction, which is the time taken for the concentration of reactant A to decrease to half of its initial value. 3. \([\mathrm{A}]_{\mathrm{b}}\): This represents the concentration of reactant A at any given time during the reaction. 4. \(k\): This is the rate constant of the reaction, which is a measure of the reaction rate and depends on factors such as temperature and catalyst presence.

Step 2: (b) Quantity for a straight line in the first-order reaction

A first-order reaction follows the rate law: \(-\frac{d[\mathrm{A}]}{dt} = k[\mathrm{A}]\), where \([\mathrm{A}]\) is the concentration of reactant A, \(t\) is time and \(k\) is the rate constant. To obtain an equation with a straight line when graphed against time, we need to linearize the above equation. This can be done by integrating and rearranging, which yields: \(\ln [\mathrm{A}] = -kt + \ln[\mathrm{A}]_{0}\) When plotting \(\ln [\mathrm{A}]\) vs. \(t\), a straight line with a slope of \(-k\) and y-intercept of \(\ln[\mathrm{A}]_{0}\) will be formed. Thus, the quantity we need to graph is the natural logarithm of the concentration of reactant A (\(\ln [\mathrm{A}]\)).

Step 3: (c) Calculating the rate constant from the graph

As mentioned in part (b), when we plot \(\ln [\mathrm{A}]\) against time, we obtain a straight line with a slope of \(-k\). To find the rate constant \(k\), we need to calculate the slope of the straight line in the graph. Using the coordinates of two points on the graph (let's call them \((t_{1}, \ln[\mathrm{A}]_{1})\) and \((t_{2}, \ln[\mathrm{A}]_{2})\)), the slope of the line can be calculated as follows: \(\text{slope} = -k = \frac{\ln[\mathrm{A}]_{2} - \ln[\mathrm{A}]_{1}}{t_{2} - t_{1}}\) So, the rate constant \(k\) can be determined by calculating the negative of the calculated slope: \(k = - \frac{\ln[\mathrm{A}]_{2} - \ln[\mathrm{A}]_{1}}{t_{2} - t_{1}}\)

Key Concepts

Chemical Kinetics

Chemical kinetics is a subfield of physical chemistry that deals with understanding the speed or rate at which chemical reactions occur and the factors affecting these rates. It examines the reaction mechanisms, the sequence of steps leading to the product formation, and the different factors like temperature, pressure, and concentration that influence the rate of a reaction.

Understanding kinetics helps chemists control processes, optimize product yields, and design better chemical reactors. It helps us to predict how changing conditions will affect the speed of a reaction which is essential in industries for making products quickly and efficiently. Moreover, it provides insights into the stability of substances and the shelf life of drugs in pharmaceuticals.

Reaction Rate Constant

The reaction rate constant, denoted by the symbol 'k', is a pivotal parameter in chemical kinetics. It quantitatively expresses the speed of a chemical reaction. For a given reaction at a constant temperature, the rate constant is fixed and used to compare the rates of different reactions.

The value of 'k' provides information about the reaction probability and the frequency of effective molecular collisions. Factors like temperature, solvent, and the presence of a catalyst can influence this constant. In the context of a textbook problem, you would often be tasked to determine the rate constant from experimental data, as it is key to understanding the reaction kinetics.

First-order Reaction

In chemical kinetics, a first-order reaction is one where the rate is directly proportional to the concentration of a single reactant. Mathematically, this can be represented as \(\frac{d[\mathrm{A}]}{dt} = -k[\mathrm{A}]\), where \([\mathrm{A}]\) is the reactant concentration, \(t\) is time, and \(k\) is the rate constant.

First-order reactions have unique characteristics, such as their half-lives being constant regardless of the initial concentration. They mostly occur in radioactive decay processes and simple isomerization reactions. The convenience with first-order kinetics lies in their simplicity; they allow easy mathematical treatment and plotting for reaction progression analysis.

Half-life of a Reaction

The half-life of a reaction, represented as \(t_{1/2}\), is the time required for the concentration of a reactant to reduce to half of its initial value. In a first-order reaction, the half-life is constant and is given by the formula \(t_{1/2} = \frac{\ln(2)}{k}\), where \(k\) is the rate constant.

This makes half-life a convenient measure to compare the speeds of different reactions. Understanding half-life is not only important in chemistry, for example, in the stabilization of potentially harmful substances, but also in fields like pharmacology, where it assists in determining dosing schedules of medications.

Concentration-Time Graph

The concentration-time graph is a visual representation that depicts how the concentration of a reactant decreases over time during a chemical reaction. For a first-order reaction, plotting the natural logarithm of the concentration of reactant \(\ln [\mathrm{A}]\) against time \(t\) gives a straight line. This linear relationship is crucial for the determination of reaction kinetics.

The slope of this line gives the negative of the rate constant \(-k\), and the intercept provides the initial concentration \(\ln [\mathrm{A}]_{0}\). This type of graph not only makes it easier to calculate the reaction rate constant but also visually illustrates the progression of the reaction and is an essential tool in understanding and explaining chemical kinetics.