Problem 39 - Chemistry The Central Science

Question

(a) For the generic reaction \(\mathrm{A} \rightarrow \mathrm{B}\) what quantity, when graphed versus time, will yield a straight line for a first- order reaction? (b) How can you calculate the rate constant for a first-order reaction from the graph you made in part (a)?

Step-by-Step Solution

Answer

(a) For a first-order reaction, plotting \(ln[\mathrm{A}]\) against time (\(t\)) will yield a straight line with a slope of \(-k\). (b) To calculate the rate constant (\(k\)), find the slope of the line \(ln[\mathrm{A}]\) versus \(t\) by selecting two points on the line and calculating the change in y divided by the change in x. The rate constant \(k\) is equal to the opposite of the slope: \[k = -\frac{ln[\mathrm{A}]_2 - ln[\mathrm{A}]_1}{t_2 - t_1} \]

Step 1: Understanding First-order Reactions

A first-order reaction is a type of chemical reaction where the rate of the reaction is directly proportional to the concentration of the reactant. Mathematically, it is represented as: \[rate = k[\mathrm{A}] \] Where \(rate\) is the reaction rate, \(k\) is the rate constant, and \([\mathrm{A}]\) represents the concentration of the reactant A.

Step 2: Determine the Quantity to be Plotted

To obtain a straight line, we need to plot a graph for which the slope is constant. Since the reaction rate relates to the concentration, we should obtain the integrated rate law for the first-order reaction. This equation shows how the concentration of the reactant A changes with time. By integrating the first-order reaction rate equation, it can be shown that: \[ln[\mathrm{A}] = ln[\mathrm{A}]_0 - kt \] Where \([\mathrm{A}]\) represents the concentration of A at a particular time \(t\), \([\mathrm{A}]_0\) is the initial concentration of A, and \(k\) is the rate constant. From this equation, we observe that plotting \(ln[\mathrm{A}]\) against \(t\) will produce a straight line with a slope of \(-k\).

Step 3: Calculate the Rate Constant from the Graph

To calculate the rate constant (\(k\)), we can use the straight line graph created in step 2. Since the slope of the line equals \(-k\), we can find the slope of the line \(ln[\mathrm{A}]\) versus \(t\) by selecting two points on the line and calculating the change in y divided by the change in x. Mathematically, the slope (\(-k\)) can be calculated as: \[-k = \frac{ln[\mathrm{A}]_2 - ln[\mathrm{A}]_1}{t_2 - t_1} \] After calculating the slope, we determine the rate constant \(k\) by switching the sign of the slope (multiplying by -1): \[k = -\frac{ln[\mathrm{A}]_2 - ln[\mathrm{A}]_1}{t_2 - t_1} \] By using the graph to find the slope, we can calculate the rate constant for the first-order reaction.

Key Concepts

Chemical Kinetics

Understanding the speed at which chemical reactions occur is essential for scientists and engineers who need to control and optimize various processes. Chemical kinetics, the branch of physical chemistry that deals with understanding the rates of chemical reactions, is crucial for designing reactors, pharmaceuticals, and even environmental systems.

At the heart of chemical kinetics is the idea that the rate of a chemical reaction is influenced by various factors, including the presence of catalysts, the concentration of reactants, and the temperature. These factors change the way molecules collide and interact, ultimately affecting how quickly products are formed from reactants.

By studying the kinetics of a reaction, one can predict how concentration changes over time and how different conditions affect the speed of the reaction, enabling us to harness chemistry for practical applications.

Reaction Rate

The reaction rate tells us how fast a reactant is being consumed or how fast a product is being created in a chemical reaction. It's measured by the change in concentration of a reactant or product per unit time. For instance, in a first-order reaction like \(\mathrm{A} \rightarrow \mathrm{B}\), the rate is directly proportional to the concentration of the reactant A. That's why the equation for the rate of a first-order reaction can be written as:
\[rate = k[\mathrm{A}] \]
The constant \(k\) in this equation is the rate constant, which is a measure of the intrinsic speed of the reaction and remains constant at a given temperature.

Understanding the reaction rate is not only about knowing how quickly a reactant disappears or a product forms but also about how those changes can be controlled and manipulated for desired outcomes, such as increasing yield in industrial processes or ensuring controlled drug release in pharmaceutical applications.

Integrated Rate Laws

Integrated rate laws are powerful tools in chemical kinetics that relate the concentrations of reactants or products to time. They are obtained by integrating the rate equations, which are based on the order of the reaction. For first-order reactions, where the rate is dependent directly on the concentration of one reactant, the integrated rate law is particularly simple and informative.

As illustrated in the exercise, the integrated rate law for a first-order reaction can be expressed as:
\[ln[\mathrm{A}] = ln[\mathrm{A}]_0 - kt \]
Here, \(ln[\mathrm{A}]_0\) is the natural logarithm of the initial concentration of A, and \(ln[\mathrm{A}]\) is the natural logarithm of the concentration of A at time \(t\). The beauty of this equation is that it gives us a straight-line relationship between \(ln[\mathrm{A}]\) and time, where the slope is equal to the negative of the rate constant \(k\). With a simple experiment and graphing \(ln[\mathrm{A}]\) against \(t\), scientists can quickly determine \(k\), providing valuable insights into the reaction's behavior over time.