Problem 39 - Chemistry The Central Science In Si Units

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) The natural logarithm of the concentration of \( A \) vs time yields a straight line. (b) The rate constant \( k \) is found from the negative slope of this line.

Step 1: Understanding First Order Reactions

In a first-order reaction, the rate of the reaction is directly proportional to the concentration of one reactant. For a generic reaction \( \mathrm{A} \rightarrow \mathrm{B} \), the rate is \( -\frac{d[\mathrm{A}]}{dt} = k [\mathrm{A}] \), where \( k \) is the rate constant.

Step 2: Identifying the Logarithmic Relationship

For first-order kinetics, we know that the concentration of reactant \( A \) over time is given by the equation \( [\mathrm{A}] = [\mathrm{A}]_0 e^{-kt} \). By taking the natural logarithm of both sides, \( \ln([\mathrm{A}]) = \ln([\mathrm{A}]_0) - kt \), which resembles the equation of a straight line, \( y = mx + b \).

Step 3: Graphing Natural Logarithm vs Time

According to the linear equation \( \ln([\mathrm{A}]) = \ln([\mathrm{A}]_0) - kt \), when \( \ln([\mathrm{A}]) \) is plotted against time \( t \), it yields a straight line where the slope \( m \) is equal to \(-k\), and the intercept \( b \) is \( \ln([\mathrm{A}]_0) \).

Step 4: Determining the Rate Constant

From the graph produced in the previous step, you can determine the rate constant \( k \) by calculating the slope. Since \( \text{slope} = -k \), the absolute value of the slope of the line gives you the rate constant \( k \).

Key Concepts

Reaction Kinetics

Reaction kinetics is an essential aspect of understanding how reactions proceed over time. When studying the kinetics of a reaction, we focus on the rate at which reactants are converted into products. For a first-order reaction, the rate is directly related to the concentration of one reactant. This means that as the concentration of the reactant decreases, the rate of the reaction also decreases proportionally. To put it simply, the speed at which the reaction occurs depends on how much reactant you have.

• First-order reactions are characterized by a single reactant whose concentration influences the rate.
• These reactions are common in chemical processes such as radioactive decay and certain enzyme reactions.

Understanding these kinetics helps predict how long a reaction will take and the concentration of reactants at different times. This knowledge is vital in fields like chemical engineering, pharmaceuticals, and environmental science.

Rate Constant

The rate constant, denoted as \( k \), is a fundamental parameter in reaction kinetics. It provides insight into the speed of a reaction under certain conditions. In the context of a first-order reaction, the rate constant can be thought of as a measure of how fast the reaction proceeds when there is a unit concentration of the reactant.

• The value of \( k \) always remains constant at a fixed temperature for a specific reaction type.
• Higher values of \( k \) generally indicate faster reactions.

For first-order reactions, calculating the rate constant involves determining the slope of a line graph. By knowing how \( k \) relates to reaction speed, you can adjust conditions to control the reaction rate, a useful skill in experimental and industrial settings. Remember, temperature and other environmental factors can affect \( k \), making it important to consider the entirety of the reaction conditions.

Concentration vs. Time Graph

In first-order reaction kinetics, a concentration vs. time graph provides a visual insight into how the concentration of a reactant changes over time. For these reactions, plotting the natural logarithm of the concentration of reactant \( [A] \) against time gives a straight line. This linear behavior is due to the logarithmic relationship within first-order kinetics, described by the equation:\[\ln([A]) = \ln([A]_0) - kt\]Where \([A]_0\) is the initial concentration, and \(t\) represents time.

• The y-axis is \( \ln([A]) \), and the x-axis is time.
• The slope of the line equals \(-k\), allowing determination of the rate constant \( k \).

Interpreting these graphs is crucial for determining reaction dynamics in case studies and real-world applications. By analyzing the slope and y-intercept, you can gain insights into the kinetic parameters and initial conditions of a reaction, making this a powerful tool in chemical analysis and engineering.