Question

Consider the following initial rate data for the decomposition of compound \(\mathrm{AB}\) to give \(\mathrm{A}\) and \(\mathrm{B}\) : Determine the half-life for the decomposition reaction initially having \(1.00 M \mathrm{AB}\) present.

Short answer

To determine the half-life for the decomposition of compound AB, first find the reaction order by creating plots for each Integrated Rate Law (zero-order, first-order, and second-order) and identify which one provides a straight line. Then, calculate the rate constant (k) using the slope of the best-fit line for the identified reaction order. Finally, use the appropriate half-life equation for the reaction order, plug in the obtained rate constant (k) and the initial concentration of 1.00 M (if required), and calculate the half-life.

Step-by-step solution

Find the Order of the Reaction

To determine the reaction order, we will use the Integrated Rate Laws for zero-order, first-order, and second-order reactions. We'll create a table for each and see which one has a straight line when we plot the values: Zero-order: \([\mathrm{AB}] = -kt+[A]_0\) First-order: \(\ln([\mathrm{A}]_0/[\mathrm{AB}]) = kt\) Second-order: \(1/[\mathrm{AB}] - 1/[\mathrm{A}]_0 = kt\) Plotting these values for each Integrated Rate Laws should give a straight line for one of them. Whichever best fits a straight line represents the true order of the reaction.

Calculate the Rate Constant

Once the reaction order has been identified by comparing the plots for Integrated Rate Laws, the rate constant can be determined by calculating the slope of the best-fit line. The rate constant (k) can be found directly from the slope of the graph for the corresponding reaction order. For zero-order: Slope = -k For first-order: Slope = k For second-order: Slope = k

Determine the Half-Life

After identifying the reaction order and rate constant (k), use the appropriate half-life equation for the given reaction order. The initial concentration of AB is 1.00 M: For zero-order: \(t_{1/2} = \frac{[\mathrm{AB}]_0}{2k}\) For first-order: \(t_{1/2} = \frac{\ln{2}}{k}\) For second-order: \(t_{1/2} = \frac{1}{k[\mathrm{AB}]_0}\) Plug in the obtained reaction rate constant (k) and initial concentration (if required) into the appropriate equation to find the half-life.

Key concepts

Reaction Order

In chemical kinetics, the reaction order is a crucial concept that helps in understanding how the concentration of reactants affects the rate of reaction.
A reaction can be zero-order, first-order, or second-order. Each reaction order has a different impact on the rate law equation.

• **Zero-order Reaction**: The rate is independent of the concentration of reactants. The rate law is expressed as \([\text{AB}]=-kt+[\text{AB}]_0\). This means that the concentration decreases linearly over time.
• **First-order Reaction**: The rate depends linearly on the concentration of one reactant. Its rate law is \(\ln([\text{AB}]_0/[\text{AB}])=kt\). Here, the logarithm of concentration changes over time.
• **Second-order Reaction**: The rate depends on the square of the concentration of one reactant or the product of two reactant concentrations. For this, the rate law is \([1/[\text{AB}] - 1/[\text{AB}]_0 = kt]\). It delves into reciprocal concentration changes over time.

You determine the reaction order by analyzing which plot of the integrated rate laws produces a straight line. That specific line indicates a match with the respective order of reaction.

Rate Constant

The rate constant, denoted as \(k\), is an important parameter in the rate equation of a chemical reaction. It provides insight into the speed and kinetics of a reaction.
For any given reaction, the rate constant is obtained by determining the slope of a linear plot corresponding to the correct reaction order.

• **For Zero-order**: Here, the rate constant is indicated by the negative slope \(\text{-k}\) of concentration versus time graph.
• **For First-order**: The rate constant is represented by the positive slope \(k\) of the plot of the natural logarithm of concentration versus time.
• **For Second-order**: The value of \(k\) is acquired from the slope of the graph of the reciprocal of concentration versus time.

The rate constant offers quantitative measures of the reaction speed and its dependency on factors such as temperature and presence of a catalyst. Each order has its unique way to determine \(k\), but each serves the same purpose—you get a clearer understanding of how fast or slow a reaction progresses under specified conditions.

Half-life

Half-life (\(t_{1/2}\)) is the time required for the concentration of a reactant to decrease to half of its initial amount. It's a critical concept in chemical reactions as it provides insight into the duration and sustainability of reactions.
Different reaction orders utilize different equations to calculate half-life:

• **Zero-order**: The half-life is calculated using \(t_{1/2} = \frac{[\text{AB}]_0}{2k}\). As the initial concentration of AB increases, the half-life also increases.
• **First-order**: Here, half-life is constant and can be expressed using \(t_{1/2} = \frac{\ln{2}}{k}\). It doesn’t depend on the initial concentration, so it's constant throughout the reaction.
• **Second-order**: The calculation \(t_{1/2} = \frac{1}{k[\text{AB}]_0}\) indicates that the half-life decreases as the initial concentration of AB increases.

Understanding the half-life of a reaction is essential for predicting how long a reaction will persist under given conditions and is especially useful in fields like pharmacokinetics and isotope decay.”

Integrated Rate Laws

Integrated Rate Laws are mathematical expressions that describe the concentration of reactants as a function of time. They are essential for determining reaction order and predicting the progression of reaction features such as half-life and remaining concentrations over time.
Each reaction order has its own integrated rate law which helps in understanding how concentration changes:

• **Zero-order**: The integrated rate law is \(\text{[AB]} = -kt+[\text{AB}]_0\), indicating that the concentration decreases at a consistent rate.
• **First-order**: Expressed by \(\ln([\text{A}]_0/[\text{AB}]) = kt\), it highlights that the natural logarithm of concentration declines linearly.
• **Second-order**: Given by \(1/[\text{AB}] - 1/[\text{A}]_0 = kt\), this shows the reciprocal concentration decreases with time in a parabolic path.

By utilizing integrated rate laws, you can calculate concentration values at any point in time, identify reaction order by observing plotted data, and apply these insights to diverse practical applications within and beyond chemistry.