Question

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

Short answer

First, find the order of the reaction (n) using the provided initial rate data: \(n = \frac{\log{(\frac{Rate_1}{Rate_2})}}{\log{(\frac{\mathrm{AB_1}}{\mathrm{AB_2}})}\) Next, calculate the rate constant (k) using the rate equation: \(k = \frac{Rate_1}{[\mathrm{AB_1}]^n}\) Finally, determine the half-life (\(t_{1/2}\)) using the appropriate formula for the reaction order: For a first-order reaction: \(t_{1/2} = \frac{0.693}{k}\) For a second-order reaction: \(t_{1/2} = \frac{1}{k[\mathrm{AB}]_0}\) Plug in the values of k and \([\mathrm{AB}]_0\) to find the half-life (\(t_{1/2}\)).

Step-by-step solution

Identify the order of the reaction

To determine the order of the reaction, analyze the initial rate data. Suppose the order of the reaction is n. Then, the rate equation will be: \(Rate = k [\mathrm{AB}]^n \) We need to find the value of n.

Create a table of initial rate data

Write down the initial concentration of AB and the corresponding initial rate in the form of a table: | Trial | Initial Concentration | Initial Rate | |-------|------------------------|--------------| | 1 | [\(\mathrm{AB}\)_1] | \(Rate_1\) | | 2 | [\(\mathrm{AB}\)_2] | \(Rate_2\) |

Determine the order of the reaction

Using the table created in step 2, analyze the change in initial rate corresponding to the change in the initial concentration. Divide the rate equation for the first trial by the rate equation for the second trial: \(\frac{Rate_1}{Rate_2} = \frac{k [\mathrm{AB}_1]^n}{k [\mathrm{AB}_2]^n}\) To determine the order of the reaction (n), you can rearrange the equation and solve for n: \(n = \frac{\log{(\frac{Rate_1}{Rate_2})}}{\log{(\frac{\mathrm{AB_1}}{\mathrm{AB_2}})}\) Plug in the values from the initial rate data table and solve for n.

Calculate the rate constant, k

Now that we know the order of the reaction, we can calculate the rate constant k. Using the rate equation and substituting the order found in step 3: \(Rate_1 = k [\mathrm{AB_1}]^n\) Solve for k: \(k = \frac{Rate_1}{[\mathrm{AB_1}]^n}\) Plug in the values from the initial rate data table and solve for k.

Determine the half-life of the reaction

Now, we have the rate constant k and the order of the reaction n. We can determine the half-life of the reaction using the following relationship between the half-life and rate constant: For a first-order reaction: \(t_{1/2} = \frac{0.693}{k}\) For a second-order reaction: \(t_{1/2} = \frac{1}{k[\mathrm{AB}]_0}\) Plug in the values of k and \([\mathrm{AB}]_0\) and solve for the half-life, \(t_{1/2}\).

Key concepts

Rate law

When studying reaction kinetics, the rate law is a mathematical expression that helps us understand how the rate of a chemical reaction depends on the concentration of the reactants. It is generally written in the form:

• Rate = k [A]^m [B]^n

Here, Rate represents the speed of the reaction, k is the rate constant, and [A] and [B] are the concentrations of the reactants. The exponents m and n indicate how the rate is affected by the concentrations of the reactants. The rate law must be determined experimentally, as it cannot be inferred directly from the balanced chemical equation.
It's worth noting that the rate constant k and the exponents (reaction orders) are specific to a particular reaction at a given temperature. Understanding the rate law is crucial because it allows chemists to predict how changes in concentration will affect the speed of the reaction, which is important for controlling processes in industrial and laboratory settings.

Reaction order

In reaction kinetics, the reaction order provides information about the dependency of the reaction rate on the concentration of the reactants. It corresponds to the sum of the exponents in the rate law equation and can be zero, whole numbers, or even fractions. The common types of reaction orders are:

• **Zero Order**: Rate is independent of the concentration of reactants. The rate does not change as the concentration changes.
• **First Order**: Rate is directly proportional to the concentration of one reactant.
• **Second Order**: Rate depends on the concentration of one reactant squared or the product of the concentrations of two different reactants.

Identifying the reaction order enables prediction of how the reaction speed changes with varied reactant concentrations. In particular scenarios, like in the study, we even use the initial rates method: by varying the concentrations and observing the effect on the rate, the appropriate exponent for each reactant can be determined, giving us the overall reaction order.

Half-life

The concept of half-life is widely used in reaction kinetics to describe how long it takes for half of a reactant to be converted into products. Each type of reaction order has its distinct formula to calculate half-life, reflecting the unique dependency each has on concentration:

• **First-order reaction**: The half-life t\(_{1/2}\) is constant and does not depend on the initial concentration. It is given by: \[ t_{1/2} = \frac{0.693}{k} \]
• **Second-order reaction**: The half-life varies with the initial concentration of the reactant: \[ t_{1/2} = \frac{1}{k[ ext{A}]_0} \]

The calculation of half-life is essential for understanding how quickly a reaction proceeds, which is particularly important in fields like pharmacology and environmental science where reaction rates can influence outcomes like drug effectiveness or pollution breakdown.

Rate constant

The rate constant, symbolized as k, is a fundamental parameter in the rate law that is specific to each reaction at a certain temperature. It provides a proportionality factor in the rate equation, establishing the relationship between the reaction rate and the concentrations of reactants.

• **Temperature Dependency**: The value of k is sensitive to temperature changes. Typically, as temperature increases, k increases, leading to a faster reaction rate.
• **Units**: The units of k vary depending on the overall reaction order and are derived from the rate units and concentration units.

To determine the value of k experimentally, we use the rate law by rearranging it to solve for k once the reaction order is known. Understanding k helps in predicting the speed of chemical reactions under different conditions, making it an essential aspect of control in chemical manufacturing and research processes.