Question

For the reaction \(A \longrightarrow\) products, the following data were obtained: \(t=0 \mathrm{s},[\mathrm{A}]=0.715 \mathrm{M} ; 22 \mathrm{s}, 0.605 \mathrm{M}\) 74 s, 0.345 M; 132 s, 0.055 M. (a) What is the order of this reaction? (b) What is the half-life of the reaction?

Short answer

The order of reaction is first order. The half-life can be calculated using the average value of k obtained from step 1 in formula \( t1/2 = 0.693 / k \).

Step-by-step solution

Calculate the Rate Constants for Each Time Interval

To find the order of the reaction, initially calculate the rate constant (k) for each time interval using the formula for a first order reaction, \( k = (1/t) * ln([A]_{initial} / [A]_{final}) \) where [A] is the molar concentration. For e.g, for \( t = 0 \) to \( t = 22 \) sec, \( k = (1/22) * ln(0.715/0.605) \) Repeat the above step for each time interval and compare the values of k.

Analyze the Rate Constants

If the rate constants are approximately the same for each time interval, this indicates that the reaction is indeed first order. If they are not, then might need to test for zero or second order reaction, but for simplicity of this explanation, assume it is a first-order reaction if k keeps approximately constant.

Calculate the Half-Life

For a first order reaction, the half-life (t1/2) is given by \( t1/2 = 0.693 / k \). Use the average k obtained from step 1 to evaluate t1/2. You might find that it matches closely with the interval of time it takes for the molar concentration of A to reduce approximately to half of its initial value, further confirming first order kinetics.

Key concepts

Reaction Order

Understanding the reaction order helps us determine how the rate of a chemical reaction depends on the concentration of its reactants. In simple terms, it tells us how the speed of the reaction changes when we change the concentration of the reactants.

* **Zero-order reactions**: For these reactions, the rate is independent of the concentration. Essentially, whenever you change the concentration, it doesn't affect the reaction rate. This means that the reaction occurs at a constant rate until the reactants are completely consumed.
* **First-order reactions**: Here, the rate is directly proportional to the concentration of one reactant. If you double the concentration, the rate also doubles. This indicates a direct relationship between concentration and rate.
* **Second-order reactions**: These depend on the concentration of one reactant squared or on two different reactants each raised to the power of one. So, if the concentration doubles, the rate becomes four times faster in some cases.

Determining the order is crucial because it helps in understanding how quickly a reaction can proceed under given conditions. In the exercise, by checking the consistency of the rate constant, it's suggested that the reaction is first-order.

Rate Constant

The rate constant, denoted as \(k\), is a crucial factor in chemical reactions. It acts as a proportionality constant in the rate equation that relates reactant concentrations to the reaction rate.

For a first-order reaction, the rate equation can be expressed as:
\[ ext{Rate} = k[A] \]

Here, *\([A]\)* represents the concentration of the reactant. The rate constant \(k\) helps quantify how fast the reaction proceeds under certain conditions. Importantly, the value of \(k\) remains constant for a given reaction at a specific temperature and is independent of the concentration.

In the problem given, the calculation of \(k\) for different time intervals allows us to determine the reaction order. If \(k\) stays relatively consistent, it gives us evidence supporting the reaction being first-order. Understanding the rate constant aids in better control and prediction of reaction outcomes.

Half-Life Calculation

Half-life in the context of chemical kinetics refers to the time it takes for half of the reactant concentration to be consumed in a reaction. It is a valuable metric because it gives a time scale for the reaction's progress.

For first-order reactions, the half-life can be calculated using the formula:
\[ t_{1/2} = \frac{0.693}{k} \]

where \( t_{1/2} \) is the half-life and \( k \) is the rate constant. This particular formula indicates that for first-order reactions, the half-life is constant regardless of the initial concentration.

In the given exercise, after determining the average rate constant \(k\), the half-life can be calculated. This value provides insight, indicating that the time required for the concentration to reduce to half its initial value matches observations, confirming the reaction's first-order nature.

First Order Reactions

First-order reactions occupy a special place in chemical kinetics due to their simplicity. In these reactions, the rate is determined by the concentration of a single reactant raised to the power of one. This makes the mathematical treatment straightforward and the predictions reliable.

Characteristics of first-order reactions include:

• The rate depends directly on the concentration of one reactant.
• The half-life is independent of the initial concentration, meaning it remains constant throughout the reaction.
• They often involve processes like radioactive decay and certain types of chemical reactions like isomerization.

In the provided exercise, the process of calculating \(k\) and the resultant consistent values suggest that the reaction is first-order. Such reactions are easier to handle due to these predictable characteristics, making them a fundamental concept in studying chemical kinetics.