Problem 42 - Chemistry: The Central Science

Question

Molecular iodine, \(\mathrm{I}_{2}(\mathrm{~g})\), dissociates into iodine atoms at \(625 \mathrm{~K}\) with a first-order rate constant of \(0.271 \mathrm{~s}^{-1}\). (a) What is the half-life for this reaction? (b) If you start with \(0.050 \mathrm{MI}_{2}\) at this temperature, how much will remain after \(5.12 \mathrm{~s}\) assuming that the iodine atoms do not recombine to form \(\mathrm{I}_{2}\) ?

Step-by-Step Solution

Answer

(a) The half-life for the dissociation of molecular iodine, \(I_2(g)\), is 2.56 s. (b) After 5.12 s, there will be 0.025 M of molecular iodine remaining, assuming no reformation of iodine atoms to form \(I_2\).

Step 1: Determine the half-life for the first-order reaction

For a first-order reaction, the half-life (t1/2) can be calculated using the following formula: \(t_{1/2} = \frac{0.693}{k}\) where: - t1/2 is the half-life, - k is the first-order rate constant. We are given that k = 0.271 s^(-1). Plugging this value into the equation, we can calculate the half-life: \(t_{1/2} = \frac{0.693}{0.271 \mathrm{~s^{-1}}} = 2.56 \mathrm{~s}\) So, the half-life for this reaction is 2.56 s.

Step 2: Determine the remaining amount of molecular iodine after 5.12 s

To find the remaining amount after 5.12 s, we can use the equation for the first-order reaction: \([A]_t = [A]_0 e^{-kt}\) where: - [A]_t is the concentration of A at time t, - [A]_0 is the initial concentration of A, - k is the rate constant, - t is the time that has elapsed. We are given the following values: - [A]_0 = 0.050 M, - k = 0.271 s^(-1), - t = 5.12 s. Plugging these values into the equation, we can calculate the remaining amount of molecular iodine: \([A]_{t} = 0.050 \mathrm{M} \cdot e^{-0.271 \mathrm{~s^{-1}} \cdot 5.12 \mathrm{~s}} = 0.050 \mathrm{M} \cdot e^{-1.386} = 0.025 \mathrm{M}\) Thus, after 5.12 s, there will be 0.025 M of molecular iodine remaining, assuming no reformation.

Key Concepts

Rate Constant

At the heart of first-order reactions lies the rate constant, denoted as 'k', which is a crucial parameter in chemical kinetics. This term helps to quantify the speed at which a reaction proceeds. Specifically, for a first-order reaction, the rate constant relates the reaction rate to the concentration of the reactant. The higher the value of 'k', the faster the reaction goes to completion.

Understanding the rate constant is essential as it remains constant at a given temperature and provides a measure of the inherent reactivity of the reaction. Calculating 'k' from experimental data allows scientists to predict how fast a reaction will proceed under similar conditions. In the case of the dissociation of molecular iodine, the rate constant is 0.271 s^-1, indicating that the reaction happens relatively quickly at 625 K.

Half-Life Formula

The half-life of a reaction is the time required for the concentration of a reactant to decrease by half. For a first-order reaction, the half-life formula is beautifully simple and is given by the expression \( t_{1/2} = \frac{0.693}{k} \), where \( t_{1/2} \) is the half-life and \( k \) is the rate constant.

This formula is independent of the initial concentration of the reactant, which is a unique characteristic of first-order kinetics. For the dissociation of molecular iodine, we've determined the half-life to be 2.56 seconds using the provided rate constant, which shows that the quantity of iodine will reduce to half its initial amount every 2.56 seconds.

Chemical Kinetics

Chemical kinetics is the study of the rates of chemical processes. For first-order reactions, the rate of the reaction is directly proportional to the concentration of one reactant. This means that as the reactant's concentration decreases, the reaction rate also decreases.

In our exercise, the dissociation of molecular iodine is described by first-order kinetics. Thus, the rate at which the iodine atoms form is directly dependent on the concentration of the molecular iodine present. Chemical kinetics not only determines the speed of reactions but also helps in understanding reaction mechanisms and conditions favoring product formation.

Reaction Concentration Over Time

For first-order reactions, the relationship between the concentration of reactants over time can be described by the equation \( [A]_t = [A]_0 e^{-kt} \), where \( [A]_t \) represents the concentration at time \( t \), \( [A]_0 \) is the initial concentration, \( k \) is the rate constant, and \( e \) is Euler’s number (approximately 2.718).

This equation allows us to predict the amount of reactant remaining after any given time. In the iodine dissociation example, we see that after 5.12 seconds, the concentration of molecular iodine decreases to half of its initial value twice, leaving us with 0.025 M. This demonstrates a practical utilization of chemical kinetics to understand how concentrations evolve during a reaction.