Question

Iodine atoms combine to form \(\mathrm{I}_{2}\) in liquid hexane solvent with a rate constant of \(1.5 \times 10^{10} \mathrm{L} / \mathrm{mol} \cdot \mathrm{s}\) . The reaction is second order in I. Since the reaction occurs so quickly, the only way to study the reaction is to create iodine atoms almost instantaneously, usually by photo- chemical decomposition of \(\mathrm{I}_{2}\) . Suppose a flash of light creates an ini- tial \([I]\) concentration of 0.0100 \(\mathrm{M}\) . How long will it take for 95\(\%\) of the newly created iodine atoms to recombine to form \(\mathrm{I}_{2} ?\)

Short answer

After calculating, the time it takes for 95% of the iodine atoms to recombine is approximately 6.67 x 10^{-5} s.

Step-by-step solution

Write down the given data

The reaction has a rate constant, k, of 1.5 x 10^{10} L/mol·s and is second order with respect to iodine atoms. The initial concentration of iodine atoms, [I]_0, is 0.0100 M. We want to find the time, t, it takes for 95% of these atoms to recombine, meaning the concentration of iodine atoms will be 5% of [I]_0, or 0.0005 M.

Express the rate of the reaction

For a second-order reaction, the rate can be expressed as the change in concentration of the reactant over time, equal to the product of the rate constant and the concentration of the reactant squared. Rate = -d[I]/dt = k[I]^2.

Set up the integrated rate law for a second-order reaction

The integrated rate law for a second-order reaction is 1/[I] - 1/[I]_0 = kt. We need to solve for t.

Input the given values and solve for t

Substitute [I] = 0.0005 M (5% of 0.0100 M), [I]_0 = 0.0100 M, and k = 1.5 x 10^{10} L/mol·s into the integrated rate law and solve for t. 1/0.0005 M - 1/0.0100 M = (1.5 x 10^{10} L/mol·s) * t.

Calculate the time, t

Rearrange the equation to solve for t: t = (1/0.0005 M - 1/0.0100 M) / (1.5 x 10^{10} L/mol·s). Perform the subtraction and division to find the value of t.

Key concepts

Chemical Kinetics

Understanding how chemical reactions occur and at what rate is essential in the study of chemistry. Chemical kinetics is the branch of physical chemistry that is concerned with understanding the rates of chemical reactions. It's a crucial area of study since it helps chemists control reactions to synthesize products efficiently, understand reaction mechanisms, and ensure safety in industrial processes.

Central to kinetics is the examination of how different conditions, such as concentration, temperature, and presence of a catalyst, affect the speed of a reaction. The rate of a reaction is typically expressed in terms of concentration change of a reactant or product per unit time. For example, a second-order reaction like the combination of iodine atoms to form \(\mathrm{I}_{2}\) in a solution, may have its rate affected significantly by the concentration of the reactant iodine atoms. In your practice problem, the concentration and the rate constant are provided, which allows you to determine the time it takes for the reaction to reach a certain completion level.

Rate Constant

The rate constant, often denoted as \(k\), is a proportionality constant in the rate equation of a chemical reaction. It is a unique value for each reaction at a given temperature, and it provides insight into the reaction's speed and mechanism.

In the exercise, the rate constant is given as \(1.5 \times 10^{10} \mathrm{L} / \mathrm{mol} \cdot \mathrm{s}\), which is unusually high, hinting at a very fast reaction under the conditions provided. The value of the rate constant can be influenced by temperature, with most reactions speeding up as temperature increases. However, the rate constant does not change with the concentration of reactants, which makes it a key indicator of inherent reaction speed. If the rate constant is known, the rate of reaction can be predicted for any given concentrations of the reactants.

Integrated Rate Law

While the rate equation tells you how fast a reaction is proceeding at any given moment, the integrated rate law lets you understand how concentrations change over time. For each order of reaction, there is a corresponding mathematical form of the integrated rate law.

For second-order reactions, such as the iodine atom recombination in your exercise, the integrated rate law is \(1/[I] - 1/[I]_0 = kt\). By integrating the rate equation over time, this formula connects the concentration of the reactant at any time point \(\([I]\)\) with the initial concentration \(\([I]_0\)\), the rate constant \(\(k\)\), and the time elapsed \(\(t\)\). In the exercise, by substituting the known concentrations and rate constant into this formula, you can solve for \(\(t\)\) to find out how long it will take for 95% of the iodine atoms to combine into \(\mathrm{I}_{2}\).

Understanding and applying the integrated rate law allows chemists to control reactions, predict how much product to expect at a given time, and design chemical processes with precision.