Question

Iodine, \(I_{2}(g),\) dissociates into iodine atoms at a particular temperature in a first-order process. If \(2.00 M \mathrm{I}_{2}\) changes to \(0.133 \mathrm{M}\) in \(10.0 \mathrm{~s}\), what is the rate constant for the reaction?

Short answer

The rate constant is approximately \(0.2708 \, s^{-1}\).

Step-by-step solution

Identify the type of reaction and equation

The reaction described is a first-order reaction. For a first-order reaction, the relation between concentration and time is given by the equation: \[ [A] = [A]_0 e^{-kt} \]Where - [A] is the concentration at time t,\- [A]_0 is the initial concentration,\- k is the rate constant, and \- t is time.

Plug in the known values

Given that the initial concentration \([A]_0\) is \(2.00 \, M\), the concentration at time t ([\(A\)]) is \(0.133 \, M\), and the time \(t\) is \(10.0 \, s\), plug these values into the equation: \[ 0.133 = 2.00 e^{-10.0k} \]

Solve for the rate constant k

First, isolate the exponential term by dividing both sides of the equation by the initial concentration: \[ \frac{0.133}{2.00} = e^{-10.0k} \]Simplify the left side: \[ 0.0665 = e^{-10.0k} \]Next, take the natural logarithm of both sides to remove the exponential: \[ \ln(0.0665) = -10.0k \]Solve for k by dividing both sides by -10.0: \[ k = -\frac{ln(0.0665)}{10.0} \]

Calculate the numerical value

Find the natural logarithm of 0.0665: \[ \ln(0.0665) \approx -2.708 \]Next, divide by -10.0: \[ k = \frac{2.708}{10.0} = 0.2708 \, s^{-1} \]

Key concepts

reaction rate constant

The reaction rate constant, denoted as ‘k’, is a crucial parameter in chemical kinetics. It quantifies the speed at which a chemical reaction occurs. For first-order reactions, the rate constant has units of reciprocal time (e.g., \(s^{-1}\)). In our exercise, we determined the rate constant by using the first-order rate law: \[ [A] = [A]_0 e^{-kt} \] Here, the rate constant reflects how quickly iodine gas dissociates into iodine atoms. Calculating the rate constant involves isolating ‘k’ and solving it mathematically. Understanding its value helps us predict reaction speeds under various conditions.

first-order kinetics

First-order kinetics describes reactions where the rate depends linearly on the concentration of one reactant. In simpler terms, if you double the concentration of the reactant, the reaction rate also doubles. For first-order reactions, the concentration-time relationship can be expressed as: \[ [A] = [A]_0 e^{-kt} \] This equation states that the concentration of the reactant decreases exponentially over time. An example is the dissociation of iodine gas into iodine atoms. Knowing that the process follows first-order kinetics lets us use this equation to determine either the rate constant or the remaining concentration at any given time.

natural logarithm

The natural logarithm (ln) is a special mathematical function that helps simplify the process of solving for variables in exponential decay scenarios, such as first-order reactions. In our example, we transformed the exponential equation: \[ \frac{0.133}{2.00} = e^{-10.0k} \] into a linear equation by taking the natural logarithm: \[ \text{ln}(0.0665) = -10.0k \] This transformation allows solving for ‘k’ directly. The natural logarithm function is crucial in simplifying exponential models, making it easier to work with large ranges in data and solving for unknowns.

dissociation reaction

A dissociation reaction is a chemical reaction in which a compound breaks down into simpler components. The breakdown of iodine gas (\(I_2(g)\)) into iodine atoms is one such reaction. This process can be represented as: \[ \text{I}_2(g) \rightarrow 2 I(g) \] In our exercise, the dissociation of iodine gas follows first-order kinetics, meaning the rate of dissociation is proportional to the concentration of iodine gas. Understanding the behavior of dissociation reactions helps us explore reaction mechanisms and predict the behavior of different substances under varying conditions.

concentration-time relationship

The concentration-time relationship is a core concept in understanding and predicting how reaction concentrations change over time. For a first-order reaction like the dissociation of iodine gas, the relationship is exponential. This is described by the equation: \[ [A] = [A]_0 e^{-kt} \] Here, \[ [A]_0 \] is the initial concentration, \[ [A] \] is the concentration at time ‘t’, ‘k’ is the rate constant, and ‘e’ is the base of the natural logarithm. By plugging in known values, we can determine how much of the reactant remains at any point. In our example, the concentration dropped from \[ 2.00 \text{ M} \] to \[ 0.133 \text{ M} \] in \[ 10.0 \text{ s} \]. This relationship is fundamental in chemical kinetics and reaction engineering.