Problem 75 - Chemistry The Science in Context

Question

The dimerization of \(\mathrm{ClO}\) $$ 2 \mathrm{ClO}(g) \rightarrow \mathrm{Cl}_{2} \mathrm{O}_{2}(g) $$ is second order in ClO. Use the following data to determine the value of \(k\) at \(298 \mathrm{K}\) $$\begin{array}{cc} \text { Time (s) } & \text { [ClO] (molecules/cm^3) } \\ \hline 0.0 & 2.60 \times 10^{11} \\ \hline 1.0 & 1.08 \times 10^{11} \\ \hline 2.0 & 6.83 \times 10^{10} \\ \hline 3.0 & 4.99 \times 10^{10} \\ \hline 4.0 & 3.93 \times 10^{10} \\ \hline \end{array}$$ Determine the half-life for the dimerization of C1O.

Step-by-Step Solution

Answer

Time (s) [ClO] (molecules/cm^3) ------------------------- 0.0 2.60 x 10^11 1.0 1.08 x 10^11 2.0 6.83 x 10^10 3.0 4.99 x 10^10 4.0 3.93 x 10^10 Answer: The half-life for the dimerization of ClO at 298 K is approximately 0.934 seconds.

Step 1: 1. Write the rate law equation for the reaction

Since the reaction is second order in \(\mathrm{ClO}\), the rate law equation can be written as: $$\text{rate} = k[\mathrm{ClO}]^2$$

Step 2: 2. Recognize that a second-order reaction has a linear form

A second-order reaction has a linear form when the reciprocals of the concentrations are plotted over time. The integrated rate law for a second-order reaction is given by: $$\frac{1}{[\mathrm{ClO}]} = kt + \frac{1}{[\mathrm{ClO}]_0}$$ We will use this equation to determine the value of \(k\).

Step 3: 3. Organize the data for calculations

Let's organize the given data for calculations. We need to find the reciprocal of the concentrations in the table. The updated data table becomes: $$\begin{array}{cc} \text { Time (s) } & \text { [ClO] (molecules/cm^3) } & \frac{1}{[\mathrm{ClO}] (\mathrm{cm}^{3}/\mathrm{molecule})} \\ \hline 0.0 & 2.60 \times 10^{11} & 3.85 \times 10^{-12} \\ \hline 1.0 & 1.08 \times 10^{11} & 9.26 \times 10^{-12} \\ \hline 2.0 & 6.83 \times 10^{10} & 1.46 \times 10^{-11} \\ \hline 3.0 & 4.99 \times 10^{10} & 2.00 \times 10^{-11} \\ \hline 4.0 & 3.93 \times 10^{10} & 2.54 \times 10^{-11} \\ \hline \end{array}$$

Step 4: 4. Plot the reciprocals of the concentrations as a function of time

Plot the reciprocals of the concentrations as a function of time. The slope of the straight line will be equal to the rate constant \(k\). In this case, as we don't have access to the plot, we will use the given data points to determine the value of the slope.

Step 5: 5. Calculate the rate constant \(k\)

Choose any two points and calculate the change in the concentration reciprocal divided by the change in time: $$k = \frac{\Delta\frac{1}{[\mathrm{ClO}]}}{\Delta t} = \frac{(2.54 - 3.85) \times 10^{-11}}{(4 - 0)\, \mathrm{s}} = 3.28 \times 10^{-11}\,\mathrm{cm}^{3}\mathrm{molecule}^{-1}\mathrm{s}^{-1}$$ Thus, the value of the rate constant \(k\) at \(298\mathrm{K}\) is approximately \(3.28 \times 10^{-11}\,\mathrm{cm}^{3}\mathrm{molecule}^{-1}\mathrm{s}^{-1}\).

Step 6: 6. Calculate the half-life

Now that we have the value of the rate constant \(k\), we can calculate the half-life for the dimerization of \(\mathrm{ClO}\). The formula for the half-life of a second-order reaction is: $$t_{1/2} = \frac{1}{k[\mathrm{ClO}]_0}$$ Since we are asked to determine the half-life for dimerization, we will use \([\mathrm{ClO}]_0 = 2.60 \times 10^{11}\,\mathrm{molecules}\cdot\mathrm{cm}^{-3}\). $$t_{1/2} = \frac{1}{(3.28 \times 10^{-11}\,\mathrm{cm}^{3}\mathrm{molecule}^{-1}\mathrm{s}^{-1}) (2.60 \times 10^{11}\,\mathrm{molecules}\cdot\mathrm{cm}^{-3})} \approx 0.934\,\mathrm{s}$$ Therefore, the half-life for the dimerization of \(\mathrm{ClO}\) is approximately \(0.934\,\mathrm{s}\).

Key Concepts

Understanding Second-Order Reactions

A second-order reaction is defined by the relationship between the rate of the reaction and the concentration of the reactants. In such a reaction, the rate is proportional to the square of the concentration of one reactant or to the product of the concentrations of two reactants.

For the dimerization of ClO, which is a classic example of a second-order reaction, we use the rate law equation \(\text{rate} = k[\mathrm{ClO}]^2\). This signifies that the rate at which ClO dimerises to form \(\mathrm{Cl}_2\mathrm{O}_2\) is proportional to the square of its concentration at any given time.

Visualizing Second-Order Kinetics

To visualize and analyze second-order kinetics, one common approach is to plot the reciprocal of the reactant concentration (\(1/[\mathrm{ClO}]\)) against time. This yields a straight line for a true second-order reaction, through which the rate constant \(k\) can be determined from the slope. This graphical method complements the mathematical approach and aids in a clearer understanding of the reaction dynamics.

Understanding the intricacies of second-order reactions is pivotal for students, as it lays the groundwork for interpreting reaction mechanisms and predicting the behavior of chemical systems.

Deciphering the Rate Law Equation

The rate law equation is a mathematical representation that relates the rate of a chemical reaction to the concentration of its reactants. It is a crucial tool in chemical kinetics, as it allows chemists to quantify the effect of reactant concentrations on the reaction rate.

The General Form of Rate Laws

A general form for the rate law is \(\text{rate} = k[\text{A}]^n[\text{B}]^m\), where \(k\) is the rate constant, \(\text{A}\) and \(\text{B}\) are the concentrations of reactants, and \(n\) and \(m\) are the orders of the reaction with respect to each reactant. For the dimerization of ClO, the rate law simplifies to \(\text{rate} = k[\mathrm{ClO}]^2\) because the reaction is second-order with respect to ClO.

Correctly applying the rate law equation allows you to predict how rapidly a reaction will occur under different concentration conditions. It's like having a crystal ball that provides insights into the pace at which a chemical transformation takes place.

Calculating Half-Life in Chemical Reactions

Half-life, often denoted by \(t_{1/2}\), is a term commonly associated with radioactive decay, but it is also widely used in chemical kinetics. In this context, half-life refers to the time required for half of a reactant to be consumed in a chemical reaction.

The Concept of Half-Life

Half-life provides a convenient way to understand the timescale of a reaction. In a second-order reaction, the half-life is inversely proportional to the initial concentration of the reactant and the rate constant \(k\) and is given by the formula \(t_{1/2} = \frac{1}{k[\mathrm{Reactant}]_0}\).

Impact of Concentration on Half-Life

For a second-order reaction, it is important to note that the half-life depends on the initial concentration of the reactant, which means the half-life changes as the reaction progresses. This is in contrast to a first-order reaction where the half-life remains constant regardless of concentration.

Understanding how to calculate the half-life for a given reaction helps students in predicting how long it will take for significant changes to occur in the concentration of a reactant, which is particularly helpful in designing chemical processes and assessing reaction efficiency.