Problem 76 - Chemistry and Chemical Reactivity

Question

The decomposition of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) to \(\mathrm{SO}_{2}\) and \(\mathrm{Cl}_{2}\) is first-order in \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) $$ \begin{array}{c} \mathrm{SO}_{2} \mathrm{Cl}_{2}(\mathrm{g}) \rightarrow \mathrm{SO}_{2}(\mathrm{g})+\mathrm{Cl}_{2}(\mathrm{g}) \\ \text { Rate }=k\left[\mathrm{SO}_{2} \mathrm{Cl}_{2}\right] \text { where } k=0.17 / \mathrm{hr} \end{array} $$ (a) What is the rate of decomposition when \(\left[\mathrm{SO}_{2} \mathrm{Cl}_{2}\right]=0.010 \mathrm{M} ?\) (b) What is the half-life of the reaction? (c) If the initial pressure of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) in a flask is \(0.050 \mathrm{atm},\) what is the pressure of all gases (i.e., the total pressure) in the flask after the reaction has proceeded for one half-life?

Step-by-Step Solution

Answer

(a) The rate is 0.0017 M/hr. (b) The half-life is 4.08 hours. (c) The total pressure is 0.075 atm.

Step 1: Understanding the Problem

The decomposition of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) is a first-order reaction. First-order reactions have a rate law in the form \(\text{Rate} = k[A]\), where \(\left[A\right]\) is the concentration of the reactant and \(k\) is the rate constant. Here, \(k = 0.17\, \text{hr}^{-1}\). We need to find the rate of decomposition, the half-life of the reaction, and the total pressure after one half-life.

Step 2: Calculate Rate of Decomposition

The rate of decomposition is given by \(\text{Rate} = k[\mathrm{SO}_{2} \mathrm{Cl}_{2}]\). Substituting the values, we have: \[\text{Rate} = 0.17 \times 0.010 = 0.0017\, \text{mol} \cdot \text{L}^{-1} \cdot \text{hr}^{-1}.\] This is the rate of decomposition of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) at \(0.010 \text{ M}.\)

Step 3: Determine the Half-life

The half-life for a first-order reaction is given by \(t_{1/2} = \frac{0.693}{k}\). Using the given \(k = 0.17\, \text{hr}^{-1}\), we can calculate: \[t_{1/2} = \frac{0.693}{0.17} \approx 4.08 \text{ hours}.\] This is the time it takes for the concentration of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) to be reduced to half.

Step 4: Calculate Pressure After One Half-life

Initially, the pressure of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) is \(0.050\, \text{atm}.\) After one half-life, half of it will have decomposed, so \(0.025\, \text{atm}\) remains as \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\). The decomposed \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) will produce 0.025 atm each of \(\mathrm{SO}_{2}\) and \(\mathrm{Cl}_{2}\), hence \(0.025 \times 2 = 0.050\, \text{atm}\) from the products. The total pressure is \[0.025 + 0.025 + 0.025 = 0.075\, \text{atm}.\]

Key Concepts

Decomposition Reaction

When discussing decomposition reactions, it's important to recognize that they involve the breakdown of a single compound into two or more products. In the case of our original exercise, \(\text{SO}_{2} \text{Cl}_{2}\) decomposes into \(\text{SO}_{2}\) and \(\text{Cl}_{2}\). Here, \(\text{SO}_{2} \text{Cl}_{2}\) is the compound that undergoes decomposition. This kind of reaction is typically described by a chemical equation indicating the reactants and the resulting products. In a decomposition reaction, energy is often required to break the bonds holding the compound together.
Features of decomposition reactions include:

• Single reactant breaking down into simpler products.
• Often requires energy input, such as heat, light, or electricity.
• Can occur naturally or be induced through laboratory techniques.

These reactions are significant in both chemical industry and biological processes, offering insights into how compounds break down into simpler substances.

Rate Law

The rate law is crucial for understanding how the concentration of reactants affects the speed of a chemical reaction. For a first-order decomposition like the one discussed, the rate law is expressed as \(\text{Rate} = k[A]\), where \(A\) is the concentration of the reactant and \(k\) is the rate constant. The rate constant \(k\) indicates how quickly the reaction proceeds under specific conditions.
In this case, \(k = 0.17 \, \text{hr}^{-1}\). This rate law implies that the rate at which \(\text{SO}_{2} \text{Cl}_{2}\) decomposes is directly proportional to its concentration. Thus, as the concentration decreases, the reaction rate diminishes accordingly. Understanding the rate law helps predict how long it will take for a reaction to reach a certain stage, an essential aspect of reaction kinetics. It also allows chemists to adjust conditions to optimize reaction rates, crucial in industrial and laboratory settings.

Half-life

Half-life in the context of chemical reactions is the time required for half of the reactant to decompose. For first-order reactions, such as \(\text{SO}_{2} \text{Cl}_{2}\)'s decomposition, the half-life is independent of its initial concentration and can be calculated using the formula \(t_{1/2} = \frac{0.693}{k}\).
In our example, using \(k = 0.17 \, \text{hr}^{-1}\), the half-life \(t_{1/2}\) is approximately 4.08 hours. This means it takes roughly 4.08 hours for the concentration of \(\text{SO}_{2} \text{Cl}_{2}\) to reduce by half.
This predictable pattern of decay is helpful in planning experimental or industrial processes. For reactions carrying potential hazards or requiring precise timing, knowing the half-life allows for controlled manipulation, ensuring safety and efficiency.

Chemical Kinetics

Chemical kinetics explores the rates of chemical processes and the factors influencing them. It encompasses the study of reaction rates, mechanisms, and conditions affecting them. In our example, understanding the kinetics of \(\text{SO}_{2} \text{Cl}_{2}\)'s decomposition provides insights into how quickly products form and relate to concentration changes.
Factors affecting reaction rates include

• Concentration of reactants: Higher concentrations usually increase reaction rates.
• Temperature: Raising temperature generally speeds up reactions.
• Catalysts: These substances can enhance reaction rates without being consumed.
• Surface area: Finer particle sizes tend to increase rates in heterogeneous reactions.

Chemical kinetics is foundational in developing new reactions and processes, crucial for everything from pharmaceuticals to environmental chemistry. By mastering kinetics, scientists can innovate and improve methods to synthesize desired compounds more efficiently.