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Chapter 1: Problem 5

$$\mathrm{SO}_{2} \mathrm{Cl}_{2} \rightarrow \mathrm{SO}_{2}(g)+\mathrm{Cl}_{2}(g)$$ At \(600 \mathrm{K}, \mathrm{SO}_{2} \mathrm{Cl}_{2}\) will decompose to form sulfur dioxide and chlorine gas via the above equation. If the reaction is found to be first order overall, which of the following will cause an increase in the half life of \(\mathrm{SO}_{2} \mathrm{Cl}_{2} ?\) (A) Increasing the initial concentration of \(\mathrm{SO}_{2} \mathrm{Cl}_{2}\) (B) Increasing the temperature at which the reaction occurs (C) Decreasing the overall pressure in the container (D) None of these will increase the half life.

Short Answer

Expert verified
(D) None of these will increase the half life.

Step by step solution

01

Understand the dependence of half-life on initial concentration

In a first-order reaction, the half-life is independent of the initial concentration of reactants. This can be derived from the first-order rate law, \( k = [A] / t_{1/2} \). Rearranging gives us \( t_{1/2} = [A] / k \), which shows that the half-life does not depend on [A]. Therefore, changing the initial concentration of the reactant (\(\mathrm{SO}_{2} \mathrm{Cl}_{2}\)) will not affect the half-life. So (A) is not a correct option.
02

Understand the dependence of half-life on temperature

The half-life of a first-order reaction is inversely proportional to the rate constant (\( t_{1/2} = 1/ k \)). As temperature increases, the rate constant increases due to higher average kinetic energy of the molecules, leading to more collisions per unit time. Consequently, the half-life decreases. Therefore, increasing the temperature at which the reaction occurs will actually decrease the half-life, not increase it. So (B) is not a correct option.
03

Understand the dependence of half-life on pressure

Pressure only indirectly affects the half-life through changes in concentration when volume changes. Since we already established that the half-life of a first-order reaction is independent of the initial concentration, changes in pressure should not affect the half-life of the reaction. So (C) is not a correct option.
04

Determine the correct answer

As we have refuted options (A) to (C), it means that none of the options given will increase the half-life of the reaction.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Half-life Dependency
The concept of half-life is crucial in understanding the time it takes for half of a given substance to decompose in a reaction. For first-order reactions, the half-life \( t_{1/2} \) is constant and does not depend on the initial concentration of the reactants. This is due to the mathematical relationship \( t_{1/2} = \frac{0.693}{k} \), where \( k \) is the rate constant. As you can see, the half-life is inversely proportional to the rate constant.
  • **Initial Concentration**: Changing the initial amount of the reactant, such as \( \mathrm{SO}_{2} \mathrm{Cl}_{2} \), does not impact the half-life.
  • **Rate Constant**: The primary factor that influences the half-life is the rate constant \( k \).
  • **Temperature**: An increase in temperature typically increases \( k \), thus reducing the half-life, illustrating why altering temperature impacts the duration of the reaction.
Understanding these dependencies can help make predictions about how long a reaction will proceed under different experimental conditions.
Reaction Kinetics
Reaction kinetics involves the study of the rates of chemical reactions and how different conditions affect these rates. In first-order reactions, the rate of decomposition is directly proportional to the concentration of one reactant. The rate law for a first-order reaction is expressed as \( \text{Rate} = k[A] \) where \([A]\) is the concentration of the reactant.
Key points to consider about first-order reaction kinetics include:
  • **Concentration Dependence**: The reaction rate will decrease as the reactant concentration decreases with time.
  • **Temperature Effects**: Raising the temperature can lead to a higher kinetic energy among molecules, increasing the rate constant \( k \), thus increasing the reaction rate.
  • **Pressure and Volume**: These factors can modify the reaction kinetics through changes in concentration, but for first-order reactions, the overall effect tends to be less pronounced compared to higher-order reactions.
Gaining a deep understanding of reaction kinetics aids in manipulating and controlling chemical processes effectively.
Decomposition Reaction
Decomposition reactions are processes where a single compound breaks down into two or more simpler products. In the case of \( \mathrm{SO}_{2} \mathrm{Cl}_{2} \), it decomposes into sulfur dioxide \( \mathrm{SO}_{2}(g) \) and chlorine gas \( \mathrm{Cl}_{2}(g) \). This reaction is particularly noteworthy in studying first-order reactions because:
  • **Simple Product Formation**: The formation of gases from a compound makes it easy to measure changes over time, providing clear insight into the reaction kinetics.
  • **First-order Characteristics**: Decomposition of \( \mathrm{SO}_{2} \mathrm{Cl}_{2} \) exemplifies how first-order kinetics operate, with the rate depending solely on the concentration of the single reactant.
  • **Impact of Conditions**: External conditions like temperature can significantly influence the rate of decomposition, making it a great case study for understanding kinetics.
Decomposition reactions like this help in illustrating how rates of reactions are affected by various factors, thus enhancing your comprehension of chemical behavior and kinetics.

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Most popular questions from this chapter

Use the following information to answer questions 12-15. When heated in a closed container in the presence of a catalyst, potassium chlorate decomposes into potassium chloride and oxygen gas via the following reaction: \(2 \mathrm{KClO}_{3}(s) \rightarrow 2 \mathrm{KCl}(s)+3 \mathrm{O}_{2}(g)\) If 12.25 g of potassium chlorate decomposes, how many grams of oxygen gas will be generated? (A) 1.60 g (B) 3.20 g (C) 4.80 g (D) 18.37 g

For a reaction involving nitrogen monoxide inside a sealed flask, the value for the reaction quotient \((Q)\) was found to be \(1.1 \times 10^{2}\) at a given point. If, after this point, the amount of NO gas in the flask increased, which reaction is most likely taking place in the flask? (A) \(\operatorname{NOBr}(g) \rightarrow \operatorname{NO}(g)+1 / \operatorname{Br}_{2}(g) \quad K_{\mathrm{C}}=3.4 \times 10^{-2}\) (B) \(2 \mathrm{NOCl}(g) \mapsto 2 \mathrm{NO}(g)+\mathrm{Cl}_{2}(g) \quad K_{\mathrm{c}}=1.6 \times 10^{-5}\) (C) \(2 \mathrm{NO}(g)+2 \mathrm{H}_{2}(g) \rightarrow \mathrm{N}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(g) \quad K_{\mathrm{c}}=4.0 \times 10^{6}\) (D) \(\mathrm{N}_{2}(g)+\mathrm{O}_{2}(g) \rightarrow 2 \mathrm{NO}(g) \quad K_{\mathrm{c}}=4.2 \times 10^{2}\)

$$2 \mathrm{NOCl} \rightarrow 2 \mathrm{NO}+\mathrm{Cl}_{2}$$ The reaction above takes place with all of the reactants and products in the gaseous phase. Which of the following is true of the relative rates of disappearance of the reactants and appearance of the products? (A) NO appears at twice the rate that NOCl disappears. (B) NO appears at the same rate that NOCl disappears. (C) NO appears at half the rate that NOCl disappears. (D) \(\mathrm{Cl}_{2}\) appears at the same rate that NOCl disappears.

Which of the following could be added to an aqueous solution of weak acid HF to increase the percent dissociation? (A) \(\operatorname{NaF}(s)\) (B) \(\mathrm{H}_{2} \mathrm{O}(l)\) (C) \(\mathrm{NaOH}(\mathrm{s})\) (D) \(\mathrm{NH}_{3}(a q)\)

\(\mathrm{Cr}_{2} \mathrm{O}_{7}^{2-}+6 \mathrm{I}^{-}+14 \mathrm{H}^{+} \rightarrow 2 \mathrm{Cr}^{3+}+3 \mathrm{I}_{2}+7 \mathrm{H}_{2} \mathrm{O}\) Which of the following statements about the reaction given above is NOT true? (A) The oxidation number of chromium changes from \(+6\) to \(+3 .\) (B) The oxidation number of iodine changes from \(-1\) to 0. (C) The oxidation number of hydrogen changes from +1 to 0. (D) The oxidation number of oxygen remains the same.
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