Consider the following elementary reactions: $$\begin{aligned} &A+B \stackrel{k_{1}}{\longrightarrow} R\\\ &R+B \stackrel{k_{2}}{\longrightarrow} S \end{aligned}$$ (a) One mole A and 3 moles B are rapidly mixed together. The reaction is very slow, allowing analysis of compositions at various times. When 2.2 moles \(\mathrm{B}\) remain unreacted, 0.2 mole \(\mathrm{S}\) is present in the mixture. What should be the composition of the mixture \((\mathrm{A}, \mathrm{B}, \mathrm{R}, \text { and } \mathrm{S})\) when the amount of S present is 0.6 mole? (b) One mole \(A\) is added bit by bit with constant stirring to 1 mole \(B\). Left overnight and then analyzed, 0.5 mole \(S\) is found. What can we say about \(k_{2} / k_{1} ?\) (c) One mole A and 1 mole B are thrown together and mixed in a flask. The reaction is very rapid and goes to completion before any rate measurements can be made. On analysis of the products of reaction 0.25 mole \(S\) is found to be present. What can we say about \(k_{2} / k_{1} ?\)

Step by step solution

01

Analyzing Composition for Part (a)

Given 0.2 mole of S, since the formation of each mole of S requires one mole of B, 0.2 mole of B will have reacted to form S and 0.2 mole of R. Therefore, initially, there were 3 - 2.2 = 0.8 moles of B that have reacted. Since 0.2 of these went to S, 0.6 moles of R have been formed. From the reaction of A and B, 0.6 moles of A have reacted too. The starting amount of A was 1 mole, thus 1 - 0.6 = 0.4 moles of A remain. Summarizing, the composition when 0.6 mole of S is present will involve: 0.4 moles of A, 2.4 moles of B, 0.6 moles of R, and 0.6 moles of S, as the formation of additional 0.4 moles of S will consume an equal amount of R and B.

02

Determining Composition for Part (a)

To determine the final composition, we just subtract 0.4 moles from B and R from the composition when 0.6 mole of S is not yet formed and add 0.4 moles to S. Final composition: A = 0.4 moles, B = 2.4 - 0.4 = 2.0 moles, R = 0.6 - 0.4 = 0.2 moles, S = 0.6 + 0.4 = 1.0 mole.

03

Analysis of Part (b)

Given that upon mixing 1 mole of A with 1 mole of B, 0.5 mole of S is found, implies that 0.5 mole of B reacted to form S. Since all of A was added and we ended with 0.5 mole of S, this means that the remaining 0.5 moles of A must have formed 0.5 moles of R. This indicates that all B has been used up in the reactions. We can deduce that upon the formation of R, all of it is readily converted to S indicating a much faster second reaction. Therefore, we can say that the rate constant k2 is much larger compared to k1, or k2/k1 >> 1.

04

Analysis of Part (c)

Considering that 0.25 mole of S is found, it means that 0.25 mole of B has reacted to form S. Since 1 mole of A and B were initially present, and only a quarter of B has reacted, there might be some leftover A, B, or R. Given that S is only a quarter of the initial amount of B, this suggests that either the reaction is not efficient in converting R to S or it is not efficient in converting A to R, implying that k2/k1 may not be much greater than 1. However, without knowing how much R was initially formed, we cannot determine the exact ratio of k2 to k1.

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Key Concepts

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

Elementary Reactions

Elementary reactions are the most basic chemical reactions, where the reactants convert to products in a single step and with specific stoichiometry. These reactions are crucial in chemical reaction engineering because they provide a foundation for understanding complex reaction mechanisms.

For instance, in the given exercise, we encounter a sequence of two elementary reactions involving species A, B, R, and S with associated rate constants. Elementary reactions are often characterized by their molecularity, indicating the number of molecules participating in a single reaction event. In most cases, like those presented in the exercise, we assume these reactions proceed in one direction for simplicity, though in reality, many elementary reactions are reversible.

Reaction Kinetics

Reaction kinetics refers to the study of the speed or rate at which chemical reactions occur. It is concerned with understanding how different variables such as concentration, temperature, and presence of a catalyst affect the speed of a reaction. The rate of reaction is usually expressed in terms of rate constants and concentrations of the reactants.

Kinetic analysis helps chemists and engineers to design reactors and processes efficiently. For example, in part (b) and (c) of the exercise, the ratio of the rate constants, often denoted as k2/k1, provides insights into which reaction is faster and to what extent. Understanding these dynamics is crucial in predicting the outcome of chemical processes and designing the systems that control them.

Reaction Composition Analysis

Reaction composition analysis is a method used to determine the quantities of various chemical species present at different stages of a reaction. It is essential for monitoring the progress of the reaction and for verifying theoretical predictions.

In the exercise, the composition analysis is used to infer the concentrations of A, B, R, and S at various times given the reactions' stoichiometry and sequence. It involves balancing the moles of reactants and products, ensuring that the law of conservation of mass is upheld. This analysis is pivotal in the field of chemical engineering for process control, optimization, and safety.

Rate Constants

Rate constants are numerical values that represent the proportionality factor between the reaction rate and the concentrations of reactants. They are a key part of the rate laws that describe how quickly a reaction occurs under given conditions.

In the provided reactions, k1 and k2 are the rate constants for the first and second elementary reactions, respectively. The magnitude of these constants, often influenced by factors like temperature and catalysts, is indicative of the reaction's propensity to take place. By comparing k2/k1, as explored in parts (b) and (c) of the exercise, chemists can predict which steps of a reaction sequence are rate-determining, which is fundamental in the design of chemical reactors and the optimization of industrial processes.