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Chapter 14: Problem 4

(a) In the reaction \(3 \mathrm{ClO}^{-}\)(aq) \(\longrightarrow 2 \mathrm{Cl}^{-}\)(aq) \(+\mathrm{ClO}_{3}{ }^{-}\)(aq), the rate of formation of \(\mathrm{Cl}^{-}\)is \(3.6 \mathrm{~mol} \cdot \mathrm{L}^{-1} \cdot \mathrm{min}^{-1}\). What is the rate of reaction of \(\mathrm{ClO}^{-}\)? (b) What is the unique rate of the reaction?

Short Answer

Expert verified
The rate of reaction of ClO- is 5.4 mol L-1 min-1, and the unique rate of the reaction is 5.4 mol L-1 min-1.

Step by step solution

01

Understand the rate of reaction

The rate of reaction is expressed in terms of the change in concentration of a reactant or product over a specific time interval. For a given balanced chemical equation, the rates of consumption of reactants and the rates of formation of products are related to the stoichiometry of the reaction.
02

Relate the rate of formation of Cl- to the rate of reaction of ClO-

The balanced equation given is: 3 ClO- -> 2 Cl- + ClO3-. According to the stoichiometry of the reaction, the ratio of the rate of reaction of ClO- to Cl- is 3:2. Using this stoichiometric relationship, the rate of reaction of ClO- can be calculated based on the provided rate of formation of Cl-.
03

Calculate the rate of reaction of ClO-

The rate of formation of Cl- is 3.6 mol L-1 min-1. Using the stoichiometric coefficients, the rate of reaction of ClO- is calculated by multiplying the rate of formation of Cl- by 3/2. Rate of reaction of ClO- = (3.6 mol L-1 min-1) * (3/2)
04

Simplify the calculation

Perform the multiplication to find the rate of reaction of ClO-: Rate of reaction of ClO- = 5.4 mol L-1 min-1.
05

Determine the unique rate of reaction

The unique rate of the reaction is the rate at which the reaction proceeds, expressed in terms of the change in concentration of one of the species per unit time. Here, it is found from the rate of disappearance of ClO-. Since it disappears, the unique rate will be negative. Unique rate = - Rate of reaction of ClO- = - 5.4 mol L-1 min-1. Therefore, the unique rate is 5.4 mol L-1 min-1.

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Key Concepts

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

Rate of Formation
Understanding the rate of formation is crucial when studying chemical reactions. It is the speed at which a chemical species is produced during a reaction. For instance, in the given reaction \(3 \mathrm{ClO}^{-}(aq) \longrightarrow 2 \mathrm{Cl}^{-}(aq) + \mathrm{ClO}_{3}^{-}(aq)\), if we say that the rate of formation of \( \mathrm{Cl}^{-} \) is \(3.6 \mathrm{~mol} \cdot \mathrm{L}^{-1} \cdot \mathrm{min}^{-1}\), we are indicating how fast chlorides are being produced.

In a classroom scenario, visualizing the process might be as simple as imagining a factory line where chloride ions are the final product being packaged up at a consistent rate, giving you a tangible sense of the pace of product creation.

Remember, the rate of formation relates directly to the concentration changes of the product per unit time. This should be determined carefully, as it can affect how you interpret the overall rate of the reaction, which we'll discuss in the next sections.
Stoichiometry of Reaction
Stoichiometry is akin to a recipe for a chemical reaction, with precise ingredients (reactants) and instructions to create the final dish (products). It tells us the proportional relationship between reactants and products. In our exercise, the stoichiometry is described by the balanced chemical equation \(3 \mathrm{ClO}^{-} \longrightarrow 2 \mathrm{Cl}^{-} + \mathrm{ClO}_{3}^{-}\), implying you need three parts of \( \mathrm{ClO}^{-}\) to produce two parts of \(\mathrm{Cl}^{-}\).

To compare rates between different species in a reaction based on stoichiometry, we establish a ratio using the coefficients from the balanced equation. This ratio allows us to calculate one component's rate from another's, ensuring we maintain the 'recipe' accurately throughout the reaction.

Application of Stoichiometry in Rate Calculation

For instance, if you know the rate of formation of one product, you can determine the rate of reaction of a reactant by using the stoichiometric coefficients as a conversion factor. This fundamental principle reinforces the idea that the amounts of substances involved in chemical reactions are always proportional to their stoichiometric coefficients.
Unique Rate of Reaction
The unique rate of reaction is a single value that represents the speed at which a chemical reaction occurs, and it's crucially important because it brings clarity to the seemingly complex dance of reactants and products. It portrays the reaction speed irrespective of individual species, providing a single measure whether we are considering the disappearance of reactants or the appearance of products.

Consider our exercise's reaction: when quantifying the unique rate, we can use the rate of disappearance of a reactant since this will be consistent with the overall progress of the reaction. The negative sign in \( - 5.4 \mathrm{mol} \cdot \mathrm{L}^{-1} \cdot \mathrm{min}^{-1} \) signifies the reactant's consumption, while the magnitude gives the rate at which the reaction moves forward.

Think of the unique rate as the universal speedometer for the reaction, taking into account all parts of the chemical equation. Whether you are a student or a researcher, the unique rate is pivotal in developing a comprehensive understanding of chemical kinetics, helping to predict how long a reaction will take under certain conditions.

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Most popular questions from this chapter

The pre-equilibrium and the steady-state approximations are two different approaches to deriving a rate law from a proposed mechanism. For the following mechanism, determine the rate law (a) by the pre-equilibrium approximation and (b) by the steady-state approximation. (c) Under what conditions do the two methods give the same answer? (d) What will the rate laws become at high concentrations of \(\mathrm{Br}^{-}\)? $$ \begin{aligned} &\mathrm{CH}_{3} \mathrm{OH}+\mathrm{H}^{+} \rightleftharpoons \mathrm{CH}_{3} \mathrm{OH}_{2}^{+} \text {(fast equilibrium) } \\ &\mathrm{CH}_{3} \mathrm{OH}_{2}^{+}+\mathrm{Br}^{-} \longrightarrow \mathrm{CH}_{3} \mathrm{Br}+\mathrm{H}_{2} \mathrm{O} \text { (slow) } \end{aligned} $$

The decomposition of gaseous hydrogen iodide, \(2 \mathrm{HI}(\mathrm{g}) \rightarrow \mathrm{H}_{2}(\mathrm{~g})+\mathrm{I}_{2}(\mathrm{~g})\), gives the data shown here for \(700 . \mathrm{K}\). $$ \begin{array}{lcccccc} \text { Time }(\mathrm{s}) & 0 . & 1000 . & 2000 . & 3000 . & 4000 . & 5000 . \\\ {[\mathrm{HI}]\left(\mathrm{mmol} \cdot \mathrm{L}^{-1}\right)} & 10.0 & 4.4 & 2.8 & 2.1 & 1.6 & 1.3 \end{array} $$ (a) Use a graphing calculator or standard graphing software, such as that on the Web site for this book, to plot the concentration of HI as a function of time. (b) Estimate the rate of decomposition of HI at each time. (c) Plot the concentrations of \(\mathrm{H}_{2}\) and \(\mathrm{I}_{2}\) as a function of time on the same graph.

In the brewing of beer, ethanal, which smells like green apples, is an intermediate in the formation of ethanol. Ethanal decomposes in the following first-order reaction: \(\mathrm{CH}_{3} \mathrm{CHO}(\mathrm{g}) \longrightarrow \mathrm{CH}_{4}(\mathrm{~g})+\) \(\mathrm{CO}(\mathrm{g})\). At an elevated temperature the rate constant for the decomposition is \(1.5 \times 10^{-3} \mathrm{~s}^{-1}\). What concentration of ethanal, which had an initial concentration of \(0.120 \mathrm{~mol} \cdot \mathrm{L}^{-1}\), remains \(20.0 \mathrm{~min}\) after the start of its decomposition at this temperature?

In the reaction \(\mathrm{CH}_{3} \mathrm{Br}(\mathrm{aq})+\mathrm{OH}^{-}(\mathrm{aq}) \rightarrow \mathrm{CH}_{3} \mathrm{OH}(\mathrm{aq})+\) \(\mathrm{Br}^{-}(\mathrm{aq})\), when the \(\mathrm{OH}^{-}\)concentration alone was doubled, the rate doubled; when the \(\mathrm{CH}_{3} \mathrm{Br}\) concentration alone was increased by a factor of \(1.2\), the rate increased by a factor of \(1.2\). Write the rate law for the reaction.

Determine the rate constant for each of the following firstorder reactions, in each case expressed for the rate of loss of \(A\) : (a) \(\mathrm{A} \longrightarrow \mathrm{B}\), given that the concentration of \(\mathrm{A}\) decreases to one-half its initial value in \(1000 . \mathrm{s} ;\) (b) \(\mathrm{A} \longrightarrow \mathrm{B}\), given that the concentration of A decreases from \(0.67 \mathrm{~mol} \cdot \mathrm{L}^{-1}\) to \(0.53 \mathrm{~mol} \cdot \mathrm{L}^{-1}\) in \(25 \mathrm{~s}\); (c) \(2 \mathrm{~A} \longrightarrow \mathrm{B}+\mathrm{C}\), given that \([\mathrm{A}]_{0}=0.153 \mathrm{~mol} \cdot \mathrm{L}^{-1}\) and that after \(115 \mathrm{~s}\) the concentration of \(B\) rises to \(0.034 \mathrm{~mol} \cdot \mathrm{L}^{-1}\).
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