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Chapter 13: Problem 42

To obtain the rate of the reaction $$3 \mathrm{I}^{-}(a q)+\mathrm{H}_{3} \mathrm{AsO}_{4}(a q)+2 \mathrm{H}^{+}(a q) \longrightarrow$$ you might follow the \(\mathrm{I}^{-}\) concentration or the \(\mathrm{I}_{3}^{-}\) concentration. How are the rates in terms of these species related?

Short Answer

Expert verified
The rate of \(\mathrm{I}^{-}\) depletion is three times that of \(\mathrm{I}_{3}^{-}\) formation.

Step by step solution

01

Understand the Reaction

In the given chemical reaction, the reactants \(3 \mathrm{I}^{-}(aq), \mathrm{H}_{3} \mathrm{AsO}_{4}(aq),\) and \(2 \mathrm{H}^{+}(aq)\) react to form an undetermined product. To find the rate in terms of \(\mathrm{I}^{-}\) and \(\mathrm{I}_{3}^{-}\), we focus on the stoichiometry of \(\mathrm{I}^{-}\) in the reaction.
02

Define the Rate of Reaction for \(\mathrm{I}^{-}\)

The rate of reaction can be defined by the change in concentration of any reactant or product per unit time. For \(\mathrm{I}^{-}\), this is expressed as:\[\text{Rate} = -\frac{1}{3} \frac{d[\mathrm{I}^{-}]}{dt}\]This equation shows that for every 3 moles of \(\mathrm{I}^{-}\) consumed, the reaction proceeds by one mole.
03

Relate \(\mathrm{I}_{3}^{-}\) Concentration Change

Assuming that triiodide ions \(\mathrm{I}_{3}^{-}\) are formed as a product and considering a simplified disproportionation reaction of \(\mathrm{I}^{-}\):\[3 \mathrm{I}^{-} \rightleftharpoons \mathrm{I}_{3}^{-}\]The rate of formation of \(\mathrm{I}_{3}^{-}\) can be related to the disappearance of \(\mathrm{I}^{-}\) as:\[\text{Rate} = \frac{d[\mathrm{I}_{3}^{-}]}{dt}\]Here, 1 mole of \(\mathrm{I}_{3}^{-}\) is formed for every 3 moles of \(\mathrm{I}^{-}\) consumed.
04

Relate the Rates

Relating the rates from \(\mathrm{I}^{-}\) consumption and \(\mathrm{I}_{3}^{-}\) formation, we see that:\[-\frac{1}{3} \frac{d[\mathrm{I}^{-}]}{dt} = \frac{d[\mathrm{I}_{3}^{-}]}{dt}\]This shows the rate of change of \(\mathrm{I}_{3}^{-}\) is exactly one-third the rate of change of \(\mathrm{I}^{-}\), but positive because it is being formed.

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Key Concepts

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

Chemical Kinetics
Chemical kinetics is a branch of chemistry that studies the speed or rate at which chemical reactions occur.
It involves understanding how different conditions such as temperature and concentration affect the rate of a reaction.
In chemical kinetics, we use the concept of a "reaction rate" to describe how quickly a reactant is consumed or a product is formed.
  • "Reaction rate" is usually measured as the change in concentration of a reactant or product over a specified period of time.
  • The rate can be affected by various factors such as temperature, pressure, and the presence of catalysts.
  • Different reactions have unique rates depending on these factors and their intrinsic properties.
Kinetics also helps us to understand the mechanism of a reaction, which is the step-by-step sequence of elementary reactions by which overall chemical change occurs. The reaction given involves iodide ions, which serve as a good example of how stoichiometry and kinetics interplay.
Reaction Stoichiometry
Reaction stoichiometry plays a crucial role in determining how the concentration of reactants and products change over time.
The stoichiometry of a reaction tells you the exact proportions in which reactants combine to form products.
  • In the provided reaction, three iodide ions (\(3 \mathrm{I}^- \)) react with other species.
  • This ratio is essential because it allows us to calculate how the concentrations of different species change over time.
  • Stoichiometric coefficients, like the 3 for \(\mathrm{I}^- \), are used to define the relationship between the rates of disappearance of reactants and appearance of products.
Understanding stoichiometry allows us to determine the relationship between rates of disappearance and appearance, which is essential for calculating reaction rates.
Rate of Formation
The rate of formation refers to the speed at which a product is generated in a chemical reaction.
It is often compared to the rate of disappearance of a reactant.
  • In the given reaction, we might be interested in how quickly \(\mathrm{I}_{3}^{-}\) is formed as iodide ions react.
  • The rate of formation can be expressed as \(\frac{d[\mathrm{I}_3^-]}{dt}\), indicating how the concentration changes with time.
  • The stoichiometry tells us that one mole of \(\mathrm{I}_{3}^{-}\) forms for every three moles of \(\mathrm{I}^-\) consumed.
The rate of formation is crucial for determining how fast a product accumulates and can significantly affect the characteristics of a chemical process.
Reactant Concentration
Reactant concentration is a central factor in determining the rate of a chemical reaction.
The concentration of reactants often influences the speed of the reaction, as a higher concentration typically leads to an increased reaction rate.
  • Initial concentrations of reactants are important in determining how the reaction proceeds over time.
  • In our example, the concentration of \(\mathrm{I}^-\) is central, as it is directly tied to the reaction's stoichiometry and overall rate.
  • The rate law of a reaction represents the relationship between the reaction rate and the concentrations of reactants.
By manipulating the concentration of reactants, chemists can control the rate at which a reaction occurs, making concentration a crucial variable in kinetic studies.

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Most popular questions from this chapter

In the presence of excess thiocyanate ion, \(\mathrm{SCN}^{-}\), the following reaction is first order in iron(III) ion, \(\mathrm{Fe}^{3+} ;\) the rate constant is \(1.27 / \mathrm{s}\). $$\mathrm{Fe}^{3+}(a q)+\mathrm{SCN}^{-}(a q) \longrightarrow \mathrm{Fe}(\mathrm{SCN})^{2+}(a q)$$ If \(90.0 \%\) reaction is required to obtain a noticeable color from the formation of the \(\mathrm{Fe}(\mathrm{SCN})^{2+}\) ion, how many seconds are required?

Benzene diazonium chloride, \(\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{NNCl}\), decomposes by a first-order rate law. $$\mathrm{C}_{6} \mathrm{H}_{5} \mathrm{NNCl} \longrightarrow \mathrm{C}_{6} \mathrm{H}_{5} \mathrm{Cl}+\mathrm{N}_{2}(g)$$ If the rate constant at \(20^{\circ} \mathrm{C}\) is \(4.3 \times 10^{-5} / \mathrm{s}\), how long will it take for \(75 \%\) of the compound to decompose?

The following data were collected for the reaction \(\mathrm{A}(g)+\mathrm{B}(g) \longrightarrow\) Products. \(\begin{array}{llll}\text { Experiment } & {[\mathbf{A}]_{0}(M)} & {[\mathrm{B}]_{0}(M)} & \text { Rate }(M / s) \\ 1 & 0.0100 & 0.100 & 1.0 \times 10^{-03} \\ 2 & 0.0300 & 0.100 & 3.0 \times 10^{-03} \\ 3 & 0.0300 & 0.300 & 2.7 \times 10^{-02}\end{array}\) a. Determine the rate law for this reaction. b. Calculate the rate constant. c. Calculate the rate when \([\mathrm{A}]=0.200 M\) and \([\mathrm{B}]=0.200 M\).

In the presence of excess thiocyanate ion, \(\mathrm{SCN}^{-}\), the following reaction is first order in iron(III) ion, \(\mathrm{Fe}^{3+}\); the rate constant is \(1.27 / \mathrm{s}\). $$\mathrm{Fe}^{3+}(a q)+\mathrm{SCN}^{-}(a q) \longrightarrow \mathrm{Fe}(\mathrm{SCN})^{2+}(a q)$$ What is the half-life in seconds? How many seconds would be required for the initial concentration of \(\mathrm{Fe}^{3+}\) to decrease to each of the following values: \(25.0 \%\) left, \(12.5 \%\) left, \(6.25 \%\) left, \(3.125 \%\) left? What is the relationship between these times and the half-life?

A reaction of the form \(a \mathrm{~A} \longrightarrow\) Products is second order with a rate constant of \(0.413 \mathrm{~L} /(\mathrm{mol} \cdot \mathrm{s}) .\) What is the half-life, in seconds, of the reaction if the initial concentration of \(\mathrm{A}\) is \(5.25 \times\) \(10^{-3} \mathrm{~mol} / \mathrm{L} ?\)
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