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A first-order reaction, \(\mathrm{A} \longrightarrow\) products, has a halflife of \(75 \mathrm{s},\) from which we can draw two conclusions. Which of the following are those two (a) the reaction goes to completion in 150 s; (b) the quantity of \(A\) remaining after 150 s is half of what remains after 75 s; (c) the same quantity of A is consumed for every 75 s of the reaction; (d) one- quarter of the original quantity of A is consumed in the first 37.5 s of the reaction; (e) twice as much A is consumed in 75 s when the initial amount of \(\mathrm{A}\) is doubled; (f) the amount of \(\mathrm{A}\) consumed in 150 s is twice as much as is consumed in 75 s.

Short Answer

Expert verified
Based on the analysis of the statements, the two correct conclusions that we can draw from the information on the reaction's half-life are: (b) the quantity of \(A\) remaining after 150 seconds is half of what remains after 75 seconds and (e) twice as much \(A\) is consumed in 75 seconds when the initial amount of \(A\) is doubled.

Step by step solution

01

Analyze Statement (a)

Statement (a) suggests the reaction completes in 150 seconds, which is twice the half-life. This isn't correct for a first-order reaction. After one half-life (75 seconds), half of the original quantity of A remains. After another half-life (another 75 seconds), only half of that remaining amount is consumed, not the entire remaining amount. Therefore, statement (a) is incorrect.
02

Analyze Statement (b)

Statement (b) suggests that the quantity of A remaining after 150 seconds is half of what remains after 75 seconds. This is correct. After 75 seconds (one half-life), half of the original amount of A remains. After another half-life (another 75 seconds), only half of the remaining amount is left. Therefore, statement (b) is correct.
03

Analyze Statement (c)

Statement (c) suggests that the same quantity of A is consumed for every 75 seconds of the reaction. This is not correct. While it is true that the reaction rate is proportional to the amount of reactant, in a first-order reaction, the actual quantity of A consumed decreases with each successive half-life. Therefore, statement (c) is incorrect.
04

Analyze Statement (d)

Statement (d) is claiming that one quarter of A is consumed in the first 37.5 seconds. This isn’t true for a first-order reaction. The half-life is defined as the time it takes for half of the reactant to be consumed, not a quarter. Therefore, statement (d) is incorrect.
05

Analyze Statement (e)

Statement (e) is asserting that if the initial amount of A is doubled, twice as much A is consumed in 75 seconds. This is true for a first-order reaction. Since the reaction rate is proportional to the concentration of the reactant, if we start with twice as much reactant, we'll consume twice as much in the same amount of time. Hence, statement (e) is correct.
06

Analyze Statement (f)

Statement (f) suggests that the amount of A consumed in 150 seconds is twice as much as is consumed in 75 seconds. It's the case with the first-order reaction that half the reactant remains after one half-life (75 seconds in this case), and half of the remaining reactant is consumed after the next half-life. Actually, the amount consumed in the first 75 seconds is more than that consumed in the next 75 seconds, so statement (f) is incorrect.

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Key Concepts

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

Reaction Kinetics
Reaction kinetics is the branch of chemistry focused on understanding the rates of chemical reactions and how different conditions affect them. It provides insights into how quickly a reaction proceeds and what factors influence this speed.

For a better understanding, consider a reaction where substance A converts to products. This is a perfect example of a first-order reaction. Here, the rate of reaction depends directly on the concentration of A. As the concentration of A changes, so does the speed of the reaction.
  • In first-order reactions, the rate is proportional to the concentration of a single reactant.
  • The rate constant ( k ext{ extbackslash}) can be determined experimentally and helps in quantifying how quickly a reaction approaches completion.
  • Every reaction has a unique rate constant depending on conditions like temperature and pressure.
When analyzing reaction kinetics, it’s crucial to pay attention to how the concentration changes over time. This helps chemists control reactions efficiently in industrial processes.
By understanding the kinetics, scientists can predict how long a reaction will take and optimize conditions to make reactions faster, safer, and more efficient.
Half-Life
Half-life is an essential concept in both chemistry and physics. It defines the time required for half of a reactant to be consumed in a reaction. For first-order reactions, the half-life is a constant value, unaffected by the initial concentration.

Take the reaction where A is being converted to products. If we know the half-life ( 75 ext{ s} extbackslash) of this first-order reaction, we can deduce the amount of A present at any given time.
  • After one half-life, half of A is left, meaning if you start with a certain amount of A, only 50% remains at the end of the half-life.
  • After two half-lives, only 25% of the original A remains, as A keeps being halved in equal time intervals.
Understanding half-life is key to predicting how quickly a reaction approaches equilibrium or completes.
In practical applications, this knowledge helps in fields such as pharmacology and radioactive decay, where knowing the duration of a substance's effectiveness or safety is critical.
Chemical Reactions
Chemical reactions are processes in which reactants are transformed into products. This transformation involves the breaking and forming of bonds and is governed by reaction kinetics and thermodynamics.

Each reaction has its unique characteristics, such as rate, energy changes, and mechanisms, which are studied intensively in chemistry. First-order reactions are particularly interesting because they have predictable rates and behaviors based on the concentration of the reactants.
  • One key feature of a first-order reaction is that the rate decreases over time as the concentration of the reactant decreases.
  • This means that in a reaction where substance A turns into a product, the amount of A diminishing follows an exponential decay curve.
  • The simplicity of first-order reactions allows for easy mathematical modeling using logarithmic functions.
To sum up, chemical reactions are fundamental processes that scientists must understand to innovate and improve the numerous applications of chemistry, from medicine to industry. By mastering the principles of reactions like those of the first-order, chemists can predict and control the outcomes of complex processes effectively.

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Most popular questions from this chapter

The first-order reaction \(A \longrightarrow\) products has a halflife, \(t_{1 / 2},\) of 46.2 min at \(25^{\circ} \mathrm{C}\) and \(2.6 \mathrm{min}\) at \(102^{\circ} \mathrm{C}.\) (a) Calculate the activation energy of this reaction. (b) At what temperature would the half-life be 10.0 min?

We have used the terms order of a reaction and molecularity of an elementary process (that is, unimolecular, bimolecular). What is the relationship, if any, between these two terms?

One example of a zero-order reaction is the decomposition of ammonia on a hot platinum wire, \(2 \mathrm{NH}_{3}(\mathrm{g}) \longrightarrow \mathrm{N}_{2}(\mathrm{g})+3 \mathrm{H}_{2}(\mathrm{g}) .\) If the concentration of ammonia is doubled, the rate of the reaction will (a) be zero; (b) double; (c) remain the same; (d) exponentially increase.

A reaction is \(50 \%\) complete in 30.0 min. How long after its start will the reaction be \(75 \%\) complete if it is (a) first order; (b) zero order?

The object is to study the kinetics of the reaction between peroxodisulfate and iodide ions. $$\begin{aligned} &\text { (a) } \mathrm{S}_{2} \mathrm{O}_{8}^{2-}(\mathrm{aq})+3 \mathrm{I}^{-}(\mathrm{aq}) \longrightarrow 2 \mathrm{SO}_{4}^{2-}(\mathrm{aq})+\mathrm{I}_{3}^{-}(\mathrm{aq}) \end{aligned}$$ The \(I_{3}^{-}\) formed in reaction (a) is actually a complex of iodine, \(\mathrm{I}_{2},\) and iodide ion, \(\mathrm{I}^{-}\). Thiosulfate ion, \(\mathrm{S}_{2} \mathrm{O}_{3}^{2-}\) also present in the reaction mixture, reacts with \(\mathrm{I}_{3}^{-}\) just as fast as it is formed. $$\text { (b) } 2 \mathrm{S}_{2} \mathrm{O}_{3}^{2-}(\mathrm{aq})+\mathrm{I}_{3}^{-}(\mathrm{aq}) \longrightarrow \mathrm{S}_{4} \mathrm{O}_{6}^{2-}+3 \mathrm{I}^{-}(\mathrm{aq})$$ When all of the thiosulfate ion present initially has been consumed by reaction (b), a third reaction occurs between \(\mathrm{I}_{3}^{-}(\mathrm{aq})\) and starch, which is also present in the reaction mixture. $$\text { (c) } \mathrm{I}_{3}^{-}(\mathrm{aq})+\operatorname{starch} \longrightarrow \text { blue complex }$$ The rate of reaction (a) is inversely related to the time required for the blue color of the starch-iodine complex to appear. That is, the faster reaction (a) proceeds, the more quickly the thiosulfate ion is consumed in reaction (b), and the sooner the blue color appears in reaction (c). One of the photographs shows the initial colorless solution and an electronic timer set at \(t=0 ;\) the other photograph shows the very first appearance of the blue complex (after 49.89 s). Tables I and II list some actual student data obtained in this study. $$\begin{array}{l} \hline\text { TABLE I } \\ \text { Reaction conditions at } 24^{\circ} \mathrm{C}: 25.0 \mathrm{mL} \text { of the } \\ \left(\mathrm{NH}_{4}\right)_{2} \mathrm{S}_{2} \mathrm{O}_{8}(\text { aq) listed, } 25.0 \mathrm{mL} \text { of the } \mathrm{KI}(\mathrm{aq}) \\ \text { listed, } 10.0 \mathrm{mL} \text { of } 0.010 \mathrm{M} \mathrm{Na}_{2} \mathrm{S}_{2} \mathrm{O}_{3}(\mathrm{aq}), \text { and } 5.0 \mathrm{mL} \\ \text { starch solution are mixed. The time is that of the } \\ \text { first appearance of the starch-iodine complex. } \\ \hline & \text { Initial Concentrations, } \mathrm{M} \\ \hline \text { Experiment } & \left(\mathrm{NH}_{4}\right)_{2} \mathrm{S}_{2} \mathrm{O}_{8} & \mathrm{KI} & \text { Time, s } \\ \hline 1 & 0.20 & 0.20 & 21 \\ 2 & 0.10 & 0.20 & 42 \\ 3 & 0.050 & 0.20 & 81 \\ 4 & 0.20 & 0.10 & 42 \\ 5 & 0.20 & 0.050 & 79 \\ \hline \end{array}$$ $$\begin{array}{l} \hline \text { TABLE II } \\ \text { Reaction conditions: those listed in Table I for } \\ \text { Experiment } 4, \text { but at the temperatures listed. } \\ \hline \text { Experiment } & \text { Temperature, }^{\circ} \mathrm{C} & \text { Time, } \mathrm{s} \\ \hline 6 & 3 & 189 \\ 7 & 13 & 88 \\ 8 & 24 & 42 \\ 9 & 33 & 21 \\ \hline \end{array}$$ (a) Use the data in Table I to establish the order of reaction (a) with respect to \(\mathrm{S}_{2} \mathrm{O}_{8}^{2-}\) and to I \(^{-}\). What is the overall reaction order? [Hint: How are the times required for the blue complex to appear related to the actual rates of reaction? (b) Calculate the initial rate of reaction in Experiment 1 expressed in \(\mathrm{M} \mathrm{s}^{-1} .\) [Hint: You must take into account the dilution that occurs when the various solutions are mixed, as well as the reaction stoichiometry indicated by equations \((a),(b), \text { and }(c) .]\) (c) Calculate the value of the rate constant, \(k,\) based on experiments 1 and 2 (d) Calculate the rate constant, \(k\), for the four different temperatures in Table II. (e) Determine the activation energy, \(E_{\mathrm{a}}\), of the peroxodisulfate- iodide ion reaction. (f) The following mechanism has been proposed for reaction (a). The first step is slow, and the others are fast. $$\begin{array}{c} \mathrm{I}^{-}+\mathrm{S}_{2} \mathrm{O}_{8}^{2-} \longrightarrow \mathrm{IS}_{2} \mathrm{O}_{8}^{3-} \\ \mathrm{IS}_{2} \mathrm{O}_{8}^{3-} \longrightarrow 2 \mathrm{SO}_{4}^{2-}+\mathrm{I}^{+} \\ \mathrm{I}^{+}+\mathrm{I}^{-} \longrightarrow \mathrm{I}_{2} \\ \mathrm{I}_{2}+\mathrm{I}^{-} \longrightarrow \mathrm{I}_{3}^{-} \end{array}$$ Show that this mechanism is consistent with both the stoichiometry and the rate law of reaction (a). Explain why it is reasonable to expect the first step in the mechanism to be slower than the others.

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