Problem 21
Question
21\. Chemical Reaction Chemical A spontaneously decomposes into chemicals \(\mathrm{B}\) and \(\mathrm{C}: \mathrm{A} \rightarrow \mathrm{B}+\mathrm{C} .\) The rate constant for this reaction is \(k\). (a) Explain why, if the amounts of \(\mathrm{A}, \mathrm{B}\), and \(\mathrm{C}\) present are denoted by \([\mathrm{A}],[\mathrm{B}]\), and \([\mathrm{C}]\) respectively, then \([\mathrm{B}]\) obeys the differential equation $$ \frac{d[\mathrm{~B}]}{d t}=k[\mathrm{~A}] $$ (b) What differential equations represent the rate of change of \([\mathrm{C}]\) and \([\mathrm{A}] ?\) (c) Suppose that at time \(t=0\) the initial amount of \(\mathrm{A}\) present is \(a\), and that there is no \(\mathrm{B}\) or \(\mathrm{C}\) present. Explain why $$ [\mathrm{A}]=a-[\mathrm{B}] $$
Step-by-Step Solution
Verified Answer
The differential equation for [B] is \( \frac{d[B]}{dt} = k[A] \), for [C] it is \( \frac{d[C]}{dt} = k[A] \), and for [A], \( \frac{d[A]}{dt} = -k[A] \). Initially, \([A] = a - [B]\) due to stoichiometry.
1Step 1: Understand the chemical reaction
The chemical reaction given is \( A \rightarrow B + C \). It is a typical first-order reaction, where the rate of decomposition of chemical \( A \) is directly proportional to its concentration \([A]\). This means the change in concentration of \( A \) with respect to time is governed by the rate constant \( k \).
2Step 2: Explain the differential equation for [B]
Since the reaction \( A \rightarrow B + C \) is first-order with respect to \( A \), the rate at which \( B \) is formed is directly proportional to the concentration of \( A \). This gives us the differential equation for \( [B] \): \( \frac{d[B]}{dt} = k[A] \). This equation states that the rate of formation of \( B \) depends on the initial concentration of \( A \) multiplied by the rate constant \( k \).
3Step 3: Find the differential equation for [C]
The formation of \( C \) happens simultaneously with \( B \), and in the same stoichiometric proportion. Therefore, \( [C] \) and \( [B] \) change at the same rate. Thus, \( \frac{d[C]}{dt} = k[A] \), because \( C \) is produced at the same rate as \( B \).
4Step 4: Find the differential equation for [A]
The rate of decrease of \([A]\) is due to its conversion to \( B \) and \( C \). Therefore, the differential equation for \( [A] \) is \( \frac{d[A]}{dt} = -k[A] \). The negative sign indicates that \([A]\) is decreasing over time, proportional to its concentration and the rate constant \( k \).
5Step 5: Initial conditions and relation between [A] and [B]
At time \( t = 0 \), the concentration \( [A] = a \) and \( [B] = [C] = 0 \). As the reaction proceeds, \( B \) and \( C \) form with concentration \([B] = [C]\). From stoichiometry, the decrease in \([A]\) exactly equals the increase in \([B]+[C]\). Since \([B]= [C]\), we get \([A] = a - [B]\), capturing the conversion from \( A \) to products.
Key Concepts
Differential EquationsFirst-order ReactionsRate Constant
Differential Equations
In chemical kinetics, differential equations play a crucial role in understanding how the concentration of substances change over time. This concept is especially important when dealing with reactions like the one given: \( A \rightarrow B + C \).
The core idea is to express the variation of concentrations with respect to time using a mathematical expression. For example, the rate of formation of \( B \) due to the decomposition of \( A \) is represented by the differential equation \( \frac{d[B]}{dt} = k[A] \). Here, \( \frac{d}{dt} \) signifies the change over time, while \( k \) is the rate constant. Such equations help us understand how quickly a reactant disappears or a product forms.
Want to know more? These equations can be solved using calculus, which allows us to predict concentrations at any point in time. Differential equations are powerful tools, especially when analyzing reactions that are not straightforward.
The core idea is to express the variation of concentrations with respect to time using a mathematical expression. For example, the rate of formation of \( B \) due to the decomposition of \( A \) is represented by the differential equation \( \frac{d[B]}{dt} = k[A] \). Here, \( \frac{d}{dt} \) signifies the change over time, while \( k \) is the rate constant. Such equations help us understand how quickly a reactant disappears or a product forms.
Want to know more? These equations can be solved using calculus, which allows us to predict concentrations at any point in time. Differential equations are powerful tools, especially when analyzing reactions that are not straightforward.
First-order Reactions
First-order reactions are a fundamental concept in chemical kinetics. In these reactions, the rate of reaction depends solely on the concentration of a single reactant. This means that the change in concentration of the reactant over time is directly proportional to its initial concentration.
In our example, the reaction \( A \rightarrow B + C \) is classified as a first-order reaction. The decomposition rate of \( A \) is expressed through the equation \( \frac{d[A]}{dt} = -k[A] \), where the negative sign indicates a decrease in \( A \). This is because \( A \) is being used up as it transforms into \( B \) and \( C \).
First-order kinetics are prevalent in many natural and industrial processes. They are easy to model mathematically, which makes them highly relevant for predictive analysis in chemistry and other scientific fields. The exponential decay model is one such application that is crucial for calculating how quickly reactions proceed.
In our example, the reaction \( A \rightarrow B + C \) is classified as a first-order reaction. The decomposition rate of \( A \) is expressed through the equation \( \frac{d[A]}{dt} = -k[A] \), where the negative sign indicates a decrease in \( A \). This is because \( A \) is being used up as it transforms into \( B \) and \( C \).
First-order kinetics are prevalent in many natural and industrial processes. They are easy to model mathematically, which makes them highly relevant for predictive analysis in chemistry and other scientific fields. The exponential decay model is one such application that is crucial for calculating how quickly reactions proceed.
Rate Constant
The rate constant, often denoted as \( k \), is a pivotal element in the study of chemical kinetics. It provides the link between reactant concentration and the rate at which products are formed. For our reaction \( A \rightarrow B + C \), \( k \) determines how quickly \( A \) converts into \( B \) and \( C \).
The value of the rate constant is dependent on factors such as temperature and the presence of catalysts. Higher values of \( k \) indicate a faster reaction, while lower values suggest a slower one. In-depth understanding of \( k \) is essential because it affects reaction speed and efficiency.
In first-order reactions, the units of \( k \) are often \( s^{-1} \), indicating its reliance on time. Recognizing the role of \( k \) not only aids in predicting reaction behavior but also in controlling conditions to optimize reactions, such as through temperature adjustments or catalyst addition.
The value of the rate constant is dependent on factors such as temperature and the presence of catalysts. Higher values of \( k \) indicate a faster reaction, while lower values suggest a slower one. In-depth understanding of \( k \) is essential because it affects reaction speed and efficiency.
In first-order reactions, the units of \( k \) are often \( s^{-1} \), indicating its reliance on time. Recognizing the role of \( k \) not only aids in predicting reaction behavior but also in controlling conditions to optimize reactions, such as through temperature adjustments or catalyst addition.
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