Question

\begin{aligned} &\text { Given the consecutive reaction A } \mathrm{k}_{1} \rightarrow \mathrm{B} \mathrm{k}_{2} \rightarrow \mathrm{C} \text { with } \mathrm{k}_{1}=\mathrm{k}_{2} \text { . }\\\ &\text { Draw a graph for the time variation of the concentrations of }\\\ &\mathrm{A}, \mathrm{B}, \text { and } \mathrm{C} \text { . } \end{aligned}

Short answer

In this consecutive reaction where \(k_1 = k_2\), the time variation of the concentrations of species A, B, and C can be described by the following expressions: - [A]: \([A] = [A]_0 e^{-k_1 t}\) - [B]: \([B] = [A]_0(\frac{1}{2} - \frac{1}{2} e^{-2k_1 t})\) - [C]: \([C] = [A]_0 (1 - e^{-k_1 t} - \frac{1}{2} + \frac{1}{2} e^{-2k_1 t})\) The graph for the time variation of these concentrations shows that the concentration of A decreases exponentially with time, while B and C initially increase and reach a maximum before decreasing with time. The concentration of C is always lower than B.

Step-by-step solution

Write down the differential rate expressions

From the given consecutive reaction, we can express the rate of change in the concentrations of A, B, and C with respect to time in the following way: For A: \(\frac{d[A]}{dt} = -k_1[A]\) For B: \(\frac{d[B]}{dt} = k_1[A] - k_2[B]\) For C: \(\frac{d[C]}{dt} = k_2[B]\)

Integrate the rate expressions

In order to find the expressions for the concentrations of A, B, and C, we need to integrate these rate expressions with respect to time. Integrated expression for A: \[ [A] = [A]_0 e^{-k_1 t} \] Integrated expression for B and C: In this case, we will first integrate the differential equation for B with respect to time: \[ \int_0^t \frac{d[B]}{dt} dt = \int_0^t (k_1[A] - k_2[B]) dt \] Since \(k_1 = k_2\), we can simplify this equation further: \[ \int_0^t \frac{d[B]}{dt} dt = \int_0^t ([A] - [B]) dt \] Using the integrated expression for A, we will replace [A] inside the integral: \[ \int_0^t \frac{d[B]}{dt} dt = \int_0^t ([A]_0 e^{-k_1 \tau} - [B]) d\tau \], where \(\tau\) is a dummy variable for time Integrating this equation will give us the concentration of B as a function of time: \[ [B](t) = [A]_0(\frac{1}{2} - \frac{1}{2} e^{-2k_1 t}) \] To find the concentration of C, we can take advantage of the conservation of mass: \[ [C](t) = [A]_0 - [A](t) - [B](t) = [A]_0 (1 - e^{-k_1 t} - \frac{1}{2} + \frac{1}{2} e^{-2k_1 t}) \]

Plot the concentration expressions as a function of time

To draw a graph of the time variation of the concentrations of A, B, and C, we will plot the expressions we obtained in step 2: - [A] vs time: \([A] = [A]_0 e^{-k_1 t}\) - [B] vs time: \([B] = [A]_0(\frac{1}{2} - \frac{1}{2} e^{-2k_1 t})\) - [C] vs time: \([C] = [A]_0 (1 - e^{-k_1 t} - \frac{1}{2} + \frac{1}{2} e^{-2k_1 t})\) These plots will show the concentration of species A decreasing as a function of time (exponential decay), while the concentrations of species B and C will initially increase and then reach a maximum before decreasing as a function of time. The concentration of species C will always be lower than that of species B due to the conservation of mass.

Key concepts

Rate of Reaction

Understanding the rate of a reaction is fundamental in chemical kinetics. It refers to the speed at which reactants are transformed into products. For instance, when considering the reaction \( A \rightarrow B \rightarrow C \), the rate at which \( A \) disappears is described by the rate law \( -\frac{d[A]}{dt} = k_1[A] \), with \( k_1 \) symbolizing the rate constant for the conversion of \( A \) to \( B \).

The negative sign indicates that the concentration of \( A \) decreases over time. The rate of reaction is directly proportional to the concentration of the reactants, thus, as \( A \) is consumed, the rate decreases. This simple relationship is characteristic of a first-order reaction, where the rate depends linearly on the concentration of one reactant.

Chemical Kinetics

Chemical kinetics is the branch of chemistry that studies the rates of chemical processes and the factors affecting these rates. It involves analyzing how different conditions, such as concentration, temperature, and presence of a catalyst, influence the speed of reactions. The study of kinetics helps chemists to deduce the reaction mechanism, that is, the step-by-step sequence of elementary steps that make up an overall reaction.

In our example of the consecutive reactions \( A \rightarrow B \rightarrow C \), we see that the kinetic analysis outputs equations that allow us to predict the concentration of each species at any given time. Through kinetics, we can also establish whether a reaction follows a simple rate law, like a first-order reaction, or a more complex one.

Integration of Rate Laws

To determine the concentration of the reactants and products as a function of time, we need to integrate the rate laws. Integration is the process of finding the antiderivative or the 'original function' when given its derivative. It allows us to transition from the rate of reaction to actual concentrations over time.

For first-order reactions, the integration is straightforward. The integrated form of the first-order rate law for \( A \rightarrow B \) is \( [A] = [A]_0 e^{-k_1 t} \) where \( [A]_0 \) is the initial concentration and \( e \) is the base of the natural logarithm. This integrated law demonstrates the exponential decay of species \( A \) with time. For more complex reactions, such as the consecutive reactions involving species \( B \) and \( C \), integration might require additional steps and considerations, such as assuming steady-state conditions or integrating over a dummy variable.

Concentration-Time Graph

The concentration-time graph portrays how the concentrations of reactants and products change over time during a chemical reaction. In the case of the consecutive reactions where \( A \rightarrow B \rightarrow C \), the graph is constructed by plotting the equations derived from integrating the rate laws against time.

For species \( A \), the concentration decreases exponentially, which is depicted as a downward curving line on the graph. For \( B \), the concentration initially rises as \( A \) is converted, then peaks, and finally decreases once \( B \) starts to get converted into \( C \). The graph for \( B \) reveals a rise and fall pattern, signifying its transient nature as an intermediate. Lastly, the concentration of \( C \) increases as time progresses, eventually leveling off as the reaction proceeds to completion. These graphical depictions provide an intuitive understanding of the dynamic changes occurring throughout a reaction sequence.