Question

The rate equation for an autocatalytic reaction \(\mathrm{A}+\mathrm{R} \stackrel{k}{\longrightarrow} \mathrm{R}+\mathrm{R}\) is \(r_{\mathrm{A}}=-\frac{\mathrm{d} C_{\mathrm{A}}}{\mathrm{d} t}=k C_{\mathrm{A}} C_{\mathrm{R}}\) The rate of disappearance of reactant \(\mathrm{A}\) is maximum when (a) \(C_{\mathrm{A}}=2 C_{\mathrm{R}}\) (b) \(C_{\mathrm{A}}=C_{\mathrm{R}}\) (c) \(C_{\mathrm{A}}=C_{\mathrm{R}} / 2\) (d) \(C_{\mathrm{A}}=\left(C_{\mathrm{R}}\right)^{1 / 2}\)

Short answer

The rate of disappearance of reactant A is maximum when \(C_{A}=2 C_{R}\), which corresponds to option (a).

Step-by-step solution

Identify Maximum Rate Condition

The maximum rate of disappearance of reactant A will occur when the product of the concentrations, \(C_A C_R\), is maximized because the rate \(r_A\) is directly proportional to this product. This becomes an optimization problem, where the maximum value of the product of \(C_A\) and \(C_R\) should be found.

Set Up Equation for Product of Concentrations

Let's define the product of the concentrations as \(P = C_A C_R\). Our goal is to find when \(P\) is maximized, which happens when its derivative with respect to one of the variables is zero, assuming the concentration of A and R are related due to the stoichiometry of the reaction.

Use Stoichiometry to Relate Concentrations

Since R is produced from A, their concentrations are related. Without additional information, we can't determine the exact relation. However, we can use each option to check if it maximizes the product.

Evaluate Options for Maximum Product

We check each given option by substituting it into the product \(P\):\(P = C_A C_R\):\(a) P_1 = (2C_R)(C_R) = 2C_R^2\)\(b) P_2 = (C_R)(C_R) = C_R^2\)\(c) P_3 = (C_R/2)(C_R) = C_R^2/2\)\(d) P_4 = (\rCR^{1/2})(C_R) = C_R^{3/2}\)We then analyze which expression has the potential to be largest considering \(C_R\) is a positive value.

Determine the Correct Option

Comparing the expressions for \(P\), we see that option (a) gives us \(2C_R^2\), which is twice as large as that in option (b) for any positive value of \(C_R\) and significantly larger than options (c) and (d). Hence, the product of concentrations is maximum when \(C_A = 2 C_R\).

Key concepts

Reaction Rate Equation

Understanding the reaction rate equation is key to grasping the dynamics of a chemical reaction. In the context of an autocatalytic reaction, the rate equation represents the speed at which reactants turn into products. It’s expressed in terms of the concentration of the reactants and a rate constant. For example, in the provided exercise, the reaction rate equation is given as
\(r_{A} = -\frac{\mathrm{d} C_{A}}{\mathrm{d} t} = k C_{A} C_{R}\)
The negative sign indicates the rate at which reactant A is being consumed. This equation is crucial because it helps us understand that the reaction rate depends on both the concentration of reactant A and the catalyst R at any given moment. Being able to manipulate and apply this equation is fundamental for studying reaction mechanisms and predicting how the reaction will progress over time.

Chemical Kinetics

Chemical kinetics is the study of the rates of chemical processes and the factors affecting these rates. It’s a cornerstone of understanding reaction mechanisms and the conditions that affect them. In our exercise, kinetics comes into play as we analyze how the reaction speeds up when substances react with each other. Autocatalytic reactions are a fascinating area of kinetics because one of the products acts as a catalyst for the reaction, thus boosting its own production as the reaction progresses. As the concentration of the autocatalyst increases, so does the reaction rate, up to a certain point. This self-accelerating nature of autocatalytic reactions makes them particularly interesting and useful in various industrial and biological processes.

Stoichiometry

Stoichiometry is all about the quantitative relationships between the amounts of reactants and products in a chemical reaction. It's grounded in the conservation of mass and the concept of moles. When dealing with reactions like the autocatalytic one in our exercise, stoichiometry becomes essential for relating the concentrations of reactants to products.
It ensures that the atoms are balanced on both sides of the reaction equation, allowing us to predict how much of each substance will be involved in the reaction. Without a clear understanding of stoichiometry, it becomes quite challenging to determine the relationship between the concentration of reactant A and the catalyst R, which is a pivotal aspect when addressing the optimization problem presented in the exercise.

Optimization Problem in Chemistry

An optimization problem in chemistry involves finding the best solution based on certain conditions and constraints. In our example, the challenge is to maximize the rate of disappearance of reactant A in an autocatalytic reaction. This type of problem is common in chemical kinetics, where we often want to speed up a reaction or increase the yield of a product.
By applying calculus, we can determine when the reaction rate will reach its maximum. To solve this, we use a systematic approach: we identify when the product of the reacting species’ concentrations will be at its highest, as this corresponds to the highest reaction rate. Through evaluating various scenarios and their impact on the rate equation's output, we can pinpoint the optimum conditions for the reaction—which, in this case, is when the concentration of A is double that of R. Thus, solving optimization problems in chemistry not only deepens understanding of reaction mechanics but also has practical applications in improving industrial processes and product creation.