Question

The greatest increase in the reaction rate for the reaction between \(\mathrm{A}\) and \(\mathrm{C}\), where rate \(=k[\mathrm{~A}]^{1 / 2}[\mathrm{C}]\), is caused by (a) doubling [A] (b) halving [C] (c) halving [A] (d) doubling [A] and [C]

Short answer

Answer: Doubling the concentrations of both A and C.

Step-by-step solution

Write down the rate equation

Write down the rate equation as given in the problem: rate \(=k[\mathrm{A}]^{1/2}[\mathrm{C}]\).

Analyze each option and calculate the new rate

We will analyze each option and calculate the new rate for each one. (a) Doubling the concentration of A: New rate = \(k[(2\mathrm{A})]^{1/2}[\mathrm{C}]\) New rate = \(k\cdot\sqrt{2}[\mathrm{A}]^{1/2}[\mathrm{C}]\) The rate increases by a factor of \(\sqrt{2}\). (b) Halving the concentration of C: New rate = \(k[\mathrm{A}]^{1/2}[(1/2)\mathrm{C}]\) New rate = \((1/2)k[\mathrm{A}]^{1/2}[\mathrm{C}]\) The rate decreases by a factor of \(1/2\). (c) Halving the concentration of A: New rate = \(k[(1/2\mathrm{A})]^{1/2}[\mathrm{C}]\) New rate = \((1/\sqrt{2})k[\mathrm{A}]^{1/2}[\mathrm{C}]\) The rate decreases by a factor of \(1/\sqrt{2}\). (d) Doubling the concentrations of both A and C: New rate = \(k[(2\mathrm{A})]^{1/2}[(2)\mathrm{C}]\) New rate = \((2\sqrt{2})k[\mathrm{A}]^{1/2}[\mathrm{C}]\) The rate increases by a factor of \(2\sqrt{2}\).

Determine the action causing the greatest increase in the reaction rate

Comparing the factors for each option, we see that option (d), doubling the concentrations of both A and C, results in the greatest increase in the reaction rate (an increase by a factor of \(2\sqrt{2}\)).

Key concepts

Understanding Reaction Kinetics

When studying chemical reactions, we delve into the realm of reaction kinetics, which is the branch of chemistry that concerns the rates at which reactions occur and the steps involved in these processes. Reaction rates can be visually observed by changes such as the formation of a precipitate, color changes, or the release of gas. On the molecular level, however, kinetics explores how different conditions, like temperature, pressure, and concentration, affect the speed of the reactions.

As we dive deeper, we come across the 'rate equation' or 'rate law', which mathematically expresses the reaction rate as a function of the concentration of reactants. This is important because it reveals the relationship between the concentration of the reactants and the overall reaction rate. For instance, our exercise presents a rate law expressed as rate = k[A]^{1/2}[C], where 'k' represents the rate constant, and [A] and [C] symbolize the reactant concentrations. A rate law can be complex, but breaking it down and analyzing how each reactant’s concentration affects the rate, as seen in the exercise's solution, gives us a clearer picture of the reaction's behavior.

Concentration Effect on Reaction Rate

The concentration of reactants plays a vital role in the speed of chemical reactions. To illustrate this, let's focus on the law of mass action, which indicates that the rate of a chemical reaction at a constant temperature depends on the concentrations of the reactants. A higher concentration often leads to more collisions between reactant particles, thus increasing the rate of reaction. Conversely, reducing the concentration typically slows the reaction down, as there are fewer particles to interact with each other.

In our exercise scenario, when the concentration of A is doubled or halved, it directly influences the rate at which the reaction between A and C occurs, albeit differently due to the square root dependence on A. Doubling [A] results in a rate increase by a factor of \(\sqrt{2}\), while halving [A] decreases the rate by a factor of \(1/\sqrt{2}\). Similarly, doubling or halving [C] changes the rate in a simple direct or inverse proportion, showcasing the profound effect concentration changes can impose on the reaction kinetics.

The Rate-Determining Step in Reactions

Within a chemical reaction involving multiple steps, one particular step usually poses a bottleneck to the overall reaction rate, known as the rate-determining step. This is the slowest step in the sequence of events that constitute the full reaction mechanism, and it essentially sets the pace for how quickly the reaction can proceed.

Think of it like the narrowest part of a funnel through which all reactants must pass. No matter how much you increase the concentration of the reactants, if they are governed by this slow step, their effects on the overall reaction rate might be limited. However, our example does not delve into the complexity of multiple step reactions and rate-determining steps. Instead, it considers a straightforward relationship between reactant concentrations and rate, which simplifies understanding how altering concentrations can affect the rate. Nonetheless, in real-world scenarios, identifying and understanding the rate-determining step is crucial for chemists seeking to optimize reaction conditions and enhance production rates in industrial applications.