Question

For the reaction \(A+B \longrightarrow C+D\), the following initial rates of reaction were found. What is the rate law for this reaction? $$\begin{array}{llll} \hline & & & \text { Initial Rate, } \\ \text { Expt } & \text { [A], M } & \text { [B], M } & \text { M min }^{-1} \\\ \hline 1 & 0.50 & 1.50 & 4.2 \times 10^{-3} \\ 2 & 1.50 & 1.50 & 1.3 \times 10^{-2} \\ 3 & 3.00 & 3.00 & 5.2 \times 10^{-2} \\ \hline \end{array}$$

Short answer

The rate law for this chemical reaction is Rate = k[A][B].

Step-by-step solution

- Compare Experiments 1 and 2

First, the change in the rate of reaction when the concentration of reactant A is altered should be noticed. Between Experiments 1 and 2, the concentration of A is changed from 0.50M to 1.50M (tripled), while the concentration of B is held constant at 1.50M. The reaction rate is increased from 4.2e-3 M min^-1 to 1.3e-2 M min^-1 (also tripled). Since tripling the concentration of A triples the rate, the reaction is first order with respect to A.

- Compare Experiments 2 and 3

Second, the change in the rate of reaction when the concentration of reactant B is altered must be examined. Between Experiments 2 and 3, the concentration of B is doubled (from 1.50M to 3.00M), and A is also doubled (from 1.50M to 3.00M). Also, the reaction rate is quadrupled (from 1.3e-2 M min^-1 to 5.2e-2 M min^-1). Again, since doubling the concentration of both A and B has doubled the rate, the reaction is first order with respect to B.

- Form the Rate Law

Once the orders of reaction with respect to A and B have been determined, these can be combined into the rate law for the reaction. Based on the results of Steps 1 and 2, the rate law should be in the form of Rate = k[A][B].

Key concepts

Reaction Order

Understanding the reaction order is crucial to determining the rate law. The reaction order indicates how the rate of the reaction is affected by the concentration of the reactants. For the reaction \(A+B \longrightarrow C+D\), we assess how changes in concentration of reactants \([A]\) and \([B]\) affect the rate of the reaction. In essence, the reaction order with respect to a particular reactant is the power to which its concentration term in the rate equation is raised.

In Step 1 of solving, we observe the effect of changing the concentration of reactant \([A]\). The concentration of \([A]\) is tripled from 0.50 M to 1.50 M, leading to a tripling of the reaction rate. This direct relationship—where the change in rate matches the change in concentration—indicates a first order reaction with respect to \([A]\).

Similarly, in Step 2, the concentration of both \([A]\) and \([B]\) is doubled, which results in the rate increasing fourfold. This implies that the reaction is first order with respect to both \([B]\).

The overall reaction order is the sum of the orders with respect to each reactant, leading us to deduce that this reaction is second order overall, with a rate law indicating dependency on both \([A]\) and \([B]\).

Initial Rates

The concept of initial rates is a key analytical tool to determine reaction kinetics. The initial rate of a reaction is measured right at the outset, where changes in concentrations are minimal. Since the reaction has just started, we assume constant conditions for the analysis, making it simpler to establish the relationship between reaction rate and concentration.

In the given exercise, initial rates for different concentrations of reactants \([A]\) and \([B]\) are provided across three experiments. By maintaining a specific style of comparison—such as holding one reactant constant while varying the other—we uncover the relationship of each reactant's concentration effects on the initial rate. This method of using initial rates allows us to systematically explore each reactant's impact, leading us to conclude the reaction order for each.

For instance, Experiment 1 and 2 show that when \([A]\) is tripled (but \([B]\) remains constant), the rate triples. This highlights the impact of \([A]\) on the initial rate, solidifying its role in determining the order of the reaction as first order.

Concentration Effect

Concentration effect uncovers the influence each reactant’s concentration has on the rate of reaction. By analyzing how changes in the concentration translate to changes in the rate, we can deduce which reactants are influencing the pace of reaction most.

The exercise utilizes this concept by having experiments where concentrations of \([A]\) and \([B]\) are systematically altered. Observing the impact: when reactant concentrations were either held steady or changed, researchers could detect these influences. Experimentation then leads to forming a rate equation formula.

For example, in the provided exercise, when \([A]\) is kept consistent, changes to \([B]\) alone are studied, and vice versa. This isolated adjustment allows for a keen understanding of its individual effect without interference. The significant finding that doubling both \([A]\) and \([B]\) doubled the rate of the reaction indicates first-order dependency for each of them. Consequently, both reactants collectively determine the rate law as \(Rate = k[A][B]\), showcasing a clear linear concentration effect for each reactant involved.