Question

The following data are for the reaction \(2 \mathrm{A}+\mathrm{B} \longrightarrow\) products. Establish the order of this reaction with respect to A and to B. $$\begin{array}{cccc} \hline \text { Expt 1, }[\mathrm{B}]=1.00 \mathrm{M} & & {\text { Expt 2, }[\mathrm{B}]=0.50 \mathrm{M}} \\ \hline \begin{array}{cccc} \text { Time, } \\ \text { min } \end{array} & \begin{array}{c} \text { [A], M } \\ \end{array} & \text { Time, } \text { min } &\text { [A], M } \\ \hline 0 & 1.000 \times 10^{-3} & 0 & 1.000 \times 10^{-3} \\ 1 & 0.951 \times 10^{-3} & 1 & 0.975 \times 10^{-3} \\ 5 & 0.779 \times 10^{-3} & 5 & 0.883 \times 10^{-3} \\ 10 & 0.607 \times 10^{-3} & 10 & 0.779 \times 10^{-3} \\ 20 & 0.368 \times 10^{-3} & 20 & 0.607 \times 10^{-3} \\ \hline \end{array}$$

Short answer

The order of the reaction is first order with respect to B and zeroth order with respect to A.

Step-by-step solution

Reactant Concentration Comparison

Examine the change in concentration of A after 1 minute in both experiments. In experiment 1 when B is 1M, A goes from 1.000 * 10-3 M to 0.951 * 10-3 M. In experiment 2, when B is .5M, A goes from 1.000 * 10-3 M to 0.975 * 10-3 M.

Rate Calculation

Calculate the ratio of the rates of reaction for the two experiments. Remember that the rate of reaction is essentially the change in concentration per unit time. For experiment 1, the rate is (1.000 - 0.951) x 10-3 M/min = 0.049 * 10-3 M/min. For experiment 2, the rate is (1.000 - 0.975) x 10-3 M/min = 0.025 * 10-3 M/min. The ratio of the rates is thus 0.049/0.025 = 1.96, which is approximately 2.

Order of Reaction Determination

Because the rate approximately doubled when we halved the concentration of B, we can deduce that the order of the reaction with respect to B is first order. Moving on to A, since the concentration of A is constant in both the experiments, the order with respect to A does not affect the result and is therefore 0. Thus, the reaction is first order with respect to B and zeroth order with respect to A.

Key concepts

Rate of Reaction

The rate of reaction tells us how fast a reaction takes place. It is determined by the change in concentration of the reactants over time. To calculate it, you need to know the initial and final concentrations of the reactants and the time taken for the change.

• In our problem, for Experiment 1, where [B] = 1.00 M, the initial concentration of [A] is 1.000 x 10^{-3} M, and it decreases to 0.951 x 10^{-3} M after 1 minute.
• In Experiment 2, where [B] = 0.50 M, the initial concentration of [A] also starts at 1.000 x 10^{-3} M but goes to 0.975 x 10^{-3} M after 1 minute.

These changes allow us to find the rate of reaction as the difference in concentration over the time interval. Understanding these rates helps to predict how quickly reactants transform into products.

Concentration Change

Concentration changes are crucial for determining the rate and order of a reaction.
In our case, we focus on how the concentration of Reactant [A] evolves over time during the experiments. Over the same time period of 1 minute, [A] decreases differently depending on the concentration of [B].

• In Experiment 1, the concentration decreases by 0.049 x 10^{-3} M.
• In Experiment 2, the concentration change is 0.025 x 10^{-3} M.

This comparison shows that the rate at which [A] decreases depends on the amount of [B] available, emphasizing the role [B] plays in the reaction's progress.

First Order Reaction

A first-order reaction indicates that the rate is directly proportional to the concentration of one reactant. This means if you double the concentration of the reactant, the reaction rate also doubles.
In this example, the experiment highlighted that when the concentration of [B] was halved from 1.00 M to 0.50 M, the rate of reaction approximately halved as well. This direct relationship implies that for this reaction:

• The order with respect to [B] is first order.
• This means [B] is essential in dictating how fast the reaction happens, as deduced from the rate calculations and concentration comparisons.

Zeroth Order Reaction

In a zeroth order reaction, the rate is independent of the concentration of the reactants involved. Even if you increase or decrease the concentration, the rate of reaction stays the same.
From our analysis, we see that [A]'s concentration is constant across both experiments, meaning changing it doesn't alter the rate of reaction.

• The reaction is thus zeroth order with respect to [A].
• This tells us that the concentration of [A] does not influence how quickly products are formed from the reaction in this specific case.

These details help clarify why [A]'s concentration doesn't affect the overall reaction pace, leading to its classification as zeroth order.