Question

Consider the generic chemical reaction: \(A+B \rightarrow A B\) . Based on experimental data, the reaction is second order in Reactant A. If the concentration of \(A\) is halved, and all other conditions remain unchanged, how does the reaction rate change?

Short answer

When the concentration of Reactant A is halved, the reaction rate becomes one-fourth (1/4) of the original rate. This is consistent with the second-order dependence on Reactant A concentration.

Step-by-step solution

Write the rate law for a second-order reaction

In a chemical reaction, the rate law is a mathematical expression that relates the reaction rate to the concentrations of the reactants. For a second-order reaction involving Reactant A, the rate law is: \[ Rate = k[A]^{2} \] where "Rate" is the reaction rate, "k" is the rate constant (which is not affected by the reactant concentrations), and "[A]" is the concentration of Reactant A.

Determine the initial reaction rate

Let us denote the initial concentration of Reactant A as \([A]_0\), and the initial reaction rate as \(Rate_0\). According to the rate law, the initial reaction rate is: \[ Rate_0 = k[A]_0^2 \]

Calculate the new concentration of Reactant A

We are told that the concentration of Reactant A is halved. This means that the new concentration of Reactant A, denoted as \([A]_1\), will be: \[ [A]_1 = \frac{1}{2} [A]_0 \]

Determine the new reaction rate

Now, let us denote the new reaction rate as \(Rate_1\). Using the rate law and the new concentration of Reactant A, we can find the new reaction rate: \[ Rate_1 = k[A]_1^2 \] Substitute \([A]_1\) with the expression in step 3: \[ Rate_1 = k \left(\frac{1}{2} [A]_0\right)^2 \] \[ Rate_1 = k \cdot \frac{1}{4} [A]_0^2 \]

Compare the initial and final reaction rates

We want to find the relationship between the initial and final reaction rates. Divide the equation for \(Rate_1\) by the equation for \(Rate_0\): \[ \frac{Rate_1}{Rate_0} = \frac{k \cdot \frac{1}{4} [A]_0^2}{k[A]_0^2} \] Simplify the equation: \[ \frac{Rate_1}{Rate_0} = \frac{1}{4} \]

Write the conclusion

When the concentration of Reactant A is halved, the reaction rate becomes one-fourth (1/4) of the original rate. So, the reaction rate decreases when the concentration of Reactant A decreases, which is consistent with the second-order dependence on Reactant A concentration.

Key concepts

Second-Order Reaction

Understanding the nature of a second-order reaction is crucial in chemical kinetics, which is the study of reaction rates. Such reactions depend on the concentrations of one second-order reactant or two first-order reactants. For a single reactant, the rate of a second-order reaction is proportional to the square of its concentration. This relationship is represented as \(Rate = k[A]^2\), where \(Rate\) is the reaction rate, \(k\) is the rate constant, and \[A\] symbolizes the concentration of the reactant.

In this type of reaction, if the concentration of the reactant is halved, the rate will decrease to a quarter of its original value, because the rate is proportional to the square of the concentration (\(\frac{1}{2}\)^2 = \(\frac{1}{4}\)). This quadratic dependence means that second-order reactions are sensitive to changes in reactant concentration, leading to larger alterations in rate than first-order reactions.

Rate Law

The rate law is an equation that quantitatively relates the rate of a chemical reaction to the concentration of its reactants. It is derived from experimental observations and not from the balanced chemical equation. The general form of the rate law for a second-order reaction is \(Rate = k[A]^2\) if it involves a single reactant or \(Rate = k[A][B]\) for two reactants A and B, each being a first-order reactant in the reaction.

The key feature of a rate law is that it must reflect the actual concentrations of the reactants used in the experiment, and it pinpoints the specific manner in which these concentrations affect the speed of the reaction. Moreover, the rate law includes a rate constant \(k\), which is unique to each reaction and dependent on factors such as temperature and catalyst presence, but it does not change with the concentration of the reactants under specific conditions.

Reactant Concentration

Reactant concentration is a term that refers to the amount of a reactant in a given volume of solution. It is often denoted by square brackets (e.g., \[A\] for reactant A). In the context of chemical kinetics, the concentration of reactants is vital because it directly influences the rate of the reaction.

According to the rate law, changes in the concentration of a reactant will affect the reaction rate as dictated by the order of the reaction with respect to that reactant. In second-order reactions, for example, the rate is proportional to the square of reactant concentration (\(Rate = k[A]^2\)). Therefore, a change in concentration will have a squared effect on the rate. This concept is critical in understanding how manipulating reactant concentrations can be used to control the rate at which reactions proceed, which is particularly important in industrial applications where reaction speed can affect production rates and costs.