Question

Consider the tabulated data showing the initial rate of a reaction (A \(\longrightarrow\) products) at several different concentrations of A. What is the order of the reaction? Write a rate law for the reac- tion, including the value of the rate constant, \(k\). $$ \begin{array}{cc} {[\mathrm{A}](\mathrm{M})} & \text { Initial Rate }(\mathrm{M} / \mathrm{s}) \\\ 0.12 & 3.89 \times 10^{-4} \\ \hline 0.18 & 8.75 \times 10^{-4} \\ \hline 0.28 & 2.12 \times 10^{-3} \\ \hline \end{array} $$

Short answer

The reaction is first order with respect to A. The rate law is rate = k[A], and the value of k is calculated using the given data.

Step-by-step solution

Observe the changes in concentration and rate

Examine how the initial rate changes as the concentration of A changes. Note the proportional relationship between concentration change and rate change to determine if the rate law is zero, first, or second order with respect to reactant A.

Calculate the rate change ratios

For each successive pair of concentration data points, calculate the ratio of the rates and the ratio of the concentrations. If the rate law is first order, the ratio of the rates will be equal to the ratio of the concentrations raised to the first power. If it is second order, the ratio will equal the square of the concentration ratios.

Determine the order of the reaction

Using the calculated ratios, determine the reaction order with respect to A. If the ratio of rate changes matches the ratio of the concentration changes, the reaction is first order. If the rate changes are the square of the concentration changes, it is second order.

Write the rate law

Based on the determined order, write the rate law for the reaction. For first order, the rate law is rate = k[A], and for second order, rate = k[A]^2.

Calculate the rate constant, k

Use one of the given data points and the rate law to solve for the rate constant, k.

Key concepts

Rate Law

Understanding the rate law is crucial for grasping the complexities of chemical reactions. A rate law expresses the relationship between the rate of a chemical reaction and the concentration of its reactants. It's usually represented in the form \(rate = k[A]^n\), where \(k\) is the rate constant, \([A]\) denotes the concentration of reactant A, and \(n\) indicates the reaction order with respect to A.

Through the rate law, we can predict how changes in concentration affect the reaction rate. If the reaction is first order, then the rate is directly proportional to the concentration of reactant A. In contrast, if the reaction is second order with respect to A, the rate is proportional to the square of the concentration of A. This information allows chemists to control reaction rates, effectively making the rate law a cornerstone concept in chemical kinetics.

Initial Rate Method

The initial rate method is a technique to determine the reaction order by measuring the initial rates of a reaction at various concentrations of the reactants. It involves conducting several experiments where only the concentration of one reactant is varied while the others are kept constant.

To effectively use this method, you'll plot the data of concentration versus initial rate and analyze how the rate changes with concentration. This approach allows you to calculate the order of the reaction by looking for a pattern in the changes of the rate with respect to changes in the concentration, as demonstrated in the textbook exercise. By doing so, you create a clearer picture of the reaction's kinetics, which in turn clarifies the mechanism by which reactants transform into products.

Rate Constant

The rate constant, symbolized by \(k\), is an essential component in the rate law equation that gives the speed at which a reaction proceeds. Its value is determined experimentally and depends on the reaction's conditions, such as temperature and the presence of a catalyst. For a given set of conditions, the rate constant is, as the name suggests, constant. It provides us with a quantitative measure of a reaction's propensity to occur.

In our exercise, calculating the rate constant involves using the rate law derived from the order of the reaction and some initial rate data. The rate constant not only bridges the conceptual gap between the reaction's macroscopic observables and the microscopic interactions, but it also offers insight into the intrinsic velocity of chemical processes, unaffected by reactant concentrations.

Chemical Kinetics

Chemical kinetics is the study of the rates of chemical reactions and the factors affecting them. It is a vast field that encompasses concepts like reaction order, rate laws, and rate constants. By examining how various variables influence the speed of reactions, kinetics provides insight into the molecular-level mechanisms underlying the observed rate laws.

In kinetics, we are not just interested in the final products of a reaction, but in how quickly they are formed. This information is pivotal for various applications such as drug development, environmental modeling, and even the food industry, where reaction speeds can directly affect product stability and efficacy. The kinetic principles help scientists to design better experiments, control reaction conditions more effectively, and predict the outcomes of chemical processes with greater accuracy.