Consider the reaction. Kinetics studies reveal a first-order rate dependence on the concentration of the \(\left(\mathrm{CH}_{3}\right)_{3} \mathrm{C}-\mathrm{Br}\) and a zero-order dependence on the concentration of \(\mathrm{H}_{2} \mathrm{O}\). (a) What happens to the reaction rate as the \(\left(\mathrm{CH}_{3}\right)_{3} \mathrm{C}-\mathrm{Br}\) concentration is changed? What happens to the reaction rate as the \(\mathrm{H}_{2} \mathrm{O}\) concentration is changed? (b) Two mechanisms for this reaction are offered below. Can you rule out either of them? Is either mechanism plausible, given the overall balanced equation and kinetic data? Explain your answer fully.

Understanding the concept of reaction order is fundamental to grasping how chemical reactions proceed. In chemical kinetics, the reaction order determines the relationship between the concentrations of the reactants and the rate at which the reaction occurs.

For a reaction where the rate is determined by the concentration of one reactant, if we say the reaction is first-order with respect to that reactant, it means the rate of the reaction is directly proportional to its concentration. Mathematically, it can be shown as:
\[ Rate = k [A] \]
where \( [A] \) is the concentration of the reactant, and \( k \) is the rate constant. When the concentration of \( A \) doubles, the rate of the reaction also doubles. This linear relationship makes it easier to predict and control reaction rates.

A zero-order reaction, on the other hand, implies that the reaction rate is independent of the concentration of the reactant; thus, increasing the reactant concentration does not affect the rate. This can be expressed as:
\[ Rate = k \]
It's significant for students to recognize that reaction order can only be determined experimentally and is not derived from the stoichiometric coefficients of the reaction equation.

Another essential parameter in chemical kinetics is the rate constant, symbolized as \( k \). The rate constant is a measure of how quickly a reaction proceeds and it is unique to each reaction at a given temperature. It relates to the intrinsic reactivity of the reactants and to the energy barrier they must overcome to react.

For a first-order reaction, we often see the rate equation written as:
\[ Rate = k [A] \]
Here, \( k \) represents the rate constant, which remains constant at a fixed temperature. The rate constant is crucial because it lets us calculate the reaction rate provided we know the concentration of the reactant. It is affected by various factors, including temperature, pressure (for gaseous reactions), and the presence of a catalyst. Understanding the rate constant's role helps us predict how changes in conditions might affect the speed of a reaction.

In the context of our exercise, the rate of the given reaction is closely tied with the concept of concentration dependence. This helps us understand and predict the outcome of changes in reactant concentrations. For a first-order reaction involving \( \left(\mathrm{CH}_3\right)_3\mathrm{C}-\mathrm{Br} \), the rate will increase as the concentration of this reactant increases; a decrease in concentration will lower the reaction rate. This occurrence is a direct result of the first-order relationship between \( \left(\mathrm{CH}_3\right)_3\mathrm{C}-\mathrm{Br} \) concentration and the rate of reaction.

However, for the water (\( \mathrm{H}_2\mathrm{O} \)) in the reaction, which exhibits zero-order dependence, its concentration will not affect the reaction rate. Such a reaction reaches a point where the reactant concentration is in great excess or the system becomes saturated, rendering the number of reactant molecules irrelevant in terms of contribution to the rate. Identifying the concentration dependence for each reactant is crucial for students to not only solve textbook problems but also to apply this knowledge in practical scenarios, such as in chemical manufacturing processes where control over reaction rate is key.

A reaction mechanism provides a step-by-step sequence of elementary reactions by which overall chemical change occurs. The mechanism includes the breaking and forming of bonds, the intermediates, transition states, and the sequence of events leading to the final products.

Understanding reaction mechanisms is vital because it explains why certain products are formed and why certain reaction conditions affect the rate in a specific way. One of the most interesting aspects of a reaction mechanism is its ability to explain the kinetic data, like reaction orders and rate constants, for complex reactions.

In the case of our reaction, we're given first-order dependence on \( \left(\mathrm{CH}_3\right)_3\mathrm{C}-\mathrm{Br} \) and zero-order dependence on \( \mathrm{H}_2\mathrm{O} \). This kinetic evidence will often provide hints about the rate-determining step—the slowest step in the mechanism that controls the overall rate. To propose a plausible mechanism, one needs to ensure that the mechanism's rate law matches the experimentally observed rate law, a task requiring critical analysis and a good understanding of the steps involved in the reaction.