The mechanism for the endothermic reaction \(\mathrm{A}+\mathrm{B} \rightarrow \mathrm{C}+\mathrm{X}\) is Step \(1: \mathrm{A}+\mathrm{A} \rightarrow \mathrm{C}+\mathrm{D}(\mathrm{slow})\) Step \(2: \mathrm{B}+\mathrm{D} \rightarrow \mathrm{X}+\mathrm{A}\) (fast) (a) Draw the reaction-energy profile for this reaction and label reactants, products, reaction intermediates, transition states, activation energies, and \(\Delta E_{\mathrm{rxn}} .\) (Hint: First draw a profile for step \(1 .\) Make it a very endothermic reaction, and remember that a slow reaction has a large value for \(E_{\mathrm{a}}\) Then draw a profile for step 2, using the line representing the step 1 products as the reactants line for step \(2 .\) Remember that a fast reaction has a small value for \(E_{\mathrm{a}}\). (b) What is the rate law for this reaction? (c) If you wanted to quadruple the rate of this reaction, by what factor would you have to increase the concentration of \(\mathrm{A}\) ?

The term endothermic reaction refers to a type of chemical process that absorbs energy, typically in the form of heat, from its surroundings. This intake of energy is necessary for the reaction to proceed, as it helps to break the bonds within the reactants, thus leading to the formation of products. In the context of a reaction-energy profile, which graphs the potential energy of a chemical system versus the progress of the reaction, endothermic reactions manifest as a net gain in energy.

For example, the exercise describes an endothermic reaction where the energy level of the products (C and D) is higher than the energy level of the reactants (A and A). This means that more energy is required to initiate the reaction than is released upon the formation of the products. The absorption of energy during an endothermic process can often be detected as a decrease in temperature of the reaction mixture, while the extent of the energy change is represented by the symbol \( \Delta E_{\mathrm{rxn}} \).

Endothermic reactions often require energy input in the form of heat to proceed, and are therefore more likely to occur at higher temperatures where the reactants have more kinetic energy. It is essential for students to understand that in an endothermic reaction, the environment becomes colder as the reaction absorbs heat, in contrast to exothermic reactions where the environment becomes warmer.

The rate law is a mathematical expression that describes the speed of a chemical reaction in relation to the concentration of its reactants. It allows us to understand how various factors, such as concentration and temperature, affect the rate of a reaction. Each reaction has its unique rate law, and this important concept helps chemists control and predict the outcome of chemical processes.

In the given exercise, we are asked to determine the rate law from a two-step mechanism. The rate-determining step (the slowest step) is identified as the first step: \( \mathrm{A} + \mathrm{A} \rightarrow \mathrm{C} + \mathrm{D} \). This step controls the overall reaction rate because it is the bottleneck of the reaction mechanism. The rate law derived from this step is \( Rate = k[A]^2 \), which indicates that the rate is dependent on the square of the concentration of reactant A.

It's vital for students to grasp that the rate-determining step essentially 'sets the pace' for the entire reaction, and understanding the rate law provides crucial insight into how altering reactant concentrations or introducing catalysts can influence the reaction rate.

The concept of activation energy (\( E_a \)), inherent in chemical kinetics, is the minimum amount of energy that reacting particles must possess for a reaction to occur. It can be visualized as a barrier that reactants must overcome to be transformed into products. In the reaction-energy profile, the activation energy is depicted as the energy difference between the reactants and the top of an energy hill, which represents the transition state.

The higher the activation energy, the fewer the number of molecules that have enough energy to react at a given temperature. This corresponds with the fact that a slow step in a mechanism (like the first step in our exercise) has a large activation energy. Conversely, a fast step has a small activation energy, indicating that reactant molecules easily achieve the transition state.

The role of activation energy in a reaction is crucial because it helps explain why certain reactions require heating, the use of catalysts, or other energy inputs to proceed. Increasing the temperature generally increases the number of molecules with energy equal to or greater than the activation energy, thus raising the rate of the reaction. In educational settings, understanding the concept of activation energy assists students in predicting how changes in conditions might influence the course of a chemical reaction.