Find the order of reaction for the rate expression rate \(=K[A][B]^{2 / 3}\). Also suggest the units of rate and rate constant for this expression.

Understanding the rate expression of a chemical reaction is vital in the field of chemical kinetics, which deals with the rates at which reactions occur. In simple terms, the rate expression, also known as the rate law, connects the rate of a reaction to the concentrations of the reactants through a specific equation: rate = K[A]ⁿ[B]^m, where K represents the rate constant, [A] and [B] are the concentrations of the reactants, and n and m indicate the orders of the reaction with respect to each reactant. The exponents n and m are derived empirically – meaning they must be determined by experimentation and cannot simply be inferred from the stoichiometric coefficients of the balanced equation.

For example, with the given rate expression rate = K[A][B]^2/3, we can immediately identify that the reaction is first order in [A] and 2/3 order in [B]. These orders tell us how sensitive the reaction rate is to changes in the concentration of each reactant: in this case, a change in concentration of A has a direct linear effect on the rate, while a change in B's concentration has a proportionally smaller influence, following the 2/3 power.

The rate constant, denoted as K, is a proportionality constant in the rate expression of a chemical reaction. Its units vary depending on the overall order of the reaction. To ensure dimensional consistency in the rate expression, the units of K must offset the units of concentration raised to the power of the reaction order. For the rate equation rate = K[A][B]^2/3, we first determine the overall order of the reaction, which is 5/3.

The rate of a reaction is typically given in moles per liter per second (mol/L*s or M/s), and thus the units of the rate constant K must cancel out the units of concentration (M) raised to the power of the overall reaction order (5/3 in our example). Therefore, the units for K would be M^-2/3/s in this case. This tells us that for every one-unit change in the rate, there is a corresponding M^2/3 change in the combined concentrations of A and B.

The order of a reaction with respect to a given reactant is a measure of how the rate is affected by the concentration of that reactant. It is often an integer but can be a fraction or even zero, indicating different types of dependency. The overall order of a reaction is found by adding the individual orders with respect to each reactant in the rate expression.

In our example, the reaction is first order with respect to reactant A and 2/3 order with respect to reactant B, resulting in an overall reaction order of 5/3 when these are added together. Understanding the order of reaction is crucial for predicting how rate changes with concentration and for determining the rate constant's units. For instance, a zero-order reaction implies that a change in the concentration of the reactant has no effect on the rate, while a second-order reaction means the rate is proportional to the square of the concentration of the reactant.

While stoichiometry is the calculation of reactants and products in chemical reactions, it is not directly used to determine the form of the rate law. The rate laws do not necessarily align with the stoichiometric coefficients as seen in a balanced chemical equation. Instead, the exponents in a rate law, which define the reaction's order with respect to each reactant, must be determined experimentally through methods such as initial rate experiments.

Stoichiometry may, however, play a role in complex reactions where the rate law is derived from a mechanism involving a series of steps. Each of these steps can have its own rate law, and the slowest step, known as the rate-determining step, controls the overall reaction rate. In these cases, the stoichiometry within the rate-determining step can be reflected in the rate law for the entire reaction process.

Chemical kinetics is the branch of chemistry that concerns the speed or rate at which chemical reactions occur, and the factors that affect these rates. It involves the study of reaction rates, the construction of models that can explain these rates, and the derivation of rate laws such as the one given in our exercise. Through kinetics, chemists can understand the pathway of a reaction, determine its mechanism, and predict the yield of products over time.

Key factors that influence reaction rates include reactant concentrations, as directly referenced in the rate expression; temperature, which often speeds up reactions as it increases; the presence of a catalyst, which lowers the activation energy required for the reaction to proceed; and the physical state of the reactants, where more homogeneous mixtures typically lead to faster reactions. By manipulating these factors, chemists can control reaction rates for various practical applications in industry and research.