From the rate expression for the following reactions, determine their order of renction and the dimensions of the rate constants. (u) \(\mathrm{WNO}_{(\mathrm{k})} \cdots \mathrm{N}_{2} \mathrm{O}_{(\mathrm{g})}+\mathrm{NO}_{2(\mathrm{~g})} ; \quad\) Rate \(=K[\mathrm{NO}]^{2}\) Rate \(=K\left[\mathrm{H}_{2} \mathrm{O}_{2}\right][\mathrm{I}]\) (c) \(\mathrm{CH}_{3} \mathrm{CHO}_{(\mathrm{g})} \longrightarrow \mathrm{CH}_{4(\mathrm{~g})}+\mathrm{CO}_{(\mathrm{g})} ;\) Rate \(=K\left[\mathrm{CH}_{3} \mathrm{CHO}\right]^{3 / 2}\) (d) \(\mathrm{CHCl}_{3(\mathrm{~g})}+\mathrm{Cl}_{2(g)} \longrightarrow \mathrm{CCl}_{4(\mathrm{~g})}+\mathrm{HCl}_{(\mathrm{g})}\) Rate \(=K\left[\mathrm{CHCl}_{3}\right]\left[\mathrm{Cl}_{2}\right]^{1 / 2}\)

The reaction order indicates the relationship between the reactant concentrations and the rate at which the reaction occurs. It sums up the exponents of the concentration terms in the rate equation. Understanding the reaction order of a chemical process is fundamental as it determines how the reaction rate changes when concentrations of reactants vary.

For example, if a rate equation is written as Rate = K[A]², where K is the rate constant and [A] is the concentration of reactant A, the reaction is second-order with respect to A. This means that if the concentration of A doubles, the rate of the reaction will increase by a factor of four, as the concentration term is squared in the rate equation. Reaction order can be integers, like 2 in the aforementioned example, or fractions, such as 3/2, as seen in reaction (c) and (d) from our exercise.

The dimensions or units of the rate constant (K) give us crucial information about the rate and order of a reaction. For a simple reaction where rate = K[A]ⁿ, the units of K can be determined through dimensional analysis by balancing the units of rate (usually M/s or mol/L·s) with the units derived from the concentration of reactants raised to the power of the reaction order.

In our exercise examples, for a second-order reaction (u) and (v), K has the units of M^-1·s^-1. For the reaction with order 3/2 (c) and (d), the units of K are M^-1/2·s^-1. This dimensional analysis is essential for consistency and helps us validate our experimental results and theoretical predictions.

A second-order reaction is characterized by a rate that is proportional to the square of the concentration of one reactant, or directly proportional to the product of the concentrations of two reactants. Looking at the initial step-by-step solution, reactions (u) and (v) are examples of second-order reactions.

In terms of kinetics, these reactions can be described by rate laws such as Rate = K[A]² or Rate = K[A][B]. The integrated rate law for a second-order reaction will yield a graph of 1/[A] versus time that is a straight line, indicating that the reaction follows second-order kinetics. This is significant in predicting how long a reaction will take and how the concentration of reactants will change over that period.

The rate determining step is the slowest step in a reaction mechanism that dictates the overall rate of the chemical reaction. It’s analogous to the narrowest point in an hourglass where the rate of sand falling is governed by the smallest opening. In a multi-step reaction, once the rate determining step is completed, the remainder of the process typically proceeds much faster.

Understanding the rate determining step is crucial for both theoretical studies and practical applications, such as in the design of catalysts and optimization of industrial processes. Identifying this bottleneck step enables chemists to modify reaction conditions or use catalysts to accelerate the overall reaction by lowering the activation energy of the rate determining step, thus increasing the reaction rate.