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Chemical kinetics is the branch of chemistry that deals with the rates of chemical reactions and the mechanisms by which they occur. It seeks to understand the steps involved in the transformation of reactants to products, the differing speeds at which reactions occur, and how various factors such as temperature, pressure, and concentration influence these rates.

To grasp chemical kinetics, consider the reaction between carbon monoxide and nitrogen dioxide in the troposphere. The speed at which this reaction progresses, and what governs this speed, are central questions answered by chemical kinetics. By dissecting the reaction step by step, kineticists can gather information about the reaction rate and deduce a rate law that scientifically defines these observations.

The rate law is a mathematical relationship that describes the reaction rate as a function of the concentration of reactants. It is unique to every chemical reaction and must be determined experimentally. In this case, the rate law for the reaction between nitrogen dioxide and carbon monoxide is second order in \(\mathrm{NO}_{2}(g)\), meaning that the rate of the reaction depends on the concentration of \(\mathrm{NO}_{2}(g)\) raised to the second power.

To put it simply, if the concentration of \(\mathrm{NO}_{2}(g)\) is doubled, the reaction rate increases by a factor of four. This quadratic relationship is key to understanding the dynamics of the reacting system and influences how one might control or optimize the reaction in various industrial or environmental settings.

Intermediates are species that appear in the mechanism of a chemical reaction but not in the overall balanced equation because they are neither reactants nor final products. They are produced in one step and consumed in another. The formation of intermediates is an important aspect because they often reveal significant details about the pathway a reaction follows.

For our given reaction, \(\mathrm{NO}_{3}(g)\) is identified as an intermediate. It is crucial in linking the reactants to the eventual products. Understanding how an intermediate is created and later utilized or destroyed gives valuable insight into the mechanism and potential side reactions that could be targeted for inhibition or enhancement in chemical processes.

Elementary reaction steps are individual reactions that happen within a complex chemical reaction mechanism. Each elementary step typically involves a small number of molecules, often only one, two, or three, and directly corresponds to a single molecular event, such as bond formation or breaking.

The proposed reaction mechanism for the combination of \(\mathrm{NO}_{2}(g)\) and \(\mathrm{CO}(g)\) hypothesizes two elementary steps. The first is the formation of \(\mathrm{NO}_{3}(g)\) and \(\mathrm{NO}(g)\) from the reaction of two \(\mathrm{NO}_{2}(g)\) molecules, and the second is the reaction of the intermediate \(\mathrm{NO}_{3}(g)\) with \(\mathrm{CO}(g)\) to produce \(\mathrm{NO}(g)\) and \(\mathrm{CO}_{2}(g)\). These steps collectively explain the observed second-order dependence on \(\mathrm{NO}_{2}(g)\) and highlight the decisive role played by the formation of the intermediate in dictating the overall reaction rate.