For the reaction; \(4 \mathrm{NH}_{3(\mathrm{~g})}+5 \mathrm{O}_{2(\mathrm{~g})} \longrightarrow 4 \mathrm{NO}_{(\mathrm{g})}+6 \mathrm{H}_{2} \mathrm{O}_{(\mathrm{g})}\), the rate of reaction in terms of disappearance of \(\mathrm{NH}_{3}\) is \(-\frac{d\left[\mathrm{NH}_{3}\right]}{d t}\), then write the rate expression in terms of concentration of \(\mathrm{O}_{2}, \mathrm{NO}\) and \(\mathrm{H}_{2} \mathrm{O}\)

Understanding reaction rate expressions is crucial in the study of chemical kinetics, a branch of physical chemistry. They allow chemists to quantify how quickly reactants are transformed into products in a chemical reaction.

For the given reaction, \(4 \text{NH}_3 (g) + 5 \text{O}_2 (g) \rightarrow 4 \text{NO} (g) + 6 \text{H}_2\text{O} (g)\), the reaction rate in terms of one reactant, ammonia (\(\text{NH}_3\)), is represented as the negative change in concentration of \(\text{NH}_3\) over time, written as \(-\frac{d[\text{NH}_3]}{dt}\).

This negative sign signifies that the concentration of \(\text{NH}_3\) decreases as the reaction progresses. To find the rate of change for other substances involved in the reaction such as \(\text{O}_2\), \(\text{NO}\), and \(\text{H}_2\text{O}\), we utilize their stoichiometric coefficients from the balanced equation to establish a direct relationship with the rate of \(\text{NH}_3\)'s disappearance. This ensures that the rate expressions are consistent in terms of the extent of reaction.

Stoichiometry provides the mathematical relationship between reactants and products in a chemical reaction, based on the law of conservation of mass. It's this balanced equation that informs us about the proportion of molecules involved and hence participates in establishing the reaction rate expressions.

In our example, stoichiometry tells us that 4 molecules of ammonia react with 5 molecules of oxygen to produce 4 molecules of nitrogen monoxide and 6 molecules of water. These stoichiometric coefficients (4 for \(\text{NH}_3\), 5 for \(\text{O}_2\), etc.) are pivotal in translating the disappearance of one reactant to the appearance of products, which leads to balanced and accurate reaction rate expressions.

Understanding how to apply these numbers to reaction rate expressions allows us to compare rates of consumption and production across different substances within the same reaction, which is essential when predicting the amounts of reactants required and products formed over time.

Understanding Rate of Disappearance

At the heart of chemical kinetics is the concept of the rate of disappearance of reactants. This rate measures the speed at which reactants are consumed in a chemical reaction. It is signed negative due to the reduction in reactant concentrations.

In our example, the rate of disappearance of ammonia can be expressed as \(-\frac{d[\text{NH}_3]}{dt}\), emphasizing that as the reaction goes on, the amount of \(\text{NH}_3\) is diminishing. When we factor in stoichiometry, this rate can be connected to other reactants. For instance, oxygen (\(\text{O}_2\)) is used up at a rate that's proportional to the rate at which ammonia disappears, but adjusted for the different stoichiometric coefficient.

This interrelationship is indispensable for calculating the concentration changes of all reactants over time, and for understanding how fast a reaction proceeds under certain conditions.

Tracking Product Formation

Parallel to the disappearance of reactants is the rate of formation of products, which measures the speed at which products are generated in a chemical reaction. Unlike reactants, this rate is positive as it corresponds to an increase in product concentration.

Continuing with the ammonia oxidation reaction as our example, we observe that nitrogen monoxide (\(\text{NO}\)) and water (\(\text{H}_2\text{O}\)) form at rates directly related to the disappearance of ammonia, again respecting their stoichiometric coefficients. For example, the rate at which \(\text{NO}\) forms is the same as the rate at which \(\text{NH}_3\) disappears because they have a 1:1 stoichiometric relationship in the balanced equation.

Understanding the rate at which products form is essential not only for predicting yields but also for designing reactors and processes in the chemical industry. It's a fundamental concept that shapes our approach to reaction conditions and optimization.