Derive the relationship between rate of reaction, rate of disappearance of \(X, Y\) and rate of formation of \(X_{2} Y_{2}\) for the reaction : $$ 2 X+3 Y \longrightarrow X_{2} Y_{3} $$

Chemical kinetics is a field of chemistry that deals with understanding the speed or rate at which a chemical reaction occurs. It involves studying how different variables such as concentration, temperature, and catalysts impact the reaction rate. A reaction rate can be defined as the change in concentration of a reactant or product per unit time.

For example, if we consider the reaction provided, \(2 X + 3 Y \longrightarrow X_{2}Y_{3}\), chemical kinetics would allow us to determine how quickly the reactants are getting consumed and how fast the product is being formed over time. By studying these rates, chemists can optimize conditions to increase the efficiency of reactions for industrial applications, understand reaction mechanisms, and predict the behavior of chemical systems.

Understanding the kinetics of a chemical reaction is fundamental in fields such as pharmaceuticals, environmental science, and energy conversion, as it can vastly affect the outcome and practicality of chemical processes.

Reaction stoichiometry involves the quantitative relationship between the amounts of reactants and products in a chemical reaction. It is based on the principle that during a chemical reaction, atoms are neither created nor destroyed—they simply rearrange. The stoichiometry of a reaction gives us the ratios of the amount of each substance involved.

In the exercise's reaction, \(2 X + 3 Y \longrightarrow X_{2}Y_{3}\), stoichiometry tells us that two moles of \(X\) will react with three moles of \(Y\) to produce one mole of \(X_{2}Y_{3}\). When determining the rate of reaction, this stoichiometric relationship is crucial as it allows us to express the rate at which reactants are consumed and products are formed in terms of each other. This balanced reaction equation is a tool for predicting how much of each substance is needed or produced, which is vital in chemical manufacturing and research settings.

Differential rate expressions relate the instantaneous rate of reaction to the concentrations of reactants and products. They are derived from the stoichiometry of the balanced chemical equation and are represented mathematically as derivatives with respect to time.

For instance, the rate of disappearance of \(X\) in the provided reaction is written as \(-\frac{d[X]}{dt}\), whereas the rate of appearance of product \(X_{2}Y_{3}\) is described by \(\frac{d[X_{2}Y_{3]}{dt}\). The negative sign in the rate of disappearance denotes that the reactant's concentration is decreasing over time.

The differential rate expression becomes a powerful tool when combined with the stoichiometry of the reaction. It allows us to establish a direct quantitative connection between the rate at which reactants are used up and the rate at which products are formed. In this reaction, the rate at which \(X\) disappears is twice the rate of formation of \(X_{2}Y_{3}\), which directly relates to the stoichiometric coefficient of \(X\) being twice as large as that of \(X_{2}Y_{3}\).