Question

For the reaction: \(2 \mathrm{HI} \rightarrow \mathrm{H}_{2}+\mathrm{I}_{2}\), the expression, \(-\frac{1}{2} \frac{\mathrm{d}[\mathrm{HI}]}{\mathrm{d} t}\) represents (a) the rate of formation of \(\mathrm{HI}\) (b) the rate of disappearance of \(\mathrm{HI}\) (c) the instantaneous rate of the reaction (d) the average rate of the reaction

Short answer

(c) the instantaneous rate of the reaction

Step-by-step solution

Identify the Reaction and the Rate Expression

First, understand that the given reaction is \(2 \mathrm{HI} \rightarrow \mathrm{H}_{2} + \mathrm{I}_{2}\), which is the breakdown of hydrogen iodide into hydrogen and iodine gas. The rate expression \(-\frac{1}{2} \frac{\mathrm{d}[\mathrm{HI}]}{\mathrm{d} t}\) includes a negative sign and is related to the concentration of hydrogen iodide, \(\mathrm{HI}\).

Understand the Negative Sign in Rate Expression

The negative sign indicates we are dealing with the rate of disappearance (consumption) of \(\mathrm{HI}\) rather than its formation. Since the reaction shows \(\mathrm{HI}\) being consumed, the rate of consumption is directly related to its rate of disappearance.

Analyze the Stoichiometry and Rate of Reaction

Stoichiometry of the reaction shows that 2 moles of \(\mathrm{HI}\) produce 1 mole of \(\mathrm{H}_{2}\) and 1 mole of \(\mathrm{I}_{2}\). The \(-\frac{1}{2}\) factor accounts for this stoichiometry, thus giving us the rate of the entire reaction as it proceeds.

Determine the Correct Option

Given the information above, the rate expression represents the rate at which \(\mathrm{HI}\) disappears. Therefore, it is not the rate of formation (a) or average rate of reaction (d). The factor \(-\frac{1}{2}\) aligns with option (c), suggesting that it is the instantaneous rate of the entire reaction.

Key concepts

Chemical Kinetics

Chemical kinetics is the branch of physical chemistry that studies the rates at which chemical reactions occur and the factors affecting those rates. The speed of a reaction is commonly expressed in terms of the change in concentration of a reactant or product per unit time. Factors influencing the reaction rate include concentration, temperature, surface area, and the presence of catalysts.

To analyze reaction rates, scientists use rate laws, which are mathematical expressions that correlate the rate of reaction to the concentrations of the reactants. For simple reactions, the rate law can often be determined directly from the stoichiometry, but more complex reactions may require experimental data to deduce the rate law. By understanding the kinetics of a chemical reaction, we can control the reaction rate to optimize industrial processes or predict how a reaction progresses over time.

Rate of Disappearance

The rate of disappearance in chemical kinetics refers to how quickly a reactant is consumed in a chemical reaction. In the expression \( -\frac{1}{2} \frac{\mathrm{d}[\mathrm{HI}]}{\mathrm{d} t} \), the rate at which hydrogen iodide (\(\mathrm{HI}\)) is used up is represented. The negative sign indicates a decrease in the concentration of \(\mathrm{HI}\) over time.

It is important to note that the rate of disappearance is equal in magnitude and opposite in sign to the rate of appearance of products. This is due to the principle of conservation of mass, where matter cannot be created or destroyed. Therefore, as \(\mathrm{HI}\) vanishes, an equivalent amount of product(s) is formed, described by the stoichiometry of the balanced chemical equation.

Stoichiometry

Stoichiometry is the mathematical study of the relationships between the amounts of reactants and products in a chemical reaction based on the law of conservation of mass. It involves using the balanced chemical equation to determine the proportions at which chemicals react.

In the given reaction, \(2 \mathrm{HI} \rightarrow \mathrm{H}_{2} + \mathrm{I}_{2}\), the stoichiometry indicates that two moles of hydrogen iodide produce one mole each of hydrogen and iodine gas. The \( -\frac{1}{2} \) factor in the rate expression correlates with this stoichiometric ratio, reflecting that for every two moles of \(\mathrm{HI}\) that disappear, the reaction proceeds by one 'stoichiometric unit'. Therefore, the stoichiometric coefficients are essential in calculating the reaction rates for each species involved in the reaction.

Instantaneous Rate of Reaction

The instantaneous rate of reaction is the rate at which a chemical reaction proceeds at a particular moment in time. Unlike the average rate, which measures the change in concentration over a broad time period, the instantaneous rate considers an infinitesimally small interval, giving a precise rate at a specific point in time.

The term \( -\frac{1}{2} \frac{\mathrm{d}[\mathrm{HI}]}{\mathrm{d} t} \) is a mathematical description of the instantaneous rate for the disappearance of \(\mathrm{HI}\) in the reaction \(2 \mathrm{HI} \rightarrow \mathrm{H}_{2} + \mathrm{I}_{2}\). The derivative \(\frac{\mathrm{d}[\mathrm{HI}]}{\mathrm{d} t}\) represents how quickly the concentration of \(\mathrm{HI}\) is changing at that exact moment, making it a snapshot of the reaction's speed at a specific point during its course.