Question

Consider the chemical reaction, \(\mathrm{N}_{2}(\mathrm{~g})+3 \mathrm{H}_{2}(\mathrm{~g}) \longrightarrow 2 \mathrm{NH}_{3}(\mathrm{~g})\) The rate of this reaction can be expressed in terms of time derivatives of concentration of \(\mathrm{N}_{2}(\mathrm{~g}), \mathrm{H}_{2}(\mathrm{~g})\) or \(\mathrm{NH}_{3}(\mathrm{~g})\). Identify the correct relationship amongst the rate expressions. (a) rate \(=-\mathrm{d}\left[\mathrm{N}_{2}\right] / \mathrm{dt}=-1 / 3 \mathrm{~d}\left[\mathrm{H}_{2}\right] / \mathrm{dt}=\mathrm{d}\left[\mathrm{NH}_{3}\right] / \mathrm{dt}\) (b) rate \(=-\mathrm{d}\left[\mathrm{N}_{2}\right] \mathrm{dt}=-3 \mathrm{~d}\left[\mathrm{H}_{2}\right] / \mathrm{dt}=2 \mathrm{~d}\left[\mathrm{NH}_{3}\right] / \mathrm{dt}\) (c) rate \(=-\mathrm{d}\left[\mathrm{N}_{2}\right] / \mathrm{dt}=-1 / 3 \mathrm{~d}\left[\mathrm{H}_{2}\right] / \mathrm{dt}=2 \mathrm{~d}\left[\mathrm{NH}_{3}\right] / \mathrm{dt}\) (d) rate \(=-\mathrm{d}\left[\mathrm{N}_{2}\right] / \mathrm{dt}=-\mathrm{d}\left[\mathrm{H}_{2}\right] / \mathrm{dt}=\mathrm{d}\left[\mathrm{NH}_{3}\right] / \mathrm{dt}\)

Short answer

The correct answer is option (c).

Step-by-step solution

Understanding Reaction Rate

The rate of a chemical reaction is determined by the change in concentration of reactants or products over time. It is common to express reaction rates in terms of derivatives of concentration.

Analyzing Stoichiometry

For the given reaction \( \mathrm{N}_{2}(\mathrm{~g})+3 \mathrm{H}_{2}(\mathrm{~g}) \longrightarrow 2 \mathrm{NH}_{3}(\mathrm{~g}) \), the stoichiometric coefficients are 1 for \( \mathrm{N}_{2} \), 3 for \( \mathrm{H}_{2} \), and 2 for \( \mathrm{NH}_{3} \). These coefficients determine the relationship among the rate expressions.

Writing Rate Expressions

The rate of disappearance of \( \mathrm{N}_{2} \) is \(-\frac{d[\mathrm{N}_{2}]}{dt}\), for \( \mathrm{H}_{2} \) it is \(-\frac{1}{3}\frac{d[\mathrm{H}_{2}]}{dt}\), and for \( \mathrm{NH}_{3} \), it is \(\frac{1}{2}\frac{d[\mathrm{NH}_{3}]}{dt}\).

Relating Rate Expressions

From stoichiometry, the rate should satisfy: \(-\frac{d[\mathrm{N}_{2}]}{dt} = -\frac{1}{3}\frac{d[\mathrm{H}_{2}]}{dt} = \frac{1}{2}\frac{d[\mathrm{NH}_{3}]}{dt}\).

Identifying Correct Option

Comparing our findings with the given options, the correct relationship is provided in option (c): rate \(=-\frac{d[\mathrm{N}_{2}]}{dt}=-\frac{1}{3}\frac{d[\mathrm{H}_{2}]}{dt}=2\frac{d[\mathrm{NH}_{3}]}{dt}\).

Key concepts

Reaction Stoichiometry

In the realm of chemistry, reaction stoichiometry plays a vital role in understanding how reactants turn into products. Simply put, it describes the quantitative relationships between substances participating in a reaction. For example, consider the reaction \( \mathrm{N}_{2}(\mathrm{~g})+3 \mathrm{H}_{2}(\mathrm{~g}) \longrightarrow 2 \mathrm{NH}_{3}(\mathrm{~g}) \).
This equation tells us that one molecule of nitrogen gas reacts with three molecules of hydrogen gas to form two molecules of ammonia gas.

• Balancing equations is crucial in stoichiometry, as it ensures that the law of conservation of mass is satisfied.
• Stoichiometric coefficients, like 1 for \( \mathrm{N}_{2} \) and 3 for \( \mathrm{H}_{2} \), reflect the proportions in which substances react and form.

Mastering stoichiometry provides the foundational tools for predicting how much product forms when given certain amounts of reactants, and it lays the groundwork for understanding reaction rates.

Rate Expressions

To delve into the speed of chemical reactions, we use rate expressions. These expressions link the change in concentration of reactants or products with time. This is framed using calculus, typically as derivatives.
Let's explore the given reaction again: \( \mathrm{N}_{2}(\mathrm{~g}) + 3 \mathrm{H}_{2}(\mathrm{~g}) \longrightarrow 2 \mathrm{NH}_{3}(\mathrm{~g}) \). Here, rate expressions would illustrate how fast \( \mathrm{N}_{2} \) and \( \mathrm{H}_{2} \) are consumed and how fast \( \mathrm{NH}_{3} \) is produced.

• The rate of disappearance of a reactant can be represented as a negative derivative, reflecting a decrease in concentration.
• The rate of appearance of a product is represented as a positive derivative.

Using such expressions helps in quantifying reaction kinetics and understanding the dynamics within chemical reactions.

Stoichiometric Coefficients

Stoichiometric coefficients are integral components of balanced chemical equations. They indicate the proportions of reactants and products involved in the reaction, acting as multipliers for molecules or moles. For example, the equation \( \mathrm{N}_{2}(\mathrm{~g}) + 3 \mathrm{H}_{2}(\mathrm{~g}) \longrightarrow 2 \mathrm{NH}_{3}(\mathrm{~g}) \) highlights these coefficients:

• 1 for \( \mathrm{N}_{2} \)
• 3 for \( \mathrm{H}_{2} \)
• 2 for \( \mathrm{NH}_{3} \)

These numbers are crucial when calculating the rate of reactions, as they often help determine the relative rate at which each reactant is consumed or each product is formed. In rate expressions, the coefficients can influence the mathematical relationship among the different derivatives of concentration.

Concentration Derivatives

The concept of concentration derivatives is central to understanding how reaction rates are expressed mathematically. A derivative in this context measures how the concentration of a substance in a reaction changes over time. For instance, in the reaction \( \mathrm{N}_{2}(\mathrm{~g}) + 3 \mathrm{H}_{2}(\mathrm{~g}) \longrightarrow 2 \mathrm{NH}_{3}(\mathrm{~g}) \), concentration derivatives indicate:

• How quickly \( \mathrm{N}_{2} \) is being consumed, represented as \(-\frac{d[\mathrm{N}_{2}]}{dt}\).
• How quickly \( \mathrm{H}_{2} \) is being used up, represented as \(-\frac{1}{3}\frac{d[\mathrm{H}_{2}]}{dt}\), taking its stoichiometric coefficient into account.
• The formation of \( \mathrm{NH}_{3} \), represented by \(\frac{1}{2}\frac{d[\mathrm{NH}_{3}]}{dt}\).

By understanding these derivatives, one can precisely identify the relationship among the rates at which reactants disappear and products form, providing deeper insights into the kinetics of the reaction.