Problem 30 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

Consider the chemical reaction, \(\mathrm{N}_{2}(\mathrm{~g})+3 \mathrm{H}_{2}(\mathrm{~g}) \rightarrow 2 \mathrm{NH}_{3}(\mathrm{~g}) .\) The rate of this reaction can be expressed in terms of time derivative 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. [2002S](a) Rate \(=-\mathrm{d}\left[\mathrm{N}_{2}\right] / \mathrm{d} t=-1 / 3 \mathrm{~d}\left[\mathrm{H}_{2}\right] / \mathrm{d} t=1 / 2 \mathrm{~d}\left[\mathrm{NH}_{3}\right] / \mathrm{d} t\) (b) Rate \(=-\mathrm{d}\left[\mathrm{N}_{2}\right] / \mathrm{d} t=-3 \mathrm{~d}\left[\mathrm{H}_{2}\right] / \mathrm{d} t=2 \mathrm{~d}\left[\mathrm{NH}_{3}\right] / \mathrm{d} t\) (c) Rate \(=\mathrm{d}\left[\mathrm{N}_{2}\right] / \mathrm{d} t=1 / 3 \mathrm{~d}\left[\mathrm{H}_{2}\right] / \mathrm{d} t=1 / 2 \mathrm{~d}\left[\mathrm{NH}_{3}\right] / \mathrm{d} t\) (d) Rate \(=-\mathrm{d}\left[\mathrm{N}_{2}\right] / \mathrm{d} t=-\mathrm{d}\left[\mathrm{H}_{2}\right] / \mathrm{d} t=\mathrm{d}\left[\mathrm{NH}_{3}\right] / \mathrm{d} t\)

Step-by-Step Solution

Answer

Correct rate expression is option (a): \(-\frac{d[\mathrm{N}_{2}]}{dt} = -\frac{1}{3}\frac{d[\mathrm{H}_{2}]}{dt} = \frac{1}{2}\frac{d[\mathrm{NH}_{3}]}{dt}\).

Step 1: Understand the Chemical Reaction

The given chemical reaction is \(\mathrm{N}_{2}(\mathrm{~g})+3 \mathrm{H}_{2}(\mathrm{~g}) \rightarrow 2 \mathrm{NH}_{3}(\mathrm{~g})\). In this reaction, one mole of nitrogen gas reacts with three moles of hydrogen gas to produce two moles of ammonia gas. Each substance has a specific stoichiometric coefficient: 1 for \(\mathrm{N}_{2}\), 3 for \(\mathrm{H}_{2}\), and 2 for \(\mathrm{NH}_{3}\).

Step 2: Define Rate of Reaction

The rate of reaction is defined as the change in concentration over time. For a reaction \(aA + bB \rightarrow cC\), the general rate can be expressed as: \(-\frac{1}{a} \frac{d[A]}{dt} = -\frac{1}{b} \frac{d[B]}{dt} = \frac{1}{c} \frac{d[C]}{dt}\). This ensures that the rate is consistent for each reactant and product, accounting for their stoichiometry.

Step 3: Apply Rate Law to Given Reaction

For \(\mathrm{N}_{2} + 3\mathrm{H}_{2} \rightarrow 2\mathrm{NH}_{3}\), the rate expressions using stoichiometry are: \(-\frac{d[\mathrm{N}_{2}]}{dt}\), \(-\frac{1}{3}\frac{d[\mathrm{H}_{2}]}{dt}\), and \(\frac{1}{2}\frac{d[\mathrm{NH}_{3}]}{dt}\). The stoichiometric coefficients are 1, 3, and 2 for \(\mathrm{N}_{2}\), \(\mathrm{H}_{2}\), and \(\mathrm{NH}_{3}\), respectively.

Step 4: Identify Correct Rate Expression

By comparing the options provided, we look for the expression that matches our derived rate law. The rate must equal \(-\frac{d[\mathrm{N}_{2}]}{dt} = -\frac{1}{3}\frac{d[\mathrm{H}_{2}]}{dt} = \frac{1}{2}\frac{d[\mathrm{NH}_{3}]}{dt}\). Option (a) fits this description. Therefore, the rate expression for the reaction is correctly identified as:\(=-\mathrm{d}\left[\mathrm{N}_{2}\right] / \mathrm{d} t=-1 / 3 \mathrm{~d}\left[\mathrm{H}_{2}\right] / \mathrm{d} t=1 / 2 \mathrm{~d}\left[\mathrm{NH}_{3}\right] / \mathrm{d} t\).

Key Concepts

Rate of Reaction

The rate of reaction is a fundamental concept in chemical kinetics, which refers to how fast or slow a reaction occurs. It is typically measured as the change in concentration of a reactant or product per unit time. This change is most often expressed in moles per liter per second (mol/L/s). When we discuss the rate of reaction, it is crucial to consider the direction of the reaction. For a reactant, which is consumed during the reaction, the rate is indicated by a negative sign to reflect the decrease in concentration over time. Conversely, for a product being formed, the rate is positive, showcasing the increase in concentration.

In practical terms, determining the rate of reaction helps chemists understand the speed of reactions and the conditions needed to optimize these reactions in industrial applications, such as in the production of ammonia through the Haber process, a real-world application of our given reaction.

• For reactants, the rate is negative: \(-\frac{d[R]}{dt}\)
• For products, the rate is positive: \(\frac{d[P]}{dt}\)

Stoichiometry

Stoichiometry is the branch of chemistry that deals with the quantitative relationships between the reactants and products in a chemical reaction. In our example, the balanced chemical equation: \(\mathrm{N}_{2} + 3 \mathrm{H}_{2} \rightarrow 2 \mathrm{NH}_{3}\), exhibits strict stoichiometric coefficients. These coefficients indicate the relative amounts of each substance involved in the reaction.

The stoichiometric coefficients are crucial for ensuring a balanced reaction. They ensure that the law of conservation of mass is adhered to, which states that mass is neither created nor destroyed in a chemical reaction. This is why, for every mole of \(\mathrm{N}_{2}\) consumed, three moles of \(\mathrm{H}_{2}\) are also consumed, resulting in the formation of two moles of \(\mathrm{NH}_{3}\).

• The coefficient 1 for \(\mathrm{N}_{2}\)
• The coefficient 3 for \(\mathrm{H}_{2}\) means three times as much \(\mathrm{H}_{2}\) is consumed
• The coefficient 2 for \(\mathrm{NH}_{3}\) means two moles are produced

This relationship is essential for calculating how much of each reactant is needed and how much product can be expected.

Reaction Rate Expressions

Reaction rate expressions provide a mathematical representation of the rate of a chemical reaction in terms of the concentration of reactants and products. These expressions involve derivatives, symbolizing the rate of change in concentration with time. For instance, in our reaction: \(\mathrm{N}_{2} + 3\mathrm{H}_{2} \rightarrow 2\mathrm{NH}_{3}\), we derive expressions for each component.

The general form of the rate expression for a reaction of the type: \(aA + bB \rightarrow cC\) is given by:
\(-\frac{1}{a} \frac{d[A]}{dt} = -\frac{1}{b} \frac{d[B]}{dt} = \frac{1}{c} \frac{d[C]}{dt}\).

• For \(\mathrm{N}_{2}\), the expression is \(-\frac{d[\mathrm{N}_{2}]}{dt}\)
• For \(\mathrm{H}_{2}\), it is \(-\frac{1}{3}\frac{d[\mathrm{H}_{2}]}{dt}\)
• For \(\mathrm{NH}_{3}\), it is \(\frac{1}{2}\frac{d[\mathrm{NH}_{3}]}{dt}\)

These rate expressions help chemists compare the rates at which different substances are consumed or produced, essential for understanding the kinetics of a reaction and facilitating the design of industrial processes.