Problem 110 - A Complete Resource Book in Chemistry for JEE Main

Question

Which one of the following equations is correct for the reaction \(\mathrm{N}_{2}(\mathrm{~g})+3 \mathrm{H}_{2}(\mathrm{~g}) \longrightarrow 2 \mathrm{NH}_{3}(\mathrm{~g}) ?\) (a) \(3 \frac{\mathrm{d}\left[\mathrm{NH}_{3}\right]}{\mathrm{dt}}=2 \frac{\mathrm{d}\left[\mathrm{H}_{2}\right]}{\mathrm{dt}}\) (b) \(2 \frac{\mathrm{d}\left[\mathrm{NH}_{3}\right]}{\mathrm{dt}}=-3 \frac{\mathrm{d}\left[\mathrm{H}_{3}\right]}{\mathrm{dt}}\) (c) \(2 \frac{\mathrm{d}\left[\mathrm{NH}_{1}\right]}{\mathrm{dt}}=\frac{\mathrm{d}\left[\mathrm{H}_{2}\right]}{\mathrm{dt}}\) (d) \(3 \frac{\mathrm{d}\left[\mathrm{NH}_{3}\right]}{\mathrm{dt}}=-2 \frac{\mathrm{d}\left[\mathrm{H}_{2}\right]}{\mathrm{dt}}\)

Step-by-Step Solution

Answer

Option (d) is correct: \(3 \frac{\mathrm{d}[\mathrm{NH}_{3}]}{\mathrm{dt}}=-2 \frac{\mathrm{d}[\mathrm{H}_{2}]}{\mathrm{dt}}\).

Step 1: Understanding the Reaction

We begin by analyzing the chemical reaction given: \( \mathrm{N}_{2}(\mathrm{~g}) + 3 \mathrm{H}_{2}(\mathrm{~g}) \rightarrow 2 \mathrm{NH}_{3}(\mathrm{~g}) \). This reaction involves the formation of ammonia (\( \mathrm{NH}_{3} \)) from nitrogen (\( \mathrm{N}_{2} \)) and hydrogen (\( \mathrm{H}_{2} \)), with 1 mole of nitrogen reacting with 3 moles of hydrogen to produce 2 moles of ammonia.

Step 2: Rate of Reaction Expression

The rate of reaction can be expressed in terms of the change in concentration of reactants or products per unit time. For the reaction \( \mathrm{N}_{2} + 3 \mathrm{H}_{2} \rightarrow 2 \mathrm{NH}_{3} \), the rate can be expressed as follows:- \( -\frac{1}{1}\frac{\mathrm{d}[\mathrm{N}_{2}]}{\mathrm{dt}} \)- \( -\frac{1}{3}\frac{\mathrm{d}[\mathrm{H}_{2}]}{\mathrm{dt}} \)- \( \frac{1}{2}\frac{\mathrm{d}[\mathrm{NH}_{3}]}{\mathrm{dt}} \)These rates should be related by the stoichiometry of the equation.

Step 3: Applying the Stoichiometry

Using stoichiometry from the balanced equation, the rate of formation of \( \mathrm{NH}_{3} \) is related to the rate of consumption of \( \mathrm{H}_{2} \). For each 3 moles of \( \mathrm{H}_{2} \) consumed, 2 moles of \( \mathrm{NH}_{3} \) are formed. Thus, the rates will relate as:- \( \frac{\mathrm{d}[\mathrm{H}_{2}]}{\mathrm{dt}} = \frac{-3}{2} \frac{\mathrm{d}[\mathrm{NH}_{3}]}{\mathrm{dt}} \)Rearranging gives the opposite relationship: \( 3 \frac{\mathrm{d}[\mathrm{NH}_{3}]}{\mathrm{dt}} = -2 \frac{\mathrm{d}[\mathrm{H}_{2}]}{\mathrm{dt}} \).

Step 4: Matching with Options

Given this relationship \( 3 \frac{\mathrm{d}[\mathrm{NH}_{3}]}{\mathrm{dt}} = -2 \frac{\mathrm{d}[\mathrm{H}_{2}]}{\mathrm{dt}} \), we can see which option matches. By comparing it with the provided options:- Option (a): Does not have a negative sign and the values are inverted.- Option (b): Has incorrect molecule and signs are incorrect.- Option (c): Incorrect substances and values.- Option (d): Matches perfectly, \( 3 \frac{\mathrm{d}[\mathrm{NH}_{3}]}{\mathrm{dt}} = -2 \frac{\mathrm{d}[\mathrm{H}_{2}]}{\mathrm{dt}} \).

Step 5: Conclusion

Option (d) is the correct equation as it accurately reflects the stoichiometric and rate relationships of the given reaction.

Key Concepts

Rate of Reaction

The rate of a chemical reaction is a measure of how quickly reactants are converted into products. In general, it is determined by observing the change in concentration of reactants and products over time.
A higher rate indicates that the reaction proceeds more rapidly, while a lower rate suggests a slower process. To describe this rate quantitatively, we use the formula:\[\text{Rate of reaction} = -\frac{\Delta [\text{Reactant}]}{\Delta t} = \frac{\Delta [\text{Product}]}{\Delta t}\]This formula shows that the rate can be expressed as the decrease in reactant concentration or the increase in product concentration per unit time. The negative sign indicates the consumption of reactants.
Different reactions have different rates and understanding these can help in controlling chemical processes efficiently.

Stoichiometry

Stoichiometry is a fundamental concept in chemistry that describes the quantitative relationship between reactants and products in a balanced chemical equation. It allows us to predict the amount of products formed from given amounts of reactants, based on the mole ratios derived from the coefficients in a balanced equation.
For example, in the reaction:\[ \mathrm{N}_{2} + 3 \mathrm{H}_{2} \rightarrow 2 \mathrm{NH}_{3} \]The stoichiometry of the reaction indicates that 1 mole of nitrogen reacts with 3 moles of hydrogen to produce 2 moles of ammonia. This means for every mole of nitrogen used, three moles of hydrogen are required, producing 2 moles of ammonia.
Understanding stoichiometry is essential in calculating the rate of reaction and ensuring that chemical processes are efficient and yield the desired outputs.

Balanced Chemical Equation

A balanced chemical equation is essential for understanding the stoichiometry and kinetics of a reaction. It shows equal numbers of each type of atom on both sides of the equation, which reflects the law of conservation of mass.
In the balanced equation for the formation of ammonia:\[ \mathrm{N}_{2} + 3 \mathrm{H}_{2} \rightarrow 2 \mathrm{NH}_{3} \]We see that there are the same number of each type of atom on both sides: 2 nitrogen atoms and 6 hydrogen atoms. This balance is crucial when calculating the rate of reaction, as it ensures that all reactants and products are accounted for correctly.
By using balanced equations, chemists can ensure that chemical reactions are efficient and predictable, minimizing waste and optimizing resource use.