Problem 106 - A Complete Resource Book in Chemistry for JEE Main

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 \(=-\mathrm{d}\left[\mathrm{N}_{2}\right] / \mathrm{dt}=-\mathrm{d}\left[\mathrm{H}_{2}\right] / \mathrm{dt}=\mathrm{d}\left[\mathrm{NH}_{3}\right] / \mathrm{dt}\)

Step-by-Step Solution

Answer

Option (a) correctly represents the rate expressions.

Step 1: Understanding Rate Expression

For a chemical reaction, the rate can be expressed as the change in concentration of reactants or products over time. In this reaction, we have reactants \(\mathrm{N}_{2}\) and \(-3 \, \mathrm{H}_{2}\), and product \(2 \, \mathrm{NH}_{3}\). The rate of depletion of reactants is negative since their concentration decreases, whereas the formation of products is positive.

Step 2: Analyzing Stoichiometry

From the balanced equation, 1 mole of \(\mathrm{N}_{2}\) reacts with 3 moles of \(\mathrm{H}_{2}\) to produce 2 moles of \(\mathrm{NH}_{3}\). This stoichiometry is crucial in setting up the people rate expression as it dictates the relationship between the concentration changes of the substances.

Step 3: Deriving Rate Relationships

The rate expression for \(\mathrm{N}_{2}\) is \(-\frac{d[\mathrm{N}_{2}]}{dt}\), for \(\mathrm{H}_{2}\) is \(-\frac{1}{3}\frac{d[\mathrm{H}_{2}]}{dt}\), and for \(\mathrm{NH}_{3}\) is \(\frac{1}{2}\frac{d[\mathrm{NH}_{3}]}{dt}\). The factor of \(\frac{1}{3}\) for \(\mathrm{H}_{2}\) corresponds to its stoichiometric coefficient, and \(\mathrm{NH}_{3}\) is divided by 2 for similar reasons.

Step 4: Checking Possible Answers

Now we compare the options to our derived rate relationships. According to stoichiometry, option (a) correctly describes \(-\frac{d[\mathrm{N}_{2}]}{dt} = -\frac{1}{3}\frac{d[\mathrm{H}_{2}]}{dt} = \frac{1}{2}\frac{d[\mathrm{NH}_{3}]}{dt}\). This confirms that option (a) is the correct rate expression relationship.

Key Concepts

Rate Expression

In chemical kinetics, the rate expression of a reaction is a fascinating way to observe how fast reactants are transformed into products. For the given reaction, \[\mathrm{N}_2(\mathrm{~g}) + 3\mathrm{H}_2(\mathrm{~g}) \to 2\mathrm{NH}_3(\mathrm{~g})\], the reaction rate can be expressed in terms of the change in concentration of either the reactants or the products over time.

• The rate for a reactant like \(\mathrm{N}_2\) or \(\mathrm{H}_2\) will be negative because it is being consumed.
• Conversely, for a product like \(\mathrm{NH}_3\), it will be positive as it is being formed.

Thus, the rate of reaction can be expressed as:\[-\frac{d[\mathrm{N}_2]}{dt} = -\frac{1}{3}\frac{d[\mathrm{H}_2]}{dt} = \frac{1}{2}\frac{d[\mathrm{NH}_3]}{dt}\]. This expression communicates the relative changes in concentrations based on stoichiometric coefficients of the balanced reaction.

Stoichiometry

Stoichiometry gives us a window into understanding how proportions of reactants and products drive a chemical reaction. In the reaction discussed, 1 mole of nitrogen gas reacts with 3 moles of hydrogen gas to form 2 moles of ammonia. This stoichiometric relationship helps us to equate the rates of change in concentration across different species in the reaction.

• The formula shows that for every 1 molecule of \(\mathrm{N}_2\) consumed, 3 molecules of \(\mathrm{H}_2\) are used and 2 molecules of \(\mathrm{NH}_3\) are formed.
• We can use these ratios to understand the mole-to-mole transformation making it possible to write a rate expression.

By setting up the reaction in this clear stoichiometric framework, it becomes easier to determine the rates of each chemical's usage with distinct coefficients (like \(-\frac{d[\mathrm{H}_2]}{dt}\) being divided by 3).

Chemical Kinetics

Chemical kinetics is the branch of physical chemistry that studies the associated rates of chemical processes. Understanding this helps us explain how reactions occur over time. Beyond simply identifying reactants and products, chemical kinetics allow us to delve into how concentration, temperature, surface area, and catalysts influence the speed of chemical reactions. For our specific reaction, \[\mathrm{N}_2(\mathrm{~g}) + 3\mathrm{H}_2(\mathrm{~g}) \to 2\mathrm{NH}_3(\mathrm{~g})\],

• The rate of a chemical reaction quantifies the speed of this transformation.
• These rates are crucial for applications in industry and pharmaceuticals where controlling the speed of reactions will affect yield and efficiency.

So by employing principles of chemical kinetics, we get a comprehensive view on how adjusting different parameters affect the progress of the reaction, allowing us to optimize it for practical use.