Problem 144 - A Complete Resource Book in Chemistry for JEE Main

Question

If a mixture of 3 mole of \(\mathrm{H}_{2}\) and 1 mole of \(\mathrm{N}_{2}\) is completely converted into \(\mathrm{NH}_{3}\), what would be the ratio of the initial and final volume at same temperature and pressure?

Step-by-Step Solution

Answer

The ratio of initial to final volume is 2:1.

Step 1: Identify the Reaction

The chemical reaction involved in this exercise is the formation of ammonia (\(\text{NH}_{3}\)) from hydrogen (\(\text{H}_{2}\)) and nitrogen (\(\text{N}_{2}\)). The balanced chemical equation for the Haber process is:\[\text{N}_{2} + 3\text{H}_{2} \rightarrow 2\text{NH}_{3}.\]

Step 2: Determine Initial Moles

Initially, we have 3 moles of \(\text{H}_{2}\) and 1 mole of \(\text{N}_{2}\). Therefore, the total initial moles of gas are:\[3 + 1 = 4 \, ext{moles}.\]

Step 3: Determine Final Moles

According to the balanced equation, 3 moles of \(\text{H}_{2}\) and 1 mole of \(\text{N}_{2}\) produce 2 moles of \(\text{NH}_{3}\). Therefore, after the reaction, there are 2 moles of gas.

Step 4: Calculate Volume Ratio

Since the reaction happens under the same temperature and pressure conditions, the volume ratio is equal to the mole ratio due to the ideal gas law (\(PV = nRT\)). The initial volume:\[V_i = n_i = 4\]The final volume:\[V_f = n_f = 2\]The ratio of initial to final volume is:\[\frac{V_i}{V_f} = \frac{4}{2} = 2:1.\]

Step 5: Verification of Conditions

It's important to verify that the conditions provided are consistent with using the ideal gas law, which states that gases at the same temperature and pressure will occupy volumes directly proportional to the number of moles.

Key Concepts

Ammonia Synthesis

Ammonia synthesis, a pivotal industrial process, revolves around the creation of ammonia (\(\text{NH}_3\)) from nitrogen (\(\text{N}_2\)) and hydrogen (\(\text{H}_2\)) gases. This reaction is known as the Haber process. It plays a critical role in the production of fertilizers, which are essential for modern agriculture. The reaction equation for this process is:

• \( \text{N}_2 + 3\text{H}_2 \rightarrow 2\text{NH}_3 \)

By carefully balancing this equation, we see how one mole of nitrogen reacts with three moles of hydrogen to produce two moles of ammonia. The Haber process requires high temperature and pressure to favor the forward reaction and maximize the yield of ammonia.
While this process is efficient, managing the conditions ensures that the reaction is both economical and safe. One must also consider catalysts, which play a vital role in increasing the reaction rate without being consumed.

Mole Concept

The mole concept is a fundamental chemical concept that helps in quantifying the amount of a substance. When working with gases like in the Haber process, it is crucial to understand how moles relate to chemical equations.

• 1 mole is equal to Avogadro's number, which is \(6.022 \times 10^{23}\) entities.
• Moles allow chemists to predict products and reactants in a reaction.

In the exercise, we start with 3 moles of hydrogen and 1 mole of nitrogen. This setup shows a typical mole ratio in the Haber process. By using 4 moles initially (3 from \(\text{H}_2\) and 1 from \(\text{N}_2\)), we can predict the final amount of ammonia produced. The mole concept simplifies the understanding of reactions by letting us focus on these basic and whole number ratios.

Ideal Gas Law

The ideal gas law is a powerful tool in chemistry and physics, often represented by the equation \(PV = nRT\). It relates

• \(P\) (pressure),
• \(V\) (volume),
• \(n\) (number of moles),
• \(R\) (ideal gas constant), and
• \(T\) (temperature)

for an ideal gas.
In the context of the given exercise, the ideal gas law explains why the volume changes correspond directly to changes in mole numbers when temperature and pressure are constant. The initial 4 moles of \(\text{N}_2\) and \(\text{H}_2\) result in a specific volume, while the final 2 moles of \(\text{NH}_3\) occupy half of this volume due to this direct relationship. This clear connection helps students understand how gases behave under ideal conditions, assuming no interactions between gas molecules.
It is a foundational concept for understanding not just the Haber process but many other phenomena in chemistry.