Problem 158 - A Complete Resource Book in Chemistry for JEE Main

Question

Consider the reaction \(\mathrm{N}_{2}+3 \mathrm{H}_{2} \rightleftharpoons 2 \mathrm{NH}_{3}\) carried out at constant temperature and pressure. If \(\Delta \mathrm{H}\) and \(\Delta \mathrm{U}\) are the enthalpy and internal energy changes for the reaction, which of the following expressions is true? (a) \(\Delta \mathrm{H}=0\) (b) \(\Delta \mathrm{H}=\Delta \mathrm{U}\) (c) \(\Delta \mathrm{H}<\Delta \mathrm{U}\) (d) \(\Delta \mathrm{H}>\Delta \mathrm{U}\)

Step-by-Step Solution

Answer

(c) \( \Delta \mathrm{H} < \Delta \mathrm{U} \)

Step 1: Analyzing the Reaction

The given reaction is \( \mathrm{N}_{2}+3 \mathrm{H}_{2} \rightleftharpoons 2 \mathrm{NH}_{3} \). It involves 4 moles of gas on the reactant side (1 mole of \( \mathrm{N}_2 \) and 3 moles of \( \mathrm{H}_2 \)) and 2 moles of gas on the product side (2 moles of \( \mathrm{NH}_3 \)). This means the number of moles of gas decreases during the reaction.

Step 2: Understanding Relationship between \( \Delta U \) and \( \Delta H \)

The relationship between the change in enthalpy (\( \Delta \mathrm{H} \)) and the change in internal energy (\( \Delta \mathrm{U} \)) is given by the equation \( \Delta \mathrm{H} = \Delta \mathrm{U} + \Delta (PV) \). For reactions at constant pressure, \( \Delta (PV) \) can be associated with \( R\Delta nT \), where \( \Delta n \) is the change in moles of gas, \( R \) is the ideal gas constant, and \( T \) is the temperature.

Step 3: Calculating \( \Delta (PV) \)

The change in moles of gas (\( \Delta n \)) is calculated as the moles of gaseous products minus the moles of gaseous reactants. In this reaction, \( \Delta n = 2 - 4 = -2 \). The \( \Delta (PV) \) term becomes \( -2RT \). Hence, \( \Delta \mathrm{H} = \Delta \mathrm{U} - 2RT \).

Step 4: Comparing \( \Delta H \) and \( \Delta U \)

Since \( \Delta \mathrm{H} = \Delta \mathrm{U} - 2RT \) and \( -2RT \) is a negative term, \( \Delta \mathrm{H} < \Delta \mathrm{U} \). This means the change in enthalpy is less than the change in internal energy.

Key Concepts

Chemical Reactions

Chemical reactions are processes in which substances, known as reactants, are transformed into different substances, known as products. During a chemical reaction, bonds between atoms in the reactants are broken and new bonds are formed to create the products. This often involves energy changes, as breaking bonds requires energy and forming bonds releases energy.
In the reaction \( \mathrm{N}_{2}+3 \mathrm{H}_{2} \rightleftharpoons 2 \mathrm{NH}_{3} \), nitrogen and hydrogen gases react to form ammonia. There is a decrease in the total number of gas molecules, going from four moles of gas on the reactant side to two moles on the product side. This indicates a change in the internal conditions of the gases, influencing pressure and volume in the system.

• Reactants: Substances consumed in the reaction.
• Products: Substances produced in the reaction.
• Molecular Changes: Corresponding to changes in the physical state, energy content, or temperature.

Understanding how reactants turn into products helps in predicting the direction and feasibility of the chemical process, particularly in industrial applications involving gas reactions.

Enthalpy Change

Enthalpy change, denoted as \( \Delta \mathrm{H} \), is a measure of the heat absorbed or released during a reaction at constant pressure. It is a crucial concept in thermodynamics, particularly for reactions involving gases.
For the reaction \( \mathrm{N}_{2}+3 \mathrm{H}_{2} \rightarrow 2 \mathrm{NH}_{3} \), the enthalpy change indicates the heat transfer associated with forming ammonia from nitrogen and hydrogen. Since this reaction results in fewer moles of gas, it often releases excess energy, making it exothermic. Exothermic reactions have a negative \( \Delta \mathrm{H} \), reflecting the system losing energy to the surroundings.

• \( \Delta \mathrm{H} < 0 \) for exothermic reactions (heat is released).
• \( \Delta \mathrm{H} > 0 \) for endothermic reactions (heat is absorbed).

Knowing \( \Delta \mathrm{H} \) aids in predicting how much energy the reaction can offer, which is valuable for controlling industrial applications where energy efficiency and sustainability are critical.

Internal Energy Change

Internal energy change, symbolized as \( \Delta \mathrm{U} \), refers to the difference in the total internal energy of a system before and after a reaction. It encompasses changes in kinetic and potential energy associated with molecular motion and interactions.
In the equation \( \Delta \mathrm{H} = \Delta \mathrm{U} + \Delta (PV) \), the internal energy change is the part of the energy transformation that occurs without any work done by or against external pressure (when \( \Delta (PV) = 0 \)). However, in our ammonia-producing reaction, \( \Delta n = -2 \) (a reduction in moles of gas), so \( \Delta (PV) = -2RT \), indicating energy removal from the system as pressure-volume work.

• \( \Delta \mathrm{U} \) reflects inherent energy transformations.
• \( \Delta (PV) \) involves changes due to pressure and volume work.
• The relationship between \( \Delta \mathrm{H} \) and \( \Delta \mathrm{U} \) is contextual.

This concept is pivotal for thermochemical equations in understanding how energy conversions sustain or inhibit chemical processes, essential for optimized reaction conditions in various environments.