Problem 7 - 43 Years JEE ADVANCED (1978-2020) + JEE MAIN Chapterwise & Topicwise Solved Papers Chemistry

Question

For which of the following reactions, \(\Delta \mathrm{H}\) is equal to \(\Delta \mathrm{U}\) ? [Main Online April 15, 2018 (I)] (a) \(\mathrm{N}_{2}(\mathrm{~g})+3 \mathrm{H}_{2}(\mathrm{~g}) \rightarrow 2 \mathrm{NH}_{3}(\mathrm{~g})\) (b) \(2 \mathrm{HI}(\mathrm{g}) \rightarrow \mathrm{H}_{2}(\mathrm{~g})+\mathrm{I}_{2}(\mathrm{~g})\) (c) \(2 \mathrm{SO}_{2}(\mathrm{~g})+\mathrm{O}_{2}(\mathrm{~g}) \rightarrow 2 \mathrm{SO}_{3}(\mathrm{~g})\) (d) \(2 \mathrm{NO}_{2}(\mathrm{~g}) \rightarrow \mathrm{N}_{2} \mathrm{O}_{4}(\mathrm{~g})\)

Step-by-Step Solution

Answer

Reaction (b) has \( \Delta \mathrm{H} = \Delta \mathrm{U} \).

Step 1: Understanding the Relationship between ΔH and ΔU

The relationship between enthalpy change (\( \Delta \mathrm{H} \)) and internal energy change (\( \Delta \mathrm{U} \)) in a chemical reaction is given by the equation \( \Delta \mathrm{H} = \Delta \mathrm{U} + \Delta nRT \), where \( \Delta n \) is the change in the number of moles of gas, \( R \) is the universal gas constant, and \( T \) is the temperature in Kelvin.

Step 2: Analyzing Reaction (a)

For the reaction \( \mathrm{N}_{2}( ext{g}) + 3 \mathrm{H}_{2}( ext{g}) \rightarrow 2 \mathrm{NH}_{3}( ext{g}) \), the number of moles of reactant gases is 4 (1 from \( \mathrm{N}_{2} \) and 3 from \( \mathrm{H}_{2} \)) and the number of moles of product gas is 2 (from \( 2 \mathrm{NH}_{3} \)). Therefore, \( \Delta n = 2 - 4 = -2 \). Since \( \Delta n eq 0 \), \( \Delta \mathrm{H} eq \Delta \mathrm{U} \).

Step 3: Analyzing Reaction (b)

For the reaction \( 2 \mathrm{HI}( ext{g}) \rightarrow \mathrm{H}_{2}( ext{g}) + \mathrm{I}_{2}( ext{g}) \), both reactants and products contain 2 moles of gases each. Thus, \( \Delta n = 2 - 2 = 0 \). This means \( \Delta \mathrm{H} = \Delta \mathrm{U} \) for this reaction.

Step 4: Analyzing Reaction (c)

For the reaction \( 2 \mathrm{SO}_{2}( ext{g}) + \mathrm{O}_{2}( ext{g}) \rightarrow 2 \mathrm{SO}_{3}( ext{g}) \), the number of moles of reactant gases is 3 (2 from \( \mathrm{SO}_{2} \) and 1 from \( \mathrm{O}_{2} \)) and the product gas is 2 moles (from \( 2 \mathrm{SO}_{3} \)). Therefore, \( \Delta n = 2 - 3 = -1 \). Since \( \Delta n eq 0 \), \( \Delta \mathrm{H} eq \Delta \mathrm{U} \).

Step 5: Analyzing Reaction (d)

For the reaction \( 2 \mathrm{NO}_{2}( ext{g}) \rightarrow \mathrm{N}_{2} \mathrm{O}_{4}( ext{g}) \), the number of moles is 2 for reactants and 1 for products. Therefore, \( \Delta n = 1 - 2 = -1 \). Since \( \Delta n eq 0 \), \( \Delta \mathrm{H} eq \Delta \mathrm{U} \).

Step 6: Conclusion

Through analyzing each reaction, it is determined that only reaction (b) meets the criterion where \( \Delta n = 0 \). Hence, \( \Delta \mathrm{H} \) is equal to \( \Delta \mathrm{U} \) for reaction (b).

Key Concepts

Thermodynamics in Chemistry

Thermodynamics plays a crucial role in understanding chemical reactions and energy transformations. In chemistry, thermodynamics helps us analyze how energy moves and transforms during reactions and phase changes. Two primary energy measures we often consider are internal energy (\( \Delta U \)) and enthalpy (\( \Delta H \)).

Internal energy refers to the total energy contained within a chemical system, which includes kinetic and potential energies at the atomic level. When a chemical reaction occurs, the internal energy may change, often through heat exchange or work done on the surroundings. Enthalpy, on the other hand, is a measure of the total heat content of a system. It's defined under constant pressure conditions, making it practically easier to measure than internal energy changes.

The relationship between enthalpy and internal energy is given by the expression\( \Delta H = \Delta U + \Delta nRT \),where\( \Delta n \)represents the change in the number of moles of gas, \( R \)is the universal gas constant, and\( T \)stands for temperature in Kelvin. This equation helps us to understand how heat is exchanged during reactions that occur at constant pressure, frequently encountered in chemical reactions.

Chemical Reactions

Chemical reactions are processes in which substances, called reactants, are transformed into different substances, known as products. Understanding the subtle energies involved in chemical transformations is vital for mastering thermodynamics and predicting reaction behavior.

Each reaction has its own characteristics and can significantly affect the enthalpy and internal energy of the system. For instance, reactions can be categorized based on their energetic processes such as endothermic and exothermic reactions.

- **Endothermic reactions** need energy intake from the surroundings and usually show a positive \( \Delta H \),meaning they gain heat.- **Exothermic reactions** release energy into the surroundings with a negative\( \Delta H \),signifying heat loss.

These changes in energy impact the temperature of the surroundings and can be quantified using the principles of thermodynamics. Analyzing each reaction implies calculating the changes in the number of gas moles to determine the relationship between \( \Delta H \)and\( \Delta U \).A key concept exemplified in the given exercise is the occurrence of \( \Delta H = \Delta U \),which happens only when there's no net change in the number of gaseous molecules.

Gas Laws

Gas laws are a set of rules that describe how gases behave, relating parameters like volume, temperature, and pressure under changing conditions. These laws are vital in understanding and applying thermodynamic principles to chemical reactions involving gaseous substances.

One of the gas laws important to thermodynamics is the ideal gas law, which states\( PV = nRT \),relating pressure (P), volume (V), number of moles (n), the universal gas constant (R), and temperature (T). These parameters can influence the enthalpy and internal energy of reactions—especially reactions involving gaseous reactants and products.

When analyzing the change in the number of gas moles in a reaction (\( \Delta n \)), the ideal gas law helps to understand how the volume and pressure might affect the enthalpy. By assessing \( \Delta n \),one can determine whether \( \Delta H \)or \( \Delta U \)is more favorable—where no change in the moles of gases (\( \Delta n = 0 \)) leads to\( \Delta H = \Delta U \).The exercise question reveals this concept by showing that for some reactions, particularly gaseous reactions where the quantity of gas stays constant, the enthalpy change equals the internal energy change. This parameter is essential in predicting the feasibility and energy distribution of chemical reactions.