Problem 20 - A Complete Resource Book in Chemistry for JEE Main

Question

Assume each reaction is carried out in an open container. For which reaction \(\Delta \mathrm{H}=\Delta \mathrm{U}\) ? (a) \(\mathrm{PCl}_{5}(\mathrm{~g}) \longrightarrow \mathrm{PCl}_{3}(\mathrm{~g})+\mathrm{Cl}_{2}(\mathrm{~g})\) (b) \(2 \mathrm{CO}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{~g}) \longrightarrow 2 \mathrm{CO}_{2}(\mathrm{~g})\) (c) \(\mathrm{H}_{2}(\mathrm{~g})+\mathrm{Br}_{2}(\mathrm{~g}) \longrightarrow 2 \mathrm{HBr}(\mathrm{g})\) (d) \(\mathrm{C}(\mathrm{s})+2 \mathrm{H}_{2} \mathrm{O}(\mathrm{g}) \longrightarrow 2 \mathrm{H}_{2}(\mathrm{~g})+\mathrm{CO}_{2}(\mathrm{~g})\)

Step-by-Step Solution

Answer

The reaction where ΔH = ΔU is (c) \( \mathrm{H}_{2} + \mathrm{Br}_{2} \rightarrow 2 \mathrm{HBr} \).

Step 1: Understanding the relationship between ΔH and ΔU

In an open system, the change in enthalpy (ΔH) can be related to the change in internal energy (ΔU) using the formula: \[ ΔH = ΔU + Δn_{g}RT \]where \( Δn_{g} \) is the change in moles of gas, \( R \) is the ideal gas constant, and \( T \) is the temperature in Kelvin.

Step 2: Determine Δnₙ (g) for each reaction

Calculate the change in moles of gas (\( Δn_{g} \)) for each reaction:- (a) \( Δn_{g} = (1 + 1) - 1 = 1 \)- (b) \( Δn_{g} = 2 - (2 + 1) = -1 \)- (c) \( Δn_{g} = 2 - (1 + 1) = 0 \)- (d) \( Δn_{g} = (2 + 1) - 2 = 1 \)

Step 3: Identify the reaction where ΔH = ΔU

For ΔH to equal ΔU, \( Δn_{g} \) must be zero because then \( Δn_{g}RT = 0 \), making the term vanish. Only reaction (c) has \( Δn_{g} = 0 \), implying that \( ΔH = ΔU \).

Key Concepts

Enthalpy

In thermodynamics, enthalpy, denoted as \( \Delta H \), is a measure of the total heat content in a system. It combines the internal energy and the effect of pressure and volume changes in the system. Enthalpy is particularly useful in chemical reactions, especially those occurring at constant pressure, which is common in an open container.

When calculating enthalpy changes, it's crucial to consider if it involves heat being absorbed (endothermic) or released (exothermic). The process involves comparing the energy states of products and reactants. A key equation in these calculations is

• \( \Delta H = \Delta U + \Delta n_{g}RT \)

where \( \Delta U \) is the change in internal energy, \( \Delta n_{g} \) represents the change in moles of gas (products minus reactants), and \( R \) is the universal gas constant.

For instance, if the products and reactants have the same amount of gas moles, like in reaction (c), the term \( \Delta n_{g}RT \) becomes zero, simplifying to \( \Delta H = \Delta U \). In this way, enthalpy serves as a pivotal concept in understanding energy changes.

Internal Energy

Internal energy, represented by \( \Delta U \), encompasses the energy within a system due to the internal motions of its molecules. This includes the kinetic and potential energy of the particles in the system. Unlike enthalpy, which hinges on external conditions like pressure and volume, internal energy focuses solely on changes within the system.

For chemical reactions, the change in internal energy can be associated with bonds breaking and forming. A stable reaction where no net energy interaction with the environment occurs might have its \( \Delta U \) closely aligned with \( \Delta H \).

Understanding \( \Delta U \) is crucial because it lays the foundation for comprehending reactions in isolated circumstances. If no work other than PV work is done and heat is exchanged with the surroundings, the enthalpy change and internal energy change can simplify their relationship, such as in open container scenarios where deviations are minimal if \( \Delta n_{g} \) is zero, ensuring \( \Delta H = \Delta U \).

Gas Laws

The gas laws form the backbone of understanding how gases behave under varying conditions of temperature, volume, and pressure. They are essential to explaining reactions, especially those involving gases in thermodynamic studies. The ideal gas law, stated as \( PV = nRT \), relates the pressure, volume, and temperature of a gas with its quantity in moles.

In reaction scenarios, changes in the number of moles of gas can significantly impact thermodynamic calculations. This is where terms like \( \Delta n_{g} \) become vital. For reactions happening in an open container—such as those in the original exercise—the change in moles of gas needs extra care since it directly affects the enthalpy calculation through the equation \( \Delta H = \Delta U + \Delta n_{g}RT \).

When examining gas reactions, it's beneficial to remember:

• Increase in gas moles leads to an increase in volume or pressure under constant conditions.
• Decrease in gas moles results in a decrease in these factors, influencing energy relationships.

By mastering the gas laws and related concepts, you can predict and calculate these changes more effectively, knowing how the reactant and product volumes affect overall energy dynamics.