Problem 120 - A Complete Resource Book in Chemistry for JEE Main

Question

For which of the following reactions, is \(\Delta \mathrm{H}\) equal to \(\Delta \mathrm{E} ?\) (a) \(\mathrm{H}_{2}(\mathrm{~g})+\mathrm{I}_{2}(\mathrm{~g}) \rightarrow 2 \mathrm{HI}(\mathrm{g})\) (b) \(\mathrm{PCl}_{5}(\mathrm{~g}) \rightarrow \mathrm{PCl}_{3}(\mathrm{~g})+\mathrm{Cl}_{2}(\mathrm{~g})\) (c) \(2 \mathrm{H}_{2} \mathrm{O}_{2}(\mathrm{~g}) \rightarrow 2 \mathrm{H}_{2} \mathrm{O}(\mathrm{g})+\mathrm{O}_{2}(\mathrm{~g})\) (d) \(\mathrm{C}(\mathrm{s})+\mathrm{O}_{2}(\mathrm{~g}) \rightarrow \mathrm{CO}_{2}(\mathrm{~g})\)

Step-by-Step Solution

Answer

Reactions (a) and (d) have \( \Delta H = \Delta E \).

Step 1: Understand the Relationship Between ΔH and ΔE

The enthalpy change, ΔH, is related to the internal energy change, ΔE, by the equation \( \Delta H = \Delta E + \Delta n_g RT \), where \( \Delta n_g \) is the change in the number of moles of gas, R is the gas constant, and T is the temperature in Kelvin.

Step 2: Analyze Reaction (a)

For \( \mathrm{H}_{2}( ext{g})+\mathrm{I}_{2}( ext{g}) \rightarrow 2 \mathrm{HI}( ext{g}) \), the change in the number of moles of gas, \( \Delta n_g = (2) - (1+1) = 0 \). Thus, \( \Delta H = \Delta E \).

Step 3: Analyze Reaction (b)

For \( \mathrm{PCl}_{5}( ext{g}) \rightarrow \mathrm{PCl}_{3}( ext{g})+\mathrm{Cl}_{2}( ext{g}) \), the change in the number of moles of gas, \( \Delta n_g = (1+1) - (1) = 1 \). So, \( \Delta H eq \Delta E \).

Step 4: Analyze Reaction (c)

For \( 2 \mathrm{H}_{2} \mathrm{O}_{2}( ext{g}) \rightarrow 2 \mathrm{H}_{2} \mathrm{O}( ext{g})+\mathrm{O}_{2}( ext{g}) \), the change in the number of moles of gas, \( \Delta n_g = (2+1) - 2 = 1 \). Hence, \( \Delta H eq \Delta E \).

Step 5: Analyze Reaction (d)

For \( \mathrm{C}( ext{s})+\mathrm{O}_{2}( ext{g}) \rightarrow \mathrm{CO}_{2}( ext{g}) \), the change in the number of moles of gas, \( \Delta n_g = (1) - (0+1) = 0 \). Therefore, \( \Delta H = \Delta E \).

Step 6: Conclusion: Select the Reactions

Both reactions (a) and (d) have \( \Delta n_g = 0 \), so for these reactions, \( \Delta H = \Delta E \).

Key Concepts

Enthalpy Change (ΔH)

In chemistry, enthalpy change (ΔH) is a crucial concept that refers to the heat change during a chemical reaction at constant pressure. It tells us whether a reaction absorbs or releases energy. If ΔH is positive, the reaction is endothermic, meaning it absorbs heat. If it's negative, the reaction is exothermic, releasing heat. ΔH is often measured in kilojoules per mole (kJ/mol), which gives a standardized way to compare different reactions.

To calculate ΔH, you can use the equation \[ ΔH = ΔE + Δn_g RT \] where ΔE is the internal energy change, Δn_g is the change in the number of gas moles, R stands for the gas constant (8.314 J/mol·K), and T is the temperature in Kelvin. This equation shows the link between internal energy and enthalpy, emphasizing how changes in gas moles can impact the heat of reaction. Understanding this relationship is essential for analyzing chemical reactions' energy shifts.

Internal Energy Change (ΔE)

Internal energy (ΔE) in a system represents the total energy within its boundaries, which is affected by the energy of particles and interactions. It's a sum of kinetic and potential energies of all particles. When a reaction occurs, there can be a change (ΔE) due to heat transfer or work done. This change is intrinsic to the system and not influenced by external pressure.

The relationship between ΔH and ΔE is explored through the formula \[ ΔH = ΔE + Δn_g RT \], where Δn_g, R, and T have their usual meanings. The internal energy change is particularly important in understanding how energy is conserved within a reaction, especially when changes in pressure or volume aren't a factor. By understanding ΔE, chemists can predict how energy flows within reactions.

Mole Concept in Gases

The mole concept in gases directly relates to the change in the number of moles (Δn_g) during a reaction. It's crucial when analyzing thermodynamic quantities like ΔH and ΔE. This concept uses Avogadro's number (approximately 6.022 x 10^23) to relate molar volume and gas molecules. Δn_g is calculated by comparing moles of gaseous products and reactants:

\[ Δn_g = moles_{products} - moles_{reactants} \]

In reactions where Δn_g = 0, like with H_2 and I_2 forming 2HI, or carbon and oxygen forming CO_2, ΔH equals ΔE. This is because the volume doesn't change with no net change in moles of gas. By understanding this, students can predict behavior and energy changes in gas-involved reactions more accurately.

Reaction Analysis

Reaction analysis involves examining chemical reactions to understand changes in enthalpy and internal energy. By analyzing the reaction's starting and ending materials, one can discern several important aspects:

• Change in moles of gases (Δn_g)
• Type of reaction (endothermic or exothermic)
• Comparisons of ΔH and ΔE

Chemical reactions have intrinsic energy changes based on bond formation and breaking. By calculating Δn_g and using the equation \[ ΔH = ΔE + Δn_g RT \], chemists can determine whether ΔH = ΔE. For instance, understanding Δn_g in complex reactions allows prediction of energy flows and temperature's role. This analysis aids in simplifying thermodynamics in chemistry, making it more understandable for students.