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) \(\mathrm{C}(\mathrm{s})+\mathrm{O}_{2}(\mathrm{~g}) \rightarrow \mathrm{CO}_{2}(\mathrm{~g})\)

Short answer

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

Step-by-step solution

Understand the Relation Between ΔH and ΔE

In thermochemistry, the relationship between the change in enthalpy (ΔH) and the change in internal energy (ΔE) for a reaction is given 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 universal gas constant, and \( T \) is the temperature in Kelvin. If \( \Delta n_g = 0 \), then \( \Delta H = \Delta E \).

Determine Change in Moles of Gas for Reaction (a)

For the reaction \( \mathrm{H}_{2}( ext{g}) + \mathrm{I}_{2}( ext{g}) \rightarrow 2 \mathrm{HI}( ext{g}) \), the change in moles of gas \( \Delta n_g \) is: \( 2 - (1 + 1) = 0 \). Hence, for this reaction, \( \Delta H = \Delta E \).

Determine Change in Moles of Gas for Reaction (b)

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

Determine Change in Moles of Gas for Reaction (c)

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

Determine Change in Moles of Gas for Reaction (d)

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

Identify Reactions Where ΔH Equals ΔE

From the analysis, the reactions where \( \Delta H = \Delta E \) are (a) \( \mathrm{H}_{2}( ext{g}) + \mathrm{I}_{2}( ext{g}) \rightarrow 2 \mathrm{HI}( ext{g}) \) and (d) \( \mathrm{C}( ext{s}) + \mathrm{O}_{2}( ext{g}) \rightarrow \mathrm{CO}_{2}( ext{g}) \).

Key concepts

Enthalpy change (ΔH)

Enthalpy is a fundamental concept in thermochemistry that describes the total heat content of a system. It is a measure of the energy change when a chemical reaction occurs at constant pressure. The enthalpy change, represented as \( \Delta H \), indicates whether the reaction is endothermic or exothermic.

• **Endothermic Reactions**: These reactions absorb energy, causing \( \Delta H \) to be positive. An example is the melting of ice.
• **Exothermic Reactions**: These reactions release energy, making \( \Delta H \) negative. Combustion reactions are a typical example.

When comparing enthalpy change to internal energy change \( \Delta E \), one must account for any work done on or by the system, particularly in gas phase reactions where volume might change. The equation linking these changes is given by \( \Delta H = \Delta E + \Delta n_g RT \), where \( \Delta n_g \) is the change in the moles of gas. If \( \Delta n_g = 0 \), it simplifies to \( \Delta H = \Delta E \), showing that no energy is lost or gained from the system via work.

Internal energy change (ΔE)

Internal energy, symbolized as \( \Delta E \), refers to the total energy change within a system during a reaction. This includes all kinetic and potential energy changes at a molecular level. Internal energy can change due to heat transfer or work done on or by the system. In chemical reactions, \( \Delta E \) is predominantly affected by formation or breaking of bonds, without accounting for the work done as in pressure-volume work in gases.

• **Closed System**: No mass crosses the boundary, only energy such as heat or work is exchanged.
• **Equation**: The fundamental relationship is \( \Delta U = q + w \), where \( q \) is heat and \( w \) is work.

When gas volumes change during reactions, there is a difference between \( \Delta E \) and \( \Delta H \). However, when no volume change occurs, \( \Delta E \) can equal \( \Delta H \), such as in reactions (a) and (d) from the exercise, where the number of gas moles remains constant, leading to \( \Delta n_g = 0 \). This equality simplifies the thermodynamic analysis of a reaction.

Gas phase reactions

Gas phase reactions involve reactants and products in the gaseous state. These reactions are highly sensitive to changes in pressure and temperature, and are typically analyzed in terms of moles of gas. A key aspect of gas phase reactions is the change in the number of moles of gas, \( \Delta n_g \), which affects the relationships between thermodynamic properties like \( \Delta H \) and \( \Delta E \).

• **Ideal Gas Behavior**: Described by the ideal gas law \( PV = nRT \), where pressure \( P \), volume \( V \), number of moles \( n \), and temperature \( T \), affect the system's behavior.
• **Molar Relationships**: For reactions where \( \Delta n_g eq 0 \), the energy change is affected by the work done by or on the gas (expansion or compression).

In the original exercise, reactions (b) and (c) have \( \Delta n_g eq 0 \), causing \( \Delta H \) to differ from \( \Delta E \). Conversely, reactions with \( \Delta n_g = 0 \) reflect situations where expansions or compressions do not occur, thereby simplifying the energy relationship and causing \( \Delta H = \Delta E \). This distinction is crucial for understanding how chemical energetics vary across different reaction types.