Question

For which of the following reactions does \(\Delta H_{\mathrm{rxn}}^{\circ}=\Delta H_{\mathrm{f}}^{\circ}\) ? (a) \(\mathrm{H}_{2}(g)+\mathrm{S}(\) rhombic \() \longrightarrow \mathrm{H}_{2} \mathrm{~S}(g)\) (b) \(\mathrm{C}(\) diamond \()+\mathrm{O}_{2}(g) \longrightarrow \mathrm{CO}_{2}(g)\) (c) \(\mathrm{H}_{2}(\mathrm{~g})+\mathrm{CuO}(s) \longrightarrow \mathrm{H}_{2} \mathrm{O}(l)+\mathrm{Cu}(s)\) (d) \(\mathrm{O}(g)+\mathrm{O}_{2}(g) \longrightarrow \mathrm{O}_{3}(g)\)

Short answer

Reaction (a) is the one where \( \Delta H_{\mathrm{rxn}}^{\circ} = \Delta H_{\mathrm{f}}^{\circ} \).

Step-by-step solution

Understand Enthalpy of Formation

The enthalpy of formation, \( \Delta H_{\mathrm{f}}^{\circ} \), is the heat change that results when one mole of a compound is formed from its elements in their standard states. It applies when the products are formed from elements in their most stable form.

Analyze Each Reaction

We need to determine if each given reaction represents the formation of one mole of a compound from its elements in their standard states and their most stable forms.

Check Reaction (a)

In reaction (a), \( \mathrm{H}_{2}(g)+\mathrm{S}(\text{rhombic}) \rightarrow \mathrm{H}_{2} \mathrm{~S}(g) \), hydrogen gas and rhombic sulfur (standard states) form hydrogen sulfide gas. This fits the criterion for \( \Delta H_{\mathrm{rxn}}^{\circ} = \Delta H_{\mathrm{f}}^{\circ} \).

Check Reaction (b)

In reaction (b), \( \mathrm{C}(\text{diamond})+\mathrm{O}_{2}(g) \rightarrow \mathrm{CO}_{2}(g) \), carbon as diamond is not in its most stable form (graphite is), thus \( \Delta H_{\mathrm{rxn}}^{\circ} eq \Delta H_{\mathrm{f}}^{\circ} \).

Check Reaction (c)

In reaction (c), \( \mathrm{H}_{2}(g)+\mathrm{CuO}(s) \rightarrow \mathrm{H}_{2} \mathrm{O}(l)+\mathrm{Cu}(s) \), neither \( \mathrm{CuO} \) nor \( \mathrm{H}_{2} \mathrm{O} \) are formed from their most stable elements; \( \Delta H_{\mathrm{rxn}}^{\circ} eq \Delta H_{\mathrm{f}}^{\circ} \).

Check Reaction (d)

In reaction (d), \( \mathrm{O}(g)+\mathrm{O}_{2}(g) \rightarrow \mathrm{O}_{3}(g) \), only ozone \( \mathrm{O}_{3} \) is formed directly from elements, but \( \mathrm{O}(g) \) is not in its most stable form (\( \mathrm{O}_{2}(g) \) is). Hence, \( \Delta H_{\mathrm{rxn}}^{\circ} eq \Delta H_{\mathrm{f}}^{\circ} \).

Conclusion

Only reaction (a) follows the conditions for the reaction's enthalpy to equal the enthalpy of formation, as it forms a compound directly from its elements in their standard states.

Key concepts

Understanding Standard States

In the world of chemistry, standard states are crucial for calculations involving thermodynamic quantities like enthalpy, entropy, and Gibbs free energy. A standard state of an element or compound is its most stable form under a set of standard conditions.
Standard conditions are defined as a pressure of 1 bar and a specific temperature, usually 298 K (25°C), although temperature can vary depending on what is being studied. For gases, the standard state is typically 1 bar pressure, for solutes in solution it is 1 M concentration, and for pure substances this means the pure liquid or solid under 1 bar pressure.

• Gases: 1 bar pressure, often 298 K.
• Solutions: 1 M concentration.
• Pure substances: 1 bar pressure in their stable form.

Therefore, a reaction involving elements in their standard states allows for a consistent baseline from which we can compare changes and measure reaction properties, such as enthalpy.

Introduction to Chemical Reactions

Chemical reactions involve the transformation of reactants into products. During this process, chemical bonds break in the reactants and new bonds form in the products. The energy changes associated with these transformations can be calculated in terms of enthalpy changes.
In a balanced chemical equation, the number and type of atoms on both sides of the reaction must be the same, reflecting the law of conservation of mass.

• Reactants: The starting substances in a chemical reaction.
• Products: The substances formed from the reaction.
• Balanced Equation: Same number of each type of atom on either side.

Overall, understanding how atoms rearrange allows chemists to predict how substances will react and what products will form, which is fundamental for controlling reactions in industrial and laboratory settings.

What is Enthalpy?

Enthalpy, represented by the symbol \(H\), is a measure of the total heat content in a chemical system at constant pressure. It encompasses both the internal energy of the system plus the energy required to make room for it (pressure-volume work).
The change in enthalpy (\( \Delta H \)) during a reaction tells us whether a reaction absorbs or releases heat.

• Exothermic Reaction: \( \Delta H < 0 \), releases heat.
• Endothermic Reaction: \( \Delta H > 0 \), absorbs heat.

Understanding enthalpy helps in catering industrial processes such as combustion and making energy-efficient systems, by predicting whether a reaction releases or requires energy.

Exploring Heat Change in Reactions

Heat change is a key aspect of chemical reactions, often dictating the feasibility and spontaneity of a process. Heat change in reactions is generally represented by the change in enthalpy (\( \Delta H \)).
For a reaction to have its \( \Delta H_{\text{rxn}}^{\circ} = \Delta H_{\text{f}}^{\circ} \), the reaction should involve the formation of one mole of a substance directly from its constituent elements in their standard states as in reaction (a).

• Positive \( \Delta H \): Reaction absorbs heat, not favorable without input of energy.
• Negative \( \Delta H \): Reaction releases heat, often favorable.

Through applying these principles, scientists and engineers can harness chemical reactions for practical applications, like energy production and material synthesis, ensuring the processes are economically and environmentally sustainable.