Question

\(\mathrm{H}_{2} \mathrm{~S}(\mathrm{~g}) \longrightarrow \mathrm{HS}(\mathrm{g})+\mathrm{H}(\mathrm{g}), \Delta \mathrm{H}^{\circ}=\mathrm{x}_{1}\), \(\Delta \mathrm{H}_{\mathrm{f}}^{\circ}\left[\mathrm{H}_{2} \mathrm{~S}(\mathrm{~g})\right]=\mathrm{x}_{2}, \Delta \mathrm{H}_{\mathrm{f}}^{\mathrm{s}}[\mathrm{H}(\mathrm{g})]=\mathrm{x}_{3}\) hence, \(\Delta \mathrm{H}_{\mathrm{f}}^{\circ}(\mathrm{HS})\) is: (a) \(x_{1}+x_{2}-x_{3}\) (b) \(\mathrm{x}_{3}-\mathrm{x}_{1}-\mathrm{x}_{2}\) (c) \(\mathrm{x}_{1}-\mathrm{x}_{2}-\mathrm{x}_{3}\) (d) \(x_{3}-x_{1}+x_{2}\)

Short answer

Option (d): \(x_3 - x_1 + x_2\) is correct.

Step-by-step solution

Understand the reaction and given data

The reaction given is the decomposition of \(\text{H}_2\text{S}(\text{g})\) into \(\text{HS}(\text{g})\) and \(\text{H}(\text{g})\). The enthalpy change \(\Delta H^{\circ}\) for this reaction is given as \(x_1\). Additionally, we are provided with the standard enthalpy of formation for \(\text{H}_2\text{S}(\text{g})\) as \(x_2\) and for \(\text{H}(\text{g})\) as \(x_3\). We need to find \(\Delta H_{\mathrm{f}}^{\circ}(\text{HS})\).

Apply Hess's Law for enthalpy calculation

According to Hess's Law, the enthalpy change of a reaction \(\Delta H^{\circ}\) can be calculated by considering the enthalpy of formation of the reactants and products. The equation to use here is: \[ \Delta H^{\circ} = \Delta H_{\mathrm{f}}^{\circ}(\text{HS}) + \Delta H_{\mathrm{f}}^{\circ}[\text{H}(\text{g})] - \Delta H_{\mathrm{f}}^{\circ}[\text{H}_2\text{S}(\text{g})] \]Substitute the given values: \[ x_1 = \Delta H_{\mathrm{f}}^{\circ}(\text{HS}) + x_3 - x_2 \]

Solve for \(\Delta H_{\mathrm{f}}^{\circ}(\text{HS})\)

Rearrange the equation from Step 2 to isolate \(\Delta H_{\mathrm{f}}^{\circ}(\text{HS})\):\[ \Delta H_{\mathrm{f}}^{\circ}(\text{HS}) = x_1 - x_3 + x_2 \]

Match with the answer choices

Compare the derived expression \(\Delta H_{\mathrm{f}}^{\circ}(\text{HS}) = x_1 - x_3 + x_2\) with the given options. It matches option (d).

Key concepts

Enthalpy Change

The enthalpy change, often denoted by \( \Delta H \), is a crucial concept in thermochemistry. It represents the heat absorbed or released during a chemical reaction at constant pressure. When thinking about reactions, it's important to remember:

\[ \Delta H = H_{\text{products}} - H_{\text{reactants}} \]

This formula signifies how the total enthalpy of the products and reactants determines whether a reaction is endothermic (absorbing heat) or exothermic (releasing heat).

• If \( \Delta H > 0 \), the reaction is endothermic.
• If \( \Delta H < 0 \), the reaction is exothermic.

Enthalpy change is pivotal in predicting reaction behavior and understanding energy transformations in chemical processes.

Enthalpy of Formation

The enthalpy of formation \( \Delta H_f^{\circ} \) is a specific type of enthalpy change, focused on forming compounds from their constituent elements. For a compound, it is defined as the energy change when one mole of a compound is formed from its elements in their standard states. This standardized value provides a benchmark for comparing how stable a compound is relative to its elements.

For instance, the enthalpy of formation of \( \text{H}_2\text{S}(\text{g}) \) is given as \( x_2 \), and that of \( \text{H}(\text{g}) \) is \( x_3 \). By using these values, together with the enthalpy change of the overall reaction, we apply Hess’s Law. This approach helps find the unknown enthalpy of formation for another product, here being \( \text{HS}(\text{g}) \). To express the unknown formation enthalpy clearly:

\[ \Delta H_{\text{f}}^{\circ}(\text{HS}) = x_1 - x_3 + x_2 \]This showcases how manipulating known formation values can inform us about less common or unknown compounds.

Thermochemistry

Thermochemistry is the study of the heat energy associated with chemical reactions and physical changes. It intricately involves concepts like enthalpy, heat capacity, and phase changes, aiming to understand energy flow in reactions.

Thermochemistry allows scientists to:

• Predict the energy requirements or releases of reactions.
• Understand reaction spontaneity and temperature effects.
• Assess energy conservation and transformation principles.

Mastering thermochemistry involves frequent practice of applying principles like Hess's Law, which states that the total enthalpy change for a reaction is the same, regardless of the pathway or steps of the reaction. Essentially, Hess's Law clarifies how different routes in a reaction mechanism do not alter the overall energy change, facilitating flexibility in calculating unknown enthalpies based upon known data. By adopting Hess’s principle, students can solve complex enthalpy problems with simple algebraic manipulation, integrating both formation and reaction enthalpies into cohesive solutions.