Question

Use Hess's law to determine \(\Delta H^{\circ}\) for the reaction \(\mathrm{C}_{3} \mathrm{H}_{4}(\mathrm{g})+2 \mathrm{H}_{2}(\mathrm{g}) \longrightarrow \mathrm{C}_{3} \mathrm{H}_{8}(\mathrm{g}),\) given that $$\mathrm{H}_{2}(\mathrm{g})+\frac{1}{2} \mathrm{O}_{2}(\mathrm{g}) \longrightarrow \mathrm{H}_{2} \mathrm{O}(\mathrm{l}) \quad \Delta H^{\circ}=-285.8 \mathrm{kJ}$$ $$\begin{aligned} \mathrm{C}_{3} \mathrm{H}_{4}(\mathrm{g})+4 \mathrm{O}_{2}(\mathrm{g}) \longrightarrow & 3 \mathrm{CO}_{2}(\mathrm{g})+2 \mathrm{H}_{2} \mathrm{O}(\mathrm{l}) \\ && \Delta H^{\circ}=-1937 \mathrm{kJ} \end{aligned}$$ $$\begin{array}{r} \mathrm{C}_{3} \mathrm{H}_{8}(\mathrm{g})+5 \mathrm{O}_{2}(\mathrm{g}) \longrightarrow 3 \mathrm{CO}_{2}(\mathrm{g})+4 \mathrm{H}_{2} \mathrm{O}(\mathrm{l}) \\ \Delta H^{\circ}=-2219.1 \mathrm{kJ} \end{array}$$

Short answer

\(\Delta H^{\circ}\) for the reaction \(\mathrm{C}_{3} \mathrm{H}_{4}(\mathrm{g})+2 \mathrm{H}_{2}(\mathrm{g}) \longrightarrow \mathrm{C}_{3} \mathrm{H}_{8}(\mathrm{g})\) is +1425.3 kJ.

Step-by-step solution

Identify the target reaction.

The target reaction is: \(\mathrm{C}_{3} \mathrm{H}_{4}(\mathrm{g})+2 \mathrm{H}_{2}(\mathrm{g}) \longrightarrow \mathrm{C}_{3} \mathrm{H}_{8}(\mathrm{g})\).

Transform the first reaction to directly generate two \(H_2O\).

If we multiply the first reaction by 4, and then reverse it, it becomes: \(4 H_2O(l) \longrightarrow 2 H_2(g) + 2 O_2(g)\), \(\Delta H^{\circ}= + 1143.2 kJ\).

Transform the second reaction so it generates the right formula

We keep the second reaction same as it is: \(C_{3}H_{4}(g) + 4 O_{2}(g) \longrightarrow 3 CO_{2}(g) + 2 H_2O(l)\), \(\Delta H^{\circ}= -1937 kJ\).

Transform the third reaction so it gets rid of the unneeded formula

The third reaction is multiplied by -1: \(3 CO_{2}(g) + 4 H_2O(l) \longrightarrow C_{3}H_{8}(g) + 5 O_{2}(g)\), \(\Delta H^{\circ} = +2219.1 kJ\).

Add up the reactions

Add the reactions from Step 2, Step 3 and Step 4 together. Once added together, the \(O_2(g)\) molecules on the right side in Reactions 2 and 4 will cancel with those on the left side in Reaction 3. The \(H_2O(l)\) molecules on right side of Reactions 2 and 3 will cancel with \(H_2O(l)\) on the left side of reaction 4, the \(CO_{2}(g)\) molecules on right side of reaction3 will cancel with \(CO_{2}(g)\) on left side of reaction 4. And we end up with the reaction we want: \(C_{3}H_{4}(g) + 2 H_{2}(g) \longrightarrow C_{3}H_{8}(g)\). The \(\Delta H^{\circ}\) of this reaction is the sum of the \(\Delta H^{\circ}\) values from the modified reactions from Steps 2, 3, and 4: \(\Delta H^{\circ} = +1143.2 kJ – 1937 kJ + 2219.1 kJ = +1425.3 kJ\).

Key concepts

Enthalpy Change

In the context of chemical reactions, the enthalpy change, denoted as \( \Delta H^{\circ} \), represents the heat absorbed or released during the reaction under standard conditions. It's a crucial component in understanding the energy profile of a chemical process. Hess's Law is particularly useful when calculating \( \Delta H^{\circ} \) because it allows us to determine the enthalpy change for a reaction even if it cannot be measured directly. This is achieved by expressing the target reaction as the sum of multiple known reactions.
The enthalpy change of a reaction can be either positive or negative.

• A positive \( \Delta H^{\circ} \) indicates an endothermic reaction, where energy is absorbed from the surroundings.
• A negative \( \Delta H^{\circ} \) denotes an exothermic reaction, where energy is released into the surroundings.

By applying Hess's Law to the compound reaction paths, we calculate the total \( \Delta H^{\circ} \) as the algebraic sum of enthalpy changes of these individual steps.
In the given problem, the enthalpy change of \( +1425.3 \mathrm{kJ} \) signifies an endothermic process for converting \( \mathrm{C}_{3} \mathrm{H}_{4} \) and \( \mathrm{H}_{2} \) into \( \mathrm{C}_{3} \mathrm{H}_{8} \).

Chemical Thermodynamics

Chemical thermodynamics involves the study of heat changes and energy transfer in chemical reactions, providing insights into reaction spontaneity and equilibrium. Hess's Law plays a pivotal role in thermodynamics, as it simplifies the computation of enthalpy changes, which are not readily available through direct experiments.
The principle behind Hess's Law lies in the conservation of energy, stating that energy in a closed system remains constant. Therefore, the total enthalpy change for a reaction sequence is independent of the pathway taken from reactants to products. This makes it possible to apply Hess's Law to circumvent the obstacles in directly determining enthalpy changes for reaction steps.
Understanding chemical thermodynamics through Hess's Law allows students to:

• Predict the heat flow and energy requirements for reactions.
• Calculate entropy and Gibbs free energy when combined with additional data.
• Optimize reaction conditions based on energetic considerations.

This knowledge forms the foundation for exploring more complex thermodynamic concepts and applications.

Reaction Pathways

The study of reaction pathways involves analyzing the individual steps that comprise a chemical reaction from start to finish. Hess's Law facilitates the understanding of these pathways by enabling the calculation of enthalpy changes for each step, even when direct measurement is difficult.

For example, in the exercise provided, the target reaction \( \mathrm{C}_{3} \mathrm{H}_{4}(\mathrm{g})+2 \mathrm{H}_{2}(\mathrm{g}) \longrightarrow \mathrm{C}_{3} \mathrm{H}_{8}(\mathrm{g}) \) is achieved by manipulating and combining three separate reactions. Each pathway adjustment, such as reversing or multiplying equations, corresponds to changes in enthalpy, required for attaining the final desired product.
By understanding reaction pathways through the lens of Hess's Law, students can better grasp:

• The stepwise process reactions undergo, from breaking and forming bonds to the generation of intermediates.
• The net energy changes associated with every pathway iteration.
• The conceptual background for more advanced reaction kinetic studies.

This approach allows for a comprehensive view of how multiple reactions contribute symbiotically to achieve a singular product formation.