Question

Using Appendix 4 and the following data, determine \(S^{\circ}\) for \(\mathrm{Fe}(\mathrm{CO})_{5}(g)\) $$ \begin{aligned} \mathrm{Fe}(s)+5 \mathrm{CO}(g) & \longrightarrow \mathrm{Fe}(\mathrm{CO})_{5}(g) & & \Delta S^{\circ}=? \\ \mathrm{Fe}(\mathrm{CO})_{5}(l) & \longrightarrow \mathrm{Fe}(\mathrm{CO})_{5}(g) & & \Delta S^{\circ}=107 \mathrm{~J} / \mathrm{K} \\ \mathrm{Fe}(s)+5 \mathrm{CO}(g) & \longrightarrow \mathrm{Fe}(\mathrm{CO})_{5}(l) & & \Delta S^{\circ}=-677 \mathrm{~J} / \mathrm{K} \end{aligned} $$

Short answer

The \(\Delta S^{\circ}\) for the target reaction \(\mathrm{Fe}(s)+5 \mathrm{CO}(g) \longrightarrow \mathrm{Fe}(\mathrm{CO})_{5}(g)\) is \(-784 \mathrm{~J}/\mathrm{K}\).

Step-by-step solution

Identify the given equations and their \(\Delta S^{\circ}\)

For this problem, we are given the following two reactions: 1. \(\mathrm{Fe}(\mathrm{CO})_{5}(l) \longrightarrow \mathrm{Fe}(\mathrm{CO})_{5}(g)\) with \(\Delta S^{\circ}=107 \mathrm{~J} / \mathrm{K}\) 2. \(\mathrm{Fe}(s)+5 \mathrm{CO}(g) \longrightarrow \mathrm{Fe}(\mathrm{CO})_{5}(l)\) with \(\Delta S^{\circ}=-677 \mathrm{~J} / \mathrm{K}\) The target reaction is: $$\mathrm{Fe}(s)+5 \mathrm{CO}(g) \longrightarrow \mathrm{Fe}(\mathrm{CO})_{5}(g)$$

Manipulate the given reactions to match the target reaction

To arrive at the target reaction, we need to manipulate the given reactions in a way that can be added together. Observe that reversing reaction 1 gives us the gaseous form of \(\mathrm{Fe}(\mathrm{CO})_{5}\), which is present in the target reaction. Let's reverse reaction 1 and change the sign of its \(\Delta S^{\circ}\): $$\mathrm{Fe}(\mathrm{CO})_{5}(g) \longrightarrow \mathrm{Fe}(\mathrm{CO})_{5}(l)$$ with \(\Delta S^{\circ}=-107 \mathrm{~J} / \mathrm{K}\) Now, we can add reaction 2 and the reversed reaction 1 to obtain the target reaction: $$\mathrm{Fe}(s)+5 \mathrm{CO}(g) \longrightarrow \mathrm{Fe}(\mathrm{CO})_{5}(l)$$ with \(\Delta S^{\circ}=-677 \mathrm{~J} / \mathrm{K}\) $$+\mathrm{Fe}(\mathrm{CO})_{5}(g) \longrightarrow \mathrm{Fe}(\mathrm{CO})_{5}(l)$$ with \(\Delta S^{\circ}=-107 \mathrm{~J} / \mathrm{K}\) $$\implies \mathrm{Fe}(s)+5 \mathrm{CO}(g) \longrightarrow \mathrm{Fe}(\mathrm{CO})_{5}(g)$$

Find the \(\Delta S^{\circ}\) for the target reaction

Now that we have the target reaction, we can calculate its \(\Delta S^{\circ}\) by adding the \(\Delta S^{\circ}\) values of the manipulated given reactions: $$\Delta S^{\circ}_{\text{target}} = \Delta S^{\circ}_{2} + \Delta S^{\circ}_{\text{reversed 1}}$$ $$\Delta S^{\circ}_{\text{target}} = -677 \mathrm{~J} / \mathrm{K} + (-107 \mathrm{~J} / \mathrm{K})$$ $$\Delta S^{\circ}_{\text{target}} = -784 \mathrm{~J} / \mathrm{K}$$ Therefore, the \(\Delta S^{\circ}\) for the target reaction \(\mathrm{Fe}(s)+5 \mathrm{CO}(g) \longrightarrow \mathrm{Fe}(\mathrm{CO})_{5}(g)\) is \(-784 \mathrm{~J}/\mathrm{K}\).

Key concepts

Entropy Change

Entropy change is an important concept in chemical thermodynamics, which measures the degree of disorder or randomness in a system during a chemical reaction or phase transition.
When a substance changes from one configuration to another, the entropy, denoted as \(\Delta S\), gives us insight into the orderliness of the system.
For a reaction or process, the change in entropy is calculated as the difference between the entropy of the final state and the initial state.In the provided exercise, we observe the transformation of iron pentacarbonyl from its liquid to gaseous phase and the formation of this compound from its elemental forms.
The entropy change for each step is calculated separately and then combined to derive the overall entropy change for the reaction.
A positive \(\Delta S\) implies the products are more disordered than reactants, while a negative \(\Delta S\) indicates a decrease in disorder, suggesting a more ordered system. In our case, the target reaction has a total \(\Delta S^{\circ}\) of \(-784\ \mathrm{~J}/\mathrm{K}\) indicating that the process leads to a more structured state.

Reaction Manipulation

Reaction manipulation involves reversing, scaling, or combining individual chemical reactions to achieve a desired target reaction.
This requires careful modifications to both the reactions and their associated thermodynamic values such as entropy, enthalpy, etc.
Each manipulation reflects upon the energy and entropy implication of the reactions.The exercise demonstrates manipulation by reversing one of the provided reactions.
Reversing a reaction changes the sign of its entropy change \(\Delta S\). This step is crucial when aligning given reactions to form the target reaction.
By manipulating the given reactions, we attain the desired target reaction, aligning all components and states with the provided objective.
This manipulation provides a direct pathway to calculate the total entropy change for the target reaction by simple algebraic addition of entropies of each individual manipulated step.

Phase Transition

Phase transition is a process where a substance changes from one state of matter to another, like solid to liquid, liquid to gas, etc.
Entropy plays a significant role in these processes as it accounts for energy dispersal and molecular disorder change during the transition.
Each phase has unique properties and entropic values affected by temperature and pressure.In the task at hand, we focus on the phase transition of \(\mathrm{Fe(CO)_5}\) from liquid to vapor.
This phase change is represented by a distinct entropy value, \(\Delta S^{\circ} = 107 \, \mathrm{J/K}.
\) During vaporization, the system moves to a state of higher randomness and energy distribution due to the increased molecular freedom in the gas phase.
Understanding phase transitions in the context of entropy equips us with insights into thermodynamic properties and helps predict system behavior under different conditions.
These insights are beneficial for controlling industrial processes that rely heavily on precise temperature and phase monitoring.