Question

Given the following data $$\begin{array}{ll}\mathrm{Fe}_{2} \mathrm{O}_{3}(s)+3 \mathrm{CO}(g) \longrightarrow 2 \mathrm{Fe}(s)+3 \mathrm{CO}_{2}(g) & \Delta H^{\circ}=-23 \mathrm{kJ} \\ 3 \mathrm{Fe}_{2} \mathrm{O}_{3}(s)+\mathrm{CO}(g) \longrightarrow 2 \mathrm{Fe}_{3} \mathrm{O}_{4}(s)+\mathrm{CO}_{2}(g) & \Delta H^{\circ}=-39 \mathrm{kJ} \\ \mathrm{Fe}_{3} \mathrm{O}_{4}(s)+\mathrm{CO}(g) \longrightarrow 3 \mathrm{FeO}(s)+\mathrm{CO}_{2}(g) & \Delta H^{\circ}=18 \mathrm{kJ} \end{array}$$ calculate \(\Delta H^{\circ}\) for the reaction $$\mathrm{FeO}(s)+\mathrm{CO}(g) \longrightarrow \mathrm{Fe}(s)+\mathrm{CO}_{2}(g)$$

Short answer

The calculated enthalpy change for the desired reaction is \(\Delta H^{\circ}_{\mathrm{FeO}(s)+\mathrm{CO}(g) \longrightarrow \mathrm{Fe}(s)+\mathrm{CO}_{2}(g)} = 9.17 \,\mathrm{kJ}\).

Step-by-step solution

Analyzing the target reaction

Compare the given reactions with the desired reaction. We can see that all the substances in the target reaction are present in the given reactions. We will use the given reactions to extract the necessary information to find the enthalpy change for the target reaction.

Manipulate the given reactions

First, we have to manipulate the given reactions to match the target reaction. We can find the desired equation by adding a combination of these reactions. We notice that to cancel out the \(\mathrm{Fe}_{2}\mathrm{O}_{3}\), we need to reverse the direction of the first two reactions. $$-\Delta H^{\circ} = -[\mathrm{Fe}_{2} \mathrm{O}_{3}(s)+3 \mathrm{CO}(g) \longrightarrow 2 \mathrm{Fe}(s)+3 \mathrm{CO}_{2}(g)]$$ $$-\Delta H^{\circ} = -[3 \mathrm{Fe}_{2} \mathrm{O}_{3}(s)+\mathrm{CO}(g) \longrightarrow 2 \mathrm{Fe}_{3} \mathrm{O}_{4}(s)+\mathrm{CO}_{2}(g)]$$ Before we add up these equations, we need to ensure the substances we don't need to cancel each other out. Thus, we can rewrite the equations as: $$\frac{1}{3}\times(-\Delta H^{\circ} = -[\mathrm{Fe}_{2} \mathrm{O}_{3}(s)+3 \mathrm{CO}(g) \longrightarrow 2 \mathrm{Fe}(s)+3 \mathrm{CO}_{2}(g)])$$ $$\frac{1}{2}\times(-\Delta H^{\circ} = -[3 \mathrm{Fe}_{2} \mathrm{O}_{3}(s)+\mathrm{CO}(g) \longrightarrow 2 \mathrm{Fe}_{3} \mathrm{O}_{4}(s)+\mathrm{CO}_{2}(g)])$$ $$\Delta H^{\circ} = [\mathrm{Fe}_{3} \mathrm{O}_{4}(s)+\mathrm{CO}(g) \longrightarrow 3 \mathrm{FeO}(s)+\mathrm{CO}_{2}(g)]$$ Now we can add up these equations getting: $$\mathrm{FeO}(s)+\mathrm{CO}(g) \longrightarrow \mathrm{Fe}(s)+\mathrm{CO}_{2}(g)$$

Calculate the enthalpy change for the target reaction

Now that we have obtained the linear combination of the given reactions to get the desired reaction, we can find the enthalpy change for this reaction by summing the enthalpy change of the given reactions after manipulating them. We have: $$\Delta H^{\circ}_{target} = \frac{-1}{3}\times(-23 \,\mathrm{kJ}) + \frac{-1}{2}\times(-39\, \mathrm{kJ}) + 18\, \mathrm{kJ}$$ $$\Delta H^{\circ}_{target} = \frac{23}{3} \,\mathrm{kJ} + \frac{39}{2}\, \mathrm{kJ} - 18\, \mathrm{kJ}$$ Calculating the enthalpy change for the target reaction: $$\Delta H^{\circ}_{target} = 7.67 \,\mathrm{kJ} + 19.5\, \mathrm{kJ} - 18\, \mathrm{kJ} =\boldsymbol{9.17\, \mathrm{kJ}}$$ The calculated enthalpy change for the desired reaction is \(\Delta H^{\circ}_{\mathrm{FeO}(s)+\mathrm{CO}(g) \longrightarrow \mathrm{Fe}(s)+\mathrm{CO}_{2}(g)} = 9.17 \,\mathrm{kJ}\).

Key concepts

Enthalpy Change

When discussing chemical reactions, one of the key concepts you'll encounter is the enthalpy change, denoted as \( \Delta H \). Enthalpy change represents the heat absorbed or released during a chemical reaction at constant pressure.
It's a crucial component because it helps us understand how energy is transferred during a reaction.Enthalpy changes can be:

• Positive: indicating an endothermic reaction where heat is absorbed.
• Negative: indicating an exothermic reaction where heat is released.

In practice, the enthalpy change for a reaction is calculated by summing the enthalpy changes of known reactions that sum to give the desired reaction. In the exercise, we use Hess's Law to calculate the enthalpy change for a complex reaction by manipulating and adding the enthalpy changes of simpler linked reactions. This demonstrates that enthalpy is a state function, where the total enthalpy change depends only on the initial and final states and not on the specific pathway.

Chemical Reactions

Chemical reactions are processes where substances, known as reactants, transform into different substances, called products. The transformation involves the breaking and forming of bonds between atoms.
Writing a balanced chemical equation is essential to understand the changes that occur in a reaction and to calculate the enthalpy change accurately. In the context of this exercise, we're dealing with reactions involving iron oxides and carbon monoxide. These reactions illustrate single replacement and combustion reactions:

• Single Replacement: where one element replaces another in a compound.
• Combustion: a substance combines with oxygen, releasing energy.

Each type of reaction has distinct energy changes, which are reflected in their enthalpy changes. By analyzing these reactions, we can apply principles like Hess's Law, a demonstration of the conservation of energy, to derive the overall energy change.

Thermochemistry

Thermochemistry is the branch of chemistry concerned with the heat energy involved in chemical and physical processes. It helps us quantify the energy changes taking place during chemical reactions.
This is important for practical applications like designing energy-efficient systems.A few key points about thermochemistry include:

• It relies heavily on the first law of thermodynamics, which states that energy cannot be created or destroyed, only transformed.
• Thermochemical equations express the energy change of reactions, often including the enthalpy change \( \Delta H \).
• Understanding the energy profiles of reactions allows chemists to predict reaction behavior and propose mechanisms.

In our specific exercise, thermochemistry provides the framework for calculating the overall enthalpy change when converting reactants to products. By manipulating and adding up known thermochemical equations, we utilize the concepts of enthalpy and Hess's Law to find unknown values.