Question

From the enthalpies of reaction $$ \begin{gathered} 2 \mathrm{C}(s)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{CO}(g) \quad \Delta H=-221.0 \mathrm{~kJ} \\ 2 \mathrm{C}(s)+\mathrm{O}_{2}(g)+4 \mathrm{H}_{2}(g) \longrightarrow 2 \mathrm{CH}_{3} \mathrm{OH}(g) \quad \Delta H=-402.4 \mathrm{~kJ} \\ \text { calculate } \Delta H \text { for the reaction } \\ \mathrm{CO}(g)+2 \mathrm{H}_{2}(g) \longrightarrow \mathrm{CH}_{3} \mathrm{OH}(g) \end{gathered} $$

Short answer

The enthalpy change for the desired reaction, CO(g) + 2 H₂(g) → CH₃OH(g), is \(\Delta H = -291.9 \,\text{kJ}\), which is calculated using Hess's Law by combining the given reactions and their respective enthalpy changes.

Step-by-step solution

Identify the Desired Reaction

We are given two reactions and their respective enthalpies: \( \begin{gathered} \text{Reaction 1: } 2 \mathrm{C}(s)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{CO}(g) \quad \Delta H_{1}=-221.0 \mathrm{~kJ} \\\ \text{Reaction 2: } 2 \mathrm{C}(s)+\mathrm{O}_{2}(g)+4 \mathrm{H}_{2}(g) \longrightarrow 2 \mathrm{CH}_{3} \mathrm{OH}(g) \quad \Delta H_{2}=-402.4 \mathrm{~kJ} \\\ \end{gathered} \) We want to find the enthalpy of the following reaction: \( \text{Desired Reaction: } \mathrm{CO}(g)+2 \mathrm{H}_{2}(g) \longrightarrow \mathrm{CH}_3\mathrm{OH}(g) \)

Use Hess's Law to Compute the Enthalpy of the Desired Reaction

Hess's Law states that if a reaction can be expressed as the sum of two or more other reactions, the enthalpy change for the overall reaction will be the sum of the enthalpy changes for those reactions. First, let's write out the given reactions, Reaction 1 and Reaction 2: \( \text{Reaction 1: } 2 \mathrm{C}(s)+\mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{CO}(g) \quad \Delta H_{1}=-221.0 \mathrm{~kJ} \) \( \text{Reaction 2: } 2 \mathrm{C}(s)+\mathrm{O}_{2}(g)+4 \mathrm{H}_{2}(g) \longrightarrow 2 \mathrm{CH}_{3} \mathrm{OH}(g) \quad \Delta H_{2}=-402.4 \mathrm{~kJ} \) Notice that if we reverse Reaction 1 and then divide it by 2, we will have a term for CO on the left side: \( \text{Reaction 1 (reversed and divided by 2): } \mathrm{CO}(g) \longrightarrow \mathrm{C}(s)+\frac{1}{2}\mathrm{O}_{2}(g) \quad \Delta H_{1}'=221.0/2 \) (The enthalpy change for the reversed reaction is the negative of the original enthalpy change) Now, we can add this modified Reaction 1 to Reaction 2, and then, subtract 2 moles of \(H_2\) from both sides to get the Desired Reaction: \( \text{ Reaction 2 + (Reaction 1 reversed and divided by 2) - 2H_2: } \mathrm{CO}(g) + 2 \mathrm{H}_2(g) \longrightarrow \mathrm{CH}_3\mathrm{OH}(g) \) Now, let's add the corresponding enthalpies: \( \Delta H = \Delta H_2 + \Delta H_{1}' = -402.4 \,\text{kJ} + 221.0/2 \,\text{kJ} \) Calculating the sum: \(\Delta H = -402.4 + 110.5 = -291.9 \,\text{kJ}\) Therefore, the enthalpy change for the Desired Reaction is: \( \Delta H = -291.9 \,\text{kJ} \)

Key concepts

Hess's Law

Understanding Hess's Law is crucial for calculating the overall enthalpy changes in chemical reactions. This principle states that the total enthalpy change in a chemical reaction is the same regardless of the number of steps or its pathway, as long as the initial and final conditions are the same.

When you're faced with complex reactions, Hess's Law allows you to break them down into simpler, hypothetical steps or to combine known reactions to determine the enthalpy change of an overall reaction. It's like a financial ledger, where you add and subtract individual transactions to find the total balance.

To apply Hess's Law effectively, remember these key points:

• Reversing a reaction changes the sign of \(\Delta H\) but not its magnitude.
• Multiplying a reaction by a coefficient also multiplies its enthalpy change.

By manipulating given reactions—reversing them, dividing or multiplying—it's possible to 'piece together' the desired reaction, like a jigsaw puzzle, and calculate its \(\Delta H\) easily.

Thermochemistry

Thermochemistry is the study of the heat involved in chemical reactions. It's centered on the concept of enthalpy (\(\Delta H\)), a critical thermodynamic property that represents the heat content of a system. The enthalpy change gives you direct insight into the energy changes associated with a chemical reaction.

There are two types of reactions based on enthalpy changes:

• Exothermic reactions, where energy is released into the surroundings, indicated by a negative \(\Delta H\).
• Endothermic reactions, where energy is absorbed from the surroundings, indicated by a positive \(\Delta H\).

Enthalpy is a state function, meaning it depends only on the state of the reactants and products, not on the path taken. This characteristic is what makes Hess's Law feasible and incredibly useful in thermochemistry.

When applying Hess's Law in thermochemistry, you align individual reaction enthalpies to deduce the overall enthalpy for a sequence of reactions, through a process that resembles balancing a chemical equation.

Chemical Reactions

Chemical reactions are processes where reactants transform into products, resulting in a rearrangement of atoms. Thermochemically, every chemical reaction comes with an energy change—captured by the enthalpy (\(\Delta H\)).

Key factors you should consider about chemical reactions include:

• The conservation of mass and energy -- atoms and energy are neither created nor destroyed.
• The bonds broken and formed -- breaking bonds requires energy (endothermic), forming bonds releases energy (exothermic).
• Stoichiometry -- the coefficients in a balanced equation tell us the proportions in which substances react and form products.

Understanding the nuances of chemical reactions is pivotal for predicting reaction behavior, and it involves a delicate balance between the enthalpies of bonds being broken and formed. By comprehending how to calculate \(\Delta H\), and applying principles like Hess's Law to chemical reactions, students can predict not only the energy change but also the favorability of chemical processes under given conditions.