Problem 76 - General Chemistry

Question

Acetic acid, \(\mathrm{CH}_{3} \mathrm{COOH}\), is contained in vinegar. Suppose acetic acid was formed from its elements, according to the following equation: $$ 2 \mathrm{C} \text { (graphite) }+2 \mathrm{H}_{2}(g)+\mathrm{O}_{2}(g) \longrightarrow \mathrm{CH}_{3} \mathrm{COOH}(l) $$ Find the enthalpy change, \(\Delta H\), for this reaction, using the following data: $$ \begin{gathered} \mathrm{CH}_{3} \mathrm{COOH}(l)+2 \mathrm{O}_{2}(g) \longrightarrow 2 \mathrm{CO}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(l) \\ \Delta H=-874 \mathrm{~kJ} \\ \mathrm{C}(\text { graphite })+\mathrm{O}_{2}(g) \longrightarrow \mathrm{CO}_{2}(g) ; \Delta H=-394 \mathrm{~kJ} \\ \mathrm{H}_{2}(g)+\frac{1}{2} \mathrm{O}_{2}(g) \longrightarrow \mathrm{H}_{2} \mathrm{O}(l) ; \Delta H=-286 \mathrm{~kJ} \end{gathered} $$

Step-by-Step Solution

Answer

The enthalpy change \( \Delta H \) for the formation of acetic acid is \(-486 \mathrm{~kJ}\).

Step 1: Understand the Target Reaction

The target reaction is the formation of acetic acid from carbon, hydrogen, and oxygen: \( 2 \mathrm{C} \text{ (graphite) } + 2 \mathrm{H}_2(g) + \mathrm{O}_2(g) \rightarrow \mathrm{CH}_3\mathrm{COOH}(l) \). We need to find the enthalpy change, \( \Delta H \), for this specific reaction.

Step 2: Analyze Given Data

You are given enthalpy changes for three reactions: \( \mathrm{CH}_3\mathrm{COOH}(l) + 2 \mathrm{O}_2(g) \rightarrow 2 \mathrm{CO}_2(g) + 2 \mathrm{H}_2\mathrm{O}(l) \) with \( \Delta H = -874 \mathrm{~kJ} \), \( \mathrm{C}(\text{graphite}) + \mathrm{O}_2(g) \rightarrow \mathrm{CO}_2(g) \) with \( \Delta H = -394 \mathrm{~kJ} \), and \( \mathrm{H}_2(g) + \frac{1}{2} \mathrm{O}_2(g) \rightarrow \mathrm{H}_2\mathrm{O}(l) \) with \( \Delta H = -286 \mathrm{~kJ} \).

Step 3: Reverse and Adjust the First Reaction

The given reaction is for the combustion of acetic acid to produce CO2 and H2O. We need the reverse reaction for forming acetic acid from CO2 and H2O. So, we reverse the reaction: \( 2 \mathrm{CO}_2(g) + 2 \mathrm{H}_2\mathrm{O}(l) \rightarrow \mathrm{CH}_3\mathrm{COOH}(l) + 2 \mathrm{O}_2(g) \) and flip the sign of \( \Delta H \) to get \( \Delta H = +874 \mathrm{~kJ} \).

Step 4: Calculate Formation Enthalpy of CO2

The formation enthalpy of CO2 from graphite carbon is given as \( \mathrm{C}(\text{graphite}) + \mathrm{O}_2(g) \rightarrow \mathrm{CO}_2(g) \) with \( \Delta H = -394 \mathrm{~kJ} \). Since the target reaction involves 2 moles of carbon, multiply \( \Delta H \) by 2 to get \( -788 \mathrm{~kJ} \).

Step 5: Calculate Formation Enthalpy of H2O

The formation enthalpy of water from H2 is given as \( \mathrm{H}_2(g) + \frac{1}{2} \mathrm{O}_2(g) \rightarrow \mathrm{H}_2\mathrm{O}(l) \) with \( \Delta H = -286 \mathrm{~kJ} \). Since the target reaction involves 2 moles of H2, we double this value to \( -572 \mathrm{~kJ} \).

Step 6: Apply Hess's Law

According to Hess's Law, the overall enthalpy change \( \Delta H \) for the formation of acetic acid can be found by adding the enthalpies from Steps 3, 4, and 5: \( \Delta H = ( +874 \mathrm{~kJ} ) + ( -788 \mathrm{~kJ} ) + ( -572 \mathrm{~kJ} ) = -486 \mathrm{~kJ} \).

Key Concepts

Hess's Law

Hess's Law is a pivotal concept in thermochemistry that simplifies calculating the enthalpy change for a reaction. It states that the total enthalpy change for a chemical reaction is the same, regardless of whether it occurs in one step or several. This principle allows us to use known enthalpy changes from related reactions to find the desired enthalpy change.
By rearranging, reversing, or combining various chemical equations, we're able to construct the target reaction indirectly, which can then give us its enthalpy change.
In the exercise with acetic acid, Hess's Law was utilized to deconstruct and then reconstruct the formation reaction of acetic acid by using the combustion data of acetic acid and the formation reactions of carbon dioxide and water. The beauty of Hess's Law is that it relies on the state function nature of enthalpy, where the path taken does not affect the total enthalpy change.

Formation Enthalpy

Formation enthalpy, often referred to as standard enthalpy of formation, is the change in enthalpy that accompanies the formation of one mole of a compound from its elements in their standard states.
Understanding this value is essential because it helps us calculate the enthalpy change for larger or complex reactions by building them from these fundamental formation reactions.
For acetic acid, the formation enthalpy is what we're indirectly calculating. We take the known formation enthalpies of carbon dioxide and water, and the reaction for burning acetic acid.
Using these, we reconstruct the formation of acetic acid, and thus derive its formation enthalpy. By manipulating known reactions and enthalpy changes, we can unravel the enthalpy of formation for compounds where direct measurement might be challenging.

Thermochemistry

Thermochemistry is the branch of chemistry that deals with the study of heat energy associated with chemical reactions and changes of state. The primary goal is to understand the enthalpy changes that accompany chemical processes.
In our example involving acetic acid, thermochemistry is at the heart of the calculation. By using known enthalpy changes—such as those from combustion reactions—we can predict or calculate the heat exchange in reactions that are not experimentally feasible to measure directly.
Key components of thermochemical calculations involve using Hess’s Law and formation enthalpies to deduce or compute unknown enthalpy changes. This makes it an indispensable tool in both academic and industrial chemistry contexts.

Acetic Acid

Acetic acid, chemically represented as \( \text{CH}_3\text{COOH} \), is a key organic compound primarily found in vinegar. It is characterized by its sour taste and pungent smell.
In chemistry, acetic acid is known for its role as a simple carboxylic acid, and it serves as a critical building block in chemical synthesis and various industrial processes.
Its formation from elemental carbon, hydrogen, and oxygen involves a specific enthalpy change, which can be calculated through thermochemical equations and principles such as Hess’s Law. Acetic acid's reactions and energy changes are instrumental in understanding broader concepts like formation enthalpies and reaction enthalpy calculations. In studying its formation, we explore both fundamental and applied principles of chemical thermodynamics.