Problem 125 - Chemistry The Science in Context

Question

Use Hess's law and the following data to calculate the standard enthalpy of formation of \(\mathrm{CH}_{4}(g)\). $$\begin{aligned} &\mathrm{C}(s)+\mathrm{O}_{2}(g) \rightarrow \mathrm{CO}_{2}(g) \quad \Delta H_{f}^{\circ}=-393.5 \mathrm{kJ} / \mathrm{mol}\\\ &\mathrm{H}_{2}(g)+\frac{1}{2} \mathrm{O}_{2}(g) \rightarrow \mathrm{H}_{2} \mathrm{O}(\ell) \quad \Delta H_{\mathrm{f}}^{\circ}=-285.9 \mathrm{kJ} / \mathrm{mol}\\\ &\begin{aligned} \mathrm{CO}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(\ell) \rightarrow & \\ \mathrm{CH}_{4}(g)+2 \mathrm{O}_{2}(g) & \Delta H_{\mathrm{rxn}}^{\circ}=-890.4 \mathrm{kJ} / \mathrm{mol} \end{aligned} \end{aligned}$$

Step-by-Step Solution

Answer

Answer: The standard enthalpy of formation of CH4(g) is -74.9 kJ/mol.

Step 1: Write the formation reaction of CH4(g)

Write down the formation reaction of \(\mathrm{CH}_{4}(g)\), which involves forming one mole of methane from its constituent elements in their standard states. $$\mathrm{C}(s) + 2\mathrm{H}_{2}(g) \rightarrow \mathrm{CH}_{4}(g)$$

Step 2: Use Hess's law to combine given reactions to obtain the formation reaction

According to Hess's law, the enthalpy change of a reaction is the same whether it occurs directly or in several steps. So, we can combine the given reactions in such a way that it results in the formation reaction: 1. Balance the number of \(\mathrm{O}_{2}\). In the formation reaction, \(\mathrm{O}_{2}\) is not involved. In the desired reaction, the second and third equations need to be multiplied by 2. 2. Since the first reaction is part of the desired formation reaction and already balanced, no modification needs to be made. 3. Reverse the third reaction as it shows the decomposition of \(\mathrm{CH}_{4}(g)\) into its products. The modified reactions are: $$\mathrm{C}(s)+\mathrm{O}_{2}(g) \rightarrow \mathrm{CO}_{2}(g) \quad \Delta H_{f}^{\circ}=-393.5 \mathrm{kJ} / \mathrm{mol}$$ $$2\left(\mathrm{H}_{2}(g) +\frac{1}{2} \mathrm{O}_{2}(g) \rightarrow \mathrm{H}_{2}\mathrm{O}(\ell) \quad \Delta H_{\mathrm{f}}^{\circ}=-285.9 \mathrm{kJ} / \mathrm{mol}\right)$$ $$\mathrm{CH}_{4}(g)+2 \mathrm{O}_{2}(g) \rightarrow \mathrm{CO}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(\ell) \quad \Delta H_{\mathrm{rxn}}^{\circ}=890.4 \mathrm{kJ} / \mathrm{mol}$$

Step 3: Add the modified reactions to obtain the formation reaction

Now, add up the modified reactions: $$\mathrm{C}(s)+\mathrm{O}_{2}(g) \rightarrow \mathrm{CO}_{2}(g)$$ $$2\left[\mathrm{H}_{2}(g)+ \frac{1}{2}\mathrm{O}_{2}(g) \rightarrow \mathrm{H}_{2}\mathrm{O}(\ell)\right]$$ $$\underline{\phantom{a}\hspace{3.83cm}+\hspace{3.83cm}\phantom{a}}$$ $$\mathrm{CH}_{4}(g) + 2\mathrm{O}_{2}(g) \rightarrow \mathrm{CO}_{2}(g)+2 \mathrm{H}_{2} \mathrm{O}(\ell)$$ $$\mathrm{C}(s) + 2\mathrm{H}_{2}(g) \rightarrow \mathrm{CH}_{4}(g)$$

Step 4: Add corresponding enthalpy changes to find the enthalpy of formation of CH4(g)

Add the corresponding enthalpy changes of the modified reactions to obtain the enthalpy change of formation of \(\mathrm{CH}_{4}(g)\): $$\Delta H_f^{\circ}(\mathrm{CH}_{4}) = -393.5 + 2(-285.9) + 890.4$$ $$\Delta H_f^{\circ}(\mathrm{CH}_{4}) = -393.5 - 571.8 + 890.4$$ $$\Delta H_f^{\circ}(\mathrm{CH}_{4}) = -965.3 + 890.4$$ Hence, the standard enthalpy of formation of \(\mathrm{CH}_{4}(g)\) is: $$\Delta H_f^{\circ}(\mathrm{CH}_{4}) = -74.9 \ \mathrm{kJ} / \mathrm{mol}$$

Key Concepts

Standard Enthalpy of Formation

The standard enthalpy of formation is a crucial concept in chemistry. It measures the energy change when one mole of a compound forms from its elements in their standard states. This helps us understand how substances release or absorb energy during their formation. The standard state of an element is its most stable form at 1 atmosphere and a specified temperature, usually 25°C.
To calculate the standard enthalpy of formation, you need data on the enthalpies of the chemical equations involved. For example, if we look at the formation of methane (\(\mathrm{CH}_4(g)\)), the data typically includes the enthalpy changes for the combustion of carbon and hydrogen to form carbon dioxide and water. Using this information, plus Hess's law, we can determine the enthalpy of formation.

• Formation reactions always create compounds from their elemental constituents.
• The standard enthalpy of formation values are usually found in tables for reference.
• Symbols: The standard enthalpy of formation is denoted as \(\Delta H_f^{\circ}\).

Having the standard enthalpy of formation provides insight not only into the possible energy exchange in formation reactions but also allows us to predict compound stability and types of bonds formed.

Chemical Thermodynamics

Chemical thermodynamics is the study of energy changes during chemical reactions. It helps us understand how energy in the form of heat is absorbed or released, which is key to predicting reaction behavior. Using Hess's law, a central principle here, thermodynamics allows the calculation of reaction enthalpy changes.
In a chemical reaction:

• Energy conservation is fundamental. The total energy of products and reactants remains constant unless exchanged with surroundings.
• Enthalpy (denoted by \(H\)) refers to the heat content of a system at constant pressure.
• Hess's law states that the total enthalpy change is the sum of enthalpy changes for each step, regardless of multiple stages or paths.

The principles of chemical thermodynamics are essential when using Hess's law. They allow chemists to break down complex reactions into simpler, manageable parts, ensuring accurate enthalpy calculations.

Chemical Equation Balancing

Balancing chemical equations is a fundamental skill in chemistry. It ensures that the number of atoms for each element is the same on both sides of a chemical reaction. Balancing is crucial as it maintains the law of conservation of mass, which states that matter cannot be created or destroyed in a chemical reaction.
To balance a chemical equation:

• Start by counting the number of atoms for each element on both sides of the equation.
• Adjust coefficients to achieve the same number of atoms for every element.
• Check your work to ensure all elements are balanced and the total charge is the same on both sides if applicable.
• Repeat the process as necessary for accuracy.

Balancing equations plays a critical role in using Hess's law. It is vital before calculating enthalpy changes because an unbalanced equation gives inaccurate energetic insights. Adequately balanced equations ensure that stoichiometric and thermal considerations are correct, leading to consistent results in thermodynamic calculations.