At \(298 \mathrm{~K}\), the standard enthalpies of formation for \(\mathrm{C}_{2} \mathrm{H}_{2}(g)\) and \(\mathrm{C}_{6} \mathrm{H}_{6}(l)\) are \(227 \mathrm{~kJ} / \mathrm{mol}\) and \(49 \mathrm{~kJ} / \mathrm{mol}\), respectively. a. Calculate \(\Delta H^{\circ}\) for $$ \mathrm{C}_{6} \mathrm{H}_{6}(l) \longrightarrow 3 \mathrm{C}_{2} \mathrm{H}_{2}(g) $$ b. Both acetylene \(\left(\mathrm{C}_{2} \mathrm{H}_{2}\right)\) and benzene \(\left(\mathrm{C}_{6} \mathrm{H}_{6}\right)\) can be used as fuels. Which compound would liberate more energy per gram when combusted in air?

Understanding the standard enthalpies of formation is essential for any student dealing with thermodynamics and chemical reactions. These values, often represented by the symbol \( \Delta H_f^\circ \), are measured under standard conditions, which usually means a temperature of 298 K (25°C) and a pressure of 1 atm (101.325 kPa).

The standard enthalpy of formation is defined as the heat change that occurs when one mole of a compound is formed from its elements in their standard states. For example, the enthalpy of formation for \( \mathrm{C}_{6} \mathrm{H}_{6}(l) \) tells us the amount of energy needed to create benzene from its elemental forms of carbon and hydrogen, under standard conditions.

When using these values to calculate the enthalpy change \( \Delta H^\circ \) of a reaction, the formula incorporates the enthalpies of formation for all reactants and products, taking into account their stoichiometric coefficients. This allows students to estimate the heat exchanged during a chemical process, which is crucial for designing reactions and evaluating their feasibility in real-world applications.

Another topic of interest is the combustion energy per gram, which is particularly relevant when comparing fuel sources. This value indicates how much energy is released as heat when a certain mass of a substance combusts completely in air (often in the presence of oxygen).

For fuels like acetylene (\( \mathrm{C}_{2} \mathrm{H}_{2} \)) and benzene (\( \mathrm{C}_{6} \mathrm{H}_{6} \)), knowing the combustion energy per gram enables us to determine which substance would provide more energy by weight when used as a fuel. Calculating this requires two pieces of information: the enthalpy change for the combustion of one mole of the substance and its molar mass.

By dividing the enthalpy change by the molar mass, we obtain the energy release per gram. This calculation assumes complete combustion, which means all the carbon in the fuel is converted to \( \mathrm{CO}_2 \) and all the hydrogen to \( \mathrm{H}_2\mathrm{O} \). In real-world applications, the combustion process might be less efficient, but these theoretical values provide a good basis for comparison.

Finally, students should be well-versed with stoichiometric coefficients. In a balanced chemical equation, these coefficients indicate the proportion of reactants consumed and products formed. They play a pivotal role in calculations involving chemical reactions, such as determining the change in enthalpy as seen in the textbook solution.

For instance, in the reaction \( \mathrm{C}_{6} \mathrm{H}_{6}(l) \rightarrow 3 \mathrm{C}_{2} \mathrm{H}_{2}(g) \), the stoichiometric coefficient of acetylene is 3, reflecting that three moles of acetylene are produced for every mole of benzene that reacts.

These coefficients are crucial for ensuring that the principle of conservation of mass is upheld in chemical reactions. Therefore, when conducting enthalpy calculations, the stoichiometric coefficients are multiplied with the standard enthalpies of formation to reflect the quantities involved in a real-life chemical scenario, leading to a comprehensive understanding of the energy profile of the reaction.