Question

Calculate \(\Delta_{\mathrm{f}} H\) for \(\mathrm{ZnSO}_{4}(\mathrm{~s})\) from the following data: \(\mathrm{ZnS}(\mathrm{s}) \rightarrow \mathrm{Zn}(\mathrm{s})+\mathrm{S}\) (rhombic), \(\Delta H_{1}\) \(=44 \mathrm{kcal} / \mathrm{mol}\) \(2 \mathrm{ZnS}(\mathrm{s})+3 \mathrm{O}_{2}(\mathrm{~g}) \rightarrow 2 \mathrm{ZnO}(\mathrm{s})+2 \mathrm{SO}_{2}(\mathrm{~g})\) \(\Delta H_{2}=-221.88 \mathrm{kcal} / \mathrm{mol}\) \(2 \mathrm{SO}_{2}(\mathrm{~g})+\mathrm{O}_{2}(\mathrm{~g}) \rightarrow 2 \mathrm{SO}_{3}(\mathrm{~g}), \quad \Delta H_{3}\) \(=-46.88 \mathrm{kcal} / \mathrm{mol}\) \(\mathrm{ZnSO}_{4}(\mathrm{~s}) \rightarrow \mathrm{ZnO}(\mathrm{s})+\mathrm{SO}_{3}(\mathrm{~g}), \Delta H_{4}\) \(=55.1 \mathrm{kcal} / \mathrm{mol}\) (a) \(-233.48 \mathrm{kcal} / \mathrm{mol}\) (b) \(-343.48 \mathrm{kcal} / \mathrm{mol}\) (c) \(-434.84 \mathrm{kcal} / \mathrm{mol}\) (d) \(-311.53 \mathrm{kcal} / \mathrm{mol}\)

Short answer

-233.48 kcal/mol

Step-by-step solution

Write the Targeted Chemical Equation

The goal is to find the enthalpy change \( \Delta_{\mathrm{f}} H \) for the formation of \( \mathrm{ZnSO}_{4}(\mathrm{s}) \). The formation reaction is \( \mathrm{Zn}(\mathrm{s}) + \mathrm{S}(\mathrm{rhombic}) + 2\mathrm{O}_{2}(\mathrm{g}) \rightarrow \mathrm{ZnSO}_{4}(\mathrm{s}) \) with unknown \( \Delta_{\mathrm{f}} H \) which we are to calculate.

Manipulate Given Reactions to Match the Target Reaction

Reverse the fourth given reaction to match the formation of \(\text{ZnSO}_4(s)\) from its elements. This will change the sign of \(\Delta H_4\).Combine the first three reactions to form the desired reaction with \(\Delta H_{\text{f}}\) as a sum of the given enthalpy changes, taking into account the stoichiometry of each reaction.

Calculate the Enthalpy Change for the Target Reaction

Use Hess's Law to add the enthalpies of the given reactions after their manipulation. \(\Delta_{\text{f}} H = \Delta H_1 + \frac{1}{2}\Delta H_2 + \Delta H_3 - \Delta H_4\)

Key concepts

Understanding Hess's Law

Hess's Law is a fundamental principle in chemical thermodynamics that simplifies the calculation of enthalpy change, which is the heat absorbed or released during a chemical reaction. It states that the total enthalpy change for a chemical reaction is the same, regardless of the number of steps the reaction is carried out in or the path taken, as long as the initial and final conditions are the same.

For instance, if you're trying to find the enthalpy change of a complex reaction, you don't need to carry out the reaction in one go. Instead, you can break it down into simpler steps, find the enthalpy changes for these individual steps, and sum them up to obtain the total enthalpy change. This principle is especially useful when the direct measurement of enthalpy change is not possible.

In the context of the provided problem, Hess's Law allows us to calculate the unknown enthalpy change of formation for \(\mathrm{ZnSO}_{4}(\mathrm{s})\) by utilizing the enthalpy changes of other reactions that involve the same compounds.

Formation Reactions and Their Significance

A formation reaction involves creating one mole of a compound from its constituent elements in their standard states. It is a specific type of chemical reaction that has considerable importance in chemical thermodynamics due to its relevance to enthalpy change, or \(\Delta_{\mathrm{f}} H\).

The standard enthalpy of formation for a compound is a reference point used to calculate the energy changes in chemical reactions. When elements combine to form a compound, they release or absorb a specific amount of energy known as the 'enthalpy of formation'. This quantity is critical in designing chemical processes and understanding reaction energetics.

In our exercise, we are tasked with calculating the enthalpy change associated with the formation of \(\mathrm{ZnSO}_{4}(\mathrm{s})\) from its elements, which is a fundamental piece of information for anyone studying the compound's thermodynamic properties.

Stoichiometry: The Mathematics of Chemical Reactions

Stoichiometry is the aspect of chemistry that deals with quantifying the amounts of reactants and products in a chemical reaction. It's the math behind the mole concept, balancing equations, and calculating yields. Applying stoichiometry allows chemists to predict how much reactant is needed to produce a desired amount of product.

When we are dealing with enthalpy changes, stoichiometry becomes essential as it dictates the proportions of reactants and products involved. In the context of the enthalpy change calculation for our exercise, we had to adjust the stoichiometric coefficients of the given reactions to ensure they lined up correctly with the formation reaction for \(\mathrm{ZnSO}_{4}(\mathrm{s})\). This adjustment lets us apply Hess's Law accurately to find the desired enthalpy of formation.

Chemical Thermodynamics: A Brief Overview

Chemical thermodynamics is the study of the interrelation of heat and work with chemical reactions or with physical changes of state within the confines of the laws of thermodynamics. It involves concepts such as enthalpy, entropy, free energy, and the laws of thermodynamics themselves.

In this field, the enthalpy change gets immense importance as it tells us how much energy is exchanged with the surroundings during a chemical reaction. Whether a reaction absorbs heat (endothermic) or releases heat (exothermic) can be quantified by calculating the enthalpy changes, providing insights into feasibility and spontaneity of reactions.

The problem and solution we discussed drew heavily on this concept, making it clear how fundamental chemical thermodynamics is to understanding the energetic aspects of reactions and the calculation of the enthalpy changes associated with them.