Question

Use the following data (in kJ/mol) to estimate \(\Delta H\) for the reaction \(S^{-}(g)+e^{-} \rightarrow S^{2-}(g)\) . Include an estimate of uncertainty. \(\begin{aligned} \mathrm{S}(s) \longrightarrow \mathrm{S}(g) & \Delta H=277 \mathrm{kJ} / \mathrm{mol} \\ \mathrm{S}(g)+\mathrm{e}^{-} \longrightarrow \mathrm{S}^{-}(g) & \Delta H=-200 \mathrm{kJ} / \mathrm{mol} \end{aligned}\) Assume that all values are known to \(\pm 1 \mathrm{kJ} / \mathrm{mol}\)

Short answer

The enthalpy change for the reaction \(S^-(g) + e^- \rightarrow S^{2-}(g)\) is \(\Delta H_\text{target} = 77\,\text{kJ/mol}\), with an estimated uncertainty of \(\pm 2\,\text{kJ/mol}\).

Step-by-step solution

Identify the target reaction and given reactions

The target reaction we want to find the enthalpy change for is \(S^-(g) + e^- \rightarrow S^{2-}(g)\). We are given two reactions: 1. \(S(s) \rightarrow S(g) \quad \Delta H = 277 \,\text{kJ/mol}\) 2. \(S(g) + e^- \rightarrow S^-(g) \quad \Delta H = -200 \,\text{kJ/mol}\)

Manipulate the given reactions to match the target reaction

Observe that if we add the two given reactions, we will get the target reaction: 1. \(S(s) \rightarrow S(g) \quad \Delta H = 277 \,\text{kJ/mol}\) 2. \(S(g) + e^- \rightarrow S^-(g) \quad \Delta H = -200 \,\text{kJ/mol}\) + -------------------------------------------- \((1 + 2)\) \(S^-(g) + e^- \rightarrow S^{2-}(g)\)

Add the enthalpy changes of the given reactions

To find the enthalpy change for the target reaction, we need to add the enthalpy changes for reactions 1 and 2: \(\Delta H_\text{target} = \Delta H_1 + \Delta H_2\) \(\Delta H_\text{target} = 277\,\text{kJ/mol} - 200\,\text{kJ/mol}\) \(\Delta H_\text{target} = 77\,\text{kJ/mol}\)

Calculate the uncertainty

We are given that the uncertainties for both reactions \(\Delta H_1\) and \(\Delta H_2\) are \(± 1\,\text{kJ/mol}\). When adding these values, we should add the uncertainties: \(\Delta H_\text{uncertainty} = \Delta H_{1, \text{uncertainty}} + \Delta H_{2, \text{uncertainty}}\) \(\Delta H_\text{uncertainty} = \pm 1\,\text{kJ/mol} + \pm 1\,\text{kJ/mol}\) \(\Delta H_\text{uncertainty} = \pm 2\,\text{kJ/mol}\)

Present the final answer

The enthalpy change for the target reaction \(S^-(g) + e^- \rightarrow S^{2-}(g)\) is \(\Delta H_\text{target} = 77\,\text{kJ/mol}\), with an estimated uncertainty of \(\pm 2\,\text{kJ/mol}\).

Key concepts

Thermodynamics

Thermodynamics is the branch of physics that deals with the relationships between heat and other forms of energy. In chemistry, it helps us understand how energy changes during chemical reactions.

An important concept in thermodynamics is the enthalpy ( egin{equation} H m{ ext{)}, which represents the total energy of a system, including both internal energy and the product of pressure and volume.

When a chemical reaction occurs, the enthalpy change ( egin{equation} ext{Δ}H m{ ext{)} indicates whether energy is absorbed or released. If egin{equation} ext{Δ}H m{ ext{)} is negative, the reaction releases energy, making it exothermic. Conversely, if egin{equation} ext{Δ}H m{ ext{)} is positive, the reaction takes in energy, making it endothermic.

Thermodynamics allows us to predict whether reactions will occur spontaneously. A spontaneous reaction generally decreases the free energy ( egin{equation} G m{ ext{)} of the system. While egin{equation} ext{Δ}H m{ ext{)} is a part of this calculation, it is not the only factor, as entropy ( egin{equation} S m{ ext{)} also plays a role. Understanding the balance between egin{equation} ext{Δ}H m{ ext{)} and egin{equation} ext{Δ}S m{ ext{)} can help predict a reaction's behavior under various conditions.

Reaction Enthalpy

Reaction enthalpy is the energy change that occurs during a chemical reaction. It tells us how much energy is absorbed or released. This energy change happens when bonds in the reactants break, and new bonds form in the products.

For example, in the exercise given, we have a target reaction: egin{equation} S^-(g) + e^- ightarrow S^{2-}(g) m{ ext{)}. To determine the enthalpy change, we combine other reactions quantitatively to see the overall energy change.

The challenge comes from calculating the enthalpy of a target reaction using known reactions. This is done by algebraically adding the enthalpy changes of each recognized reaction, keeping in mind the direction of the reaction and if one needs to be reversed. Always remember, reversing a reaction changes the sign of its enthalpy change.

By working through these analyses, students grasp how complex reactions are built out of simpler steps, and how we use known quantities to make educated predictions about unknown energies.

Uncertainty in Measurements

Uncertainties in measurements are common in scientific experiments and calculations. They reflect the precision of the measurements and the reliability of the calculated results.

In the context of enthalpy calculations, uncertainties arise from the accuracy of enthalpy values used in the steps of the calculation. For the exercise, the target reaction's uncertainty derives from the cumulative uncertainties of the individual reactions used to calculate it.

Generally, when adding or subtracting measured values, their uncertainties are also added. For example, if each known reaction enthalpy has an uncertainty of ± 1 kJ/mol, the total uncertainty for their sum would be ± 2 kJ/mol.

Understanding and computing uncertainties helps in evaluating how precise the enthalpy change calculations are. Even though we can estimate energy changes reasonably well, it is essential to acknowledge these uncertainties to express results more accurately. This practice allows us to critically assess how close to the actual values our results might be.