Question

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

Short answer

The enthalpy change for the reaction \(\mathrm{S}^{-}(g)+\mathrm{e}^{-} \rightarrow \mathrm{S}^{2-}(g)\) is \(\Delta H = -477\,\mathrm{kJ/mol}\) with an uncertainty of \(\pm 2\,\mathrm{kJ/mol}\).

Step-by-step solution

Write the target reaction

Write the target reaction which we want to find the enthalpy change for: $$ \mathrm{S}^{-}(g)+\mathrm{e}^{-}\rightarrow \mathrm{S}^{2-}(g) $$

Create the reaction chain

We will create a reaction chain (Hess's Law) using the given data and combining them to form the target reaction. We will reverse the first given reaction since we want to start from the \(\mathrm{S}^{-}(g)\) species: (i) $$ \mathrm{S}(g) \rightarrow \mathrm{S}(s) $$ (Reversing the 1st given reaction) $$ \Delta H_1 = -277\,\mathrm{kJ/mol} $$ (ii) $$ \mathrm{S}(g) + \mathrm{e}^{-} \rightarrow \mathrm{S}^{-}(g) $$ (2nd given reaction) $$ \Delta H_2 = -200\,\mathrm{kJ/mol} $$

Combine the reactions and calculate ΔH for the target reaction

Now add up reactions (i) and (ii) to get the target reaction and calculate the enthalpy change, \(\Delta H\): $$ \begin{aligned} \text{(i)} ~~~~~~\mathrm{S}(g) &\rightarrow \mathrm{S}(s) &\Delta H_1 = &-277\,\mathrm{kJ/mol} \\ \text{(ii)} ~~~~~~\mathrm{S}(g) + \mathrm{e}^{-} &\rightarrow \mathrm{S}^{-}(g) &\Delta H_2 = &-200\,\mathrm{kJ/mol} \\ \hline \text{Target} ~~~~\mathrm{S}^{-}(g)+\mathrm{e}^{-} &\rightarrow \mathrm{S}^{2-}(g) &\Delta H = &\Delta H_1 + \Delta H_2 \\ \end{aligned} $$ $$ \Delta H = (-277\,\mathrm{kJ/mol}) + (-200\,\mathrm{kJ/mol}) = -477\,\mathrm{kJ/mol} $$ The enthalpy change for the target reaction is \(\Delta H = -477\,\mathrm{kJ/mol}\).

Estimate the uncertainty

All values are known to \(\pm 1\,\mathrm{kJ/mol}\). As we have added the given \(\Delta H\) values, we also sum the uncertainties: $$ \text{Uncertainty} = 1\,\mathrm{kJ/mol} + 1\,\mathrm{kJ/mol} = 2\,\mathrm{kJ/mol} $$ The estimated uncertainty for the enthalpy change of the target reaction is \(\pm 2\,\mathrm{kJ/mol}\).

Final Answer

The enthalpy change for the reaction \(\mathrm{S}^{-}(g)+\mathrm{e}^{-} \rightarrow \mathrm{S}^{2-}(g)\) is \(\Delta H = -477\,\mathrm{kJ/mol}\) with an uncertainty of \(\pm 2\,\mathrm{kJ/mol}\).

Key concepts

Enthalpy Change

The concept of enthalpy change is a cornerstone in the study of chemical reactions and energy changes. It represents the heat exchanged with the surroundings at constant pressure during a chemical process. This is denoted by \( \Delta H \) and measured in kilojoules per mole (kJ/mol). Positive values of \( \Delta H \) indicate an endothermic reaction where heat is absorbed, while negative values indicate an exothermic reaction with the release of heat.

Understanding the enthalpy change is crucial because it helps scientists and engineers calculate the energy requirements or outputs of chemical reactions, such as those in industrial processes or new energy resources development. It is also vital in predicting the spontaneity of a reaction along with entropy considerations—another thermodynamic function.

Chemical Thermodynamics

Delving into chemical thermodynamics allows us to describe the energy changes that accompany chemical reactions. It's the arena where we apply the laws of thermodynamics to chemistry. Chemical thermodynamics provides us the tools to predict whether a reaction can occur spontaneously and to what extent it will proceed.

One of these tools is Hess's Law, which states that the total enthalpy change in a chemical reaction is the same, regardless of the number of steps taken to complete the reaction. This principle is employed when direct measurement of enthalpy change is tricky, allowing us to calculate it indirectly from other known reactions.

Revisiting Hess's Law

Using Hess's Law, we can combine several reactions whose \( \Delta H \) values are known to determine the \( \Delta H \) for a complex reaction. Each step correlates to a change in enthalpy that, when summed, gives the total \( \Delta H \) for the desired process, reflecting the conservation of energy.

Reaction Stoichiometry

Reaction stoichiometry refers to the quantitative relationship between the reactants and products in a balanced chemical equation. By using the coefficients from the balanced reaction, we can determine the proportions of substances consumed and formed. This aspect is not just theoretical; it has practical applications in predicting the amounts of substances needed and produced in chemical processes.

In the context of thermodynamics and Hess's Law, stoichiometry allows us to align the equations correctly so the enthalpy changes can be added or subtracted according to the stoichiometric multipliers of each species in the reaction. Only when reactions are correctly stoichiometrically balanced can we apply Hess's Law to find the desired \( \Delta H \) of a reaction.

Practical Application

When approaching exercises, evaluating the stoichiometry ensures accuracy in our calculations and precision in predicting energy changes. The combination of these concepts gives a comprehensive view enabling students to tackle complex thermodynamic problems confidently.