Question

The following data compare the standard enthalpies and free energies of formation of some crystalline ionic substances and aqueous solutions of the substances: $$ \begin{array}{lrr} \text { Substance } & \Delta \boldsymbol{H}_{f}^{\circ}(\mathbf{k} \mathbf{J} / \mathbf{m o l}) & \Delta \mathbf{G}_{f}^{\circ}(\mathbf{k J} / \mathbf{m o l}) \\ \hline \mathrm{AgNO}_{3}(s) & -124.4 & -33.4 \\ \mathrm{AgNO}_{3}(a q) & -101.7 & -34.2 \\ \mathrm{MgSO}_{4}(s) & -1283.7 & -1169.6 \\ \mathrm{MgSO}_{4}(a q) & -1374.8 & -1198.4 \end{array} $$ (a) Write the formation reaction for \(\mathrm{AgNO}_{3}(s) .\) Based on this reaction, do you expect the entropy of the system to increase or decrease upon the formation of \(\mathrm{AgNO}_{3}(s) ?\) (b) Use \(\Delta H_{f}^{\circ}\) and \(\Delta G_{f}^{\circ}\) of \(\mathrm{AgNO}_{3}(s)\) to determine the entropy change upon formation of the substance. Is your answer consistent with your reasoning in part (a)? (c) Is dissolving \(\mathrm{AgNO}_{3}\) in water an exothermic or endothermic process? What about dissolving \(\mathrm{MgSO}_{4}\) in water? (d) For both \(\mathrm{AgNO}_{3}\) and \(\mathrm{MgSO}_{4},\) use the data to calculate the entropy change when the solid is dissolved in water. (e) Discuss the results from part (d) with reference to material presented in this chapter and in the "A Closer Look" box on page 540 .

Short answer

In summary, the formation of AgNO3(s) involves a decrease in entropy, as evidenced by the formation reaction and the calculated negative entropy change. Dissolving both AgNO3 and MgSO4 in water leads to an increase in entropy, with AgNO3 dissolution being endothermic and MgSO4 dissolution being exothermic. These results are consistent with the general trends of ionic substances dissolving in water, as dissolution typically results in increased disorder and entropy in the system.

Step-by-step solution

(a) Writing the formation reaction for AgNO3(s) and predicting entropy change

To write the formation reaction for AgNO3(s), we need to consider its constituent elements in their standard states and react them to form one mole of AgNO3(s): $$ \mathrm{Ag(s)} + \frac{1}{2}\mathrm{N_2(g)} + \frac{3}{2}\mathrm{O_2(g)} \rightarrow \mathrm{AgNO_3(s)} $$ We can observe that the reaction involves a transition from gaseous molecules (N2 and O2) to a solid ionic compound (AgNO3). As gases have higher entropy than solids, we expect the entropy of the system to decrease upon the formation of AgNO3(s).

(b) Calculating the entropy change and checking consistency

We can use the standard enthalpy of formation, \(\Delta H_{f}^{\circ}\), and the standard free energy of formation, \(\Delta G_{f}^{\circ}\), to calculate the entropy change, \(\Delta S_{f}^{\circ}\), upon the formation of AgNO3(s). This relationship can be described using the formula: $$ \Delta G_{f}^{\circ} = \Delta H_{f}^{\circ} - T\Delta S_{f}^{\circ} $$ We will assume standard conditions, so the temperature T = 298 K. Rearranging the formula to solve for entropy, we have: $$ \Delta S_{f}^{\circ} = \frac{\Delta H_{f}^{\circ} - \Delta G_{f}^{\circ}}{T} $$ Plugging in the given values for AgNO3(s), we get: $$ \Delta S_{f}^{\circ} = \frac{-124.4 - (-33.4)}{298} = -0.305\,\mathrm{kJ/mol\,K} $$ This negative value indicates a decrease in entropy, which is consistent with our reasoning in part (a).

(c) Determining exo- or endothermic dissolving processes

To determine whether dissolving AgNO3 and MgSO4 in water is an exothermic or endothermic process, we can simply compare the standard enthalpies of formation for the solid and the aqueous states: For AgNO3: $$ \Delta H_{aq} - \Delta H_{s} = -101.7 - (-124.4) = 22.7\,\mathrm{kJ/mol} $$ Since the value is positive, dissolving AgNO3 in water is an endothermic process. For MgSO4: $$ \Delta H_{aq} - \Delta H_{s} = -1374.8 - (-1283.7) = -91.1\,\mathrm{kJ/mol} $$ Since the value is negative, dissolving MgSO4 in water is an exothermic process.

(d) Calculating entropy change for dissolving

We can use the same method as in part (b) to calculate the entropy change, \(\Delta S\), for dissolving the ionic solids in water. This time, we will use the values for the aqueous states: For AgNO3: $$ \Delta S_{AgNO_3} = \frac{\Delta H_{aq} - \Delta G_{aq}}{T} = \frac{-101.7 - (-34.2)}{298} = 0.227\,\mathrm{kJ/mol\,K} $$ For MgSO4: $$ \Delta S_{MgSO_4} = \frac{\Delta H_{aq} - \Delta G_{aq}}{T} = \frac{-1374.8 - (-1198.4)}{298} = 0.592\,\mathrm{kJ/mol\,K} $$ Both values are positive, indicating an increase in entropy upon dissolving the ionic solids in water.

(e) Discussing the results

The results from part (d) show that dissolving AgNO3 and MgSO4 in water leads to entropy increases. This is consistent with processes involving the dissolution of ionic compounds because dissolving ionic substances typically leads to an increase in disorder due to the separation of ions in solution. This increase in disorder is manifested as an increase in entropy, which is evident from the positive values calculated for \(\Delta S\) in part (d). The differences in the magnitudes of the various entropy changes can be attributed to the specific structures and bonding of the ionic substances, which can affect the degree of disorder in solution. In general, entropy changes can provide valuable insight into the spontaneity and energetics of chemical reactions and processes, as demonstrated in this exercise.

Key concepts

Standard Enthalpy of Formation

The standard enthalpy of formation, denoted as \( \Delta H_{f}^{\circ} \), is a thermodynamic quantity that measures the heat change when one mole of a compound is formed from its elements at standard conditions (25°C, 1 atmosphere pressure). These standard conditions allow chemists to compare the thermochemical properties of different substances uniformly.

For instance, when considering the formation of \( \text{AgNO}_3(s) \), the \( \Delta H_{f}^{\circ} \), which indicates the heat absorbed or released during the formation, is a crucial indicator of whether the process is exothermic or endothermic. A negative value implies that the reaction releases heat, thus being exothermic, as in the formation of \( \text{AgNO}_3(s) \) (-124.4 kJ/mol).

By knowing the \( \Delta H_{f}^{\circ} \) value, we get a glimpse of the energetic aspect of forming a compound, which is essential when predicting reaction behavior and stability. Students can use enthalpy values to infer whether a substance forms readily or needs additional energy input.

Standard Free Energy of Formation

Standard free energy of formation, denoted as \( \Delta G_{f}^{\circ} \), provides insight into the spontaneity of a formation reaction under standard conditions. It integrates both enthalpic and entropic factors to predict whether a process will occur without external energy input.

Based on the given data for \( \text{AgNO}_3(s) \), where \( \Delta G_{f}^{\circ} \) is -33.4 kJ/mol, we can conclude that the formation of \( \text{AgNO}_3 \) from its elements is a spontaneous process at standard conditions. When \( \Delta G_{f}^{\circ} \) is negative, it denotes spontaneity whereas a positive value would indicate non-spontaneity, requiring an energy input to proceed.

Understanding \( \Delta G_{f}^{\circ} \) is fundamental because it guides us in predicting the feasibility of chemical reactions and also helps us assess the influence of temperature and entropy on a reaction's spontaneity.

Entropy Change

Entropy, symbolized as \( S \), is a measure of disorder or randomness in a system. An entropy change, \( \Delta S \), occurs when there is a difference in entropy between the initial and final states of a reaction or a process. In chemistry, an increase in entropy typically corresponds to more disorder, such as when a solid melts into a liquid or dissolves into a solution.

For the formation of \( \text{AgNO}_3(s) \), we expect a decrease in entropy since we're forming a solid from gaseous reactants, which means there's less disorder in the system. This expectation is confirmed when we calculate \( \Delta S_{f}^{\circ} \), yielding a negative value, indicative of the decrease in entropy upon formation.

Understanding entropy changes is crucial for grasping how energy disperses in a chemical reaction. It can also help explain why certain reactions, like the dissolving of ionic compounds, typically increase in entropy due to increased dispersion of ions.

Dissolving Ionic Compounds

When ionic compounds dissolve in water, the ions are surrounded by water molecules, a process which greatly increases the entropy of the system. This is because the solid crystalline structure, which has a low entropy due to its orderly arrangement, is broken down into individual ions that move freely in solution, leading to a higher state of disorder.

As observed in AgNO3 and MgSO4, dissolving these compounds in water results in positive entropy changes, indicating increased disorder. For instance, AgNO3's \( \Delta S \) of 0.227 kJ/mol·K reflects this rise in entropy. Such processes can be endothermic or exothermic, with endothermic processes like dissolving AgNO3 requiring heat because the process absorbs energy to break the lattice structure of the solid. Conversely, exothermic processes, such as the dissolving of MgSO4, release energy.

Students should note that while the change in entropy is a central factor, understanding the overall energy change, involving both enthalpy and free energy, is vital for a complete thermodynamic analysis of dissolving ionic compounds.