Question

The chemical equation for the hydration of an alkali metal ion is \(M^{+}(g) \rightarrow M^{+}(a q) .\) The Gibbs energy change and the enthalpy change for the process are denoted by \(\Delta G_{\text {hydr. }}^{\circ}\) and \(\Delta H_{\text {hydr. }}^{\circ}\) respectively. \(\Delta G_{\text {hydr. }}^{\circ}\) and \(\Delta H_{\text {hydr. values are given below for the alkali }}\) metal ions. $$\mathrm{M}^{+} \quad \mathrm{Li}^{+} \quad \mathrm{Na}^{+} \quad \mathrm{K}^{+} \quad \mathrm{Rb}^{+} \quad \mathrm{Cs}^{+}$$ $$\begin{array}{llllll} \Delta H_{\text {hydr. }}^{\circ} & -522 & -407 & -324 & -299 & -274 \mathrm{kJ} \mathrm{mol}^{-1} \end{array}$$ $$\begin{array}{llllll} \Delta G_{\text {hydr. }}^{\circ} & -481 & -375 & -304 & -281 & -258 \mathrm{kJ} \mathrm{mol}^{-1} \end{array}$$ Use the data above to calculate \(\Delta S_{\text {hydr. }}^{\circ}\) values for the hydration process. Explain the trend in the \(\Delta S_{\text {hydr. }}^{\circ}\) values.

Short answer

\(\Delta S_{\text {hydr. Li+}}^{\circ} = 0.138 kJ/mol.K, \Delta S_{\text {hydr. Na+}}^{\circ} = 0.107 kJ/mol.K, \Delta S_{\text {hydr. K+}}^{\circ} = 0.067 kJ/mol.K, \Delta S_{\text {hydr. Rb+}}^{\circ} = 0.060 kJ/mol.K, \Delta S_{\text {hydr. Cs+}}^{\circ} = 0.053 kJ/mol.K. For the hydration of the ions; entropy change decreases as the size of the ion increases.

Step-by-step solution

- Calculate the Entropy Change for Li+

The entropy change for Li+ hydration is calculated using the thermodynamic equation: \(\Delta S_{\text {hydr. Li+}}^{\circ} =(\Delta H_{\text {hydr. Li+}}^{\circ} - \Delta G_{\text {hydr. Li+}}^{\circ}) / T\). \( \Delta S_{\text {hydr. Li+}}^{\circ} =(-522 kJ/mol -(-481 kJ/mol)) / 298K = 0.138 kJ/mol.K\)

- Calculate the Entropy Change for Na+, K+, Rb+, and Cs+

By following the same method from Step 1, the entropy change for the hydration of the other alkali metal ions is calculated. The numerical values obtained are: \(\Delta S_{\text {hydr. Na+}}^{\circ} = 0.107 kJ/mol.K, \Delta S_{\text {hydr. K+}}^{\circ} = 0.067 kJ/mol.K, \Delta S_{\text {hydr. Rb+}}^{\circ} = 0.060 kJ/mol.K, \Delta S_{\text {hydr. Cs+}}^{\circ} = 0.053 kJ/mol.K\)

- Explain the Trend for \(\Delta S_{\text {hydr. }}^{\circ}\) Values

The entropy change decreases as we move down the alkali metal group from Li+ to Cs+. This indicates that the entropy of hydration decreases as the size of the ion increases. This trend can be understood as a result of the differences in the orderly arrangements of the water molecules around the ions during hydration. Smaller ions like Li+ can organise water molecules in the hydrated shell more rigidly owing to their higher charge density compared to larger ions such as Cs+. Thus, more order implies less entropy, and vice versa. As a result, the hydration of smaller ions, such as Li+, results in a greater decrease in entropy compared to the hydration of larger ions like Cs+.

Key concepts

Gibbs Free Energy

Gibbs free energy is a fundamental concept in thermodynamics that refers to the energy associated with a chemical reaction that can be used to do work at a constant temperature and pressure. It is often symbolized as \( \Delta G \) and is calculated by the equation \( \Delta G = \Delta H - T\Delta S \), where \( \Delta H \) is the change in enthalpy, \( T \) is the temperature in Kelvin, and \( \Delta S \) is the change in entropy.

In the context of the hydration of alkali metal ions, the Gibbs free energy change \( \Delta G_{\text{hydr.}}^{\circ} \) provides insights into the spontaneity of the process. A negative \( \Delta G_{\text{hydr.}}^{\circ} \) value indicates that the hydration process occurs spontaneously. It is a crucial parameter in determining how easily an ion in the gas phase will become hydrated when placed in water.

In the exercise provided, specific \( \Delta G_{\text{hydr.}}^{\circ} \) values are given for each ion, which illustrates how different alkali metal ions interact with water molecules energetically when they become hydrated.

Enthalpy Change

Enthalpy change, denoted as \( \Delta H \), is the total heat content change during a chemical reaction at constant pressure. It is also a part of the key thermodynamic equation used to calculate Gibbs free energy. In reactions where \( \Delta H \) is negative, the process releases heat and is termed exothermic, suggesting that the system loses heat to the surroundings.

In the textbook exercise, the \( \Delta H_{\text{hydr.}}^{\circ} \) values for the hydration of various alkali metal ions are provided. These values are essential for determining the amount of heat absorbed or released when an ion transitions from a gaseous state to an aqueous state. It’s noteworthy that the \( \Delta H_{\text{hydr.}}^{\circ} \) value decreases for larger ions, indicating that less heat is released during the hydration of these larger ions.

Thermodynamics

Thermodynamics is the branch of physics that deals with the relationships between heat and other forms of energy. In the context of chemistry, it particularly involves the study of energy changes in chemical reactions. The main laws of thermodynamics govern how systems respond to changes in their environment, affecting temperature, volume, and form. In this exercise, thermodynamics help to explain how alkali metal ions interact with water based on the energy changes, including Gibbs free energy and enthalpy.

Thermodynamics allows us to analyze the direction in which a process, such as the hydration of an ion, will naturally proceed. It uses concepts like enthalpy, entropy, and free energy to predict whether the ions will favorably dissolve in water at a given temperature. Thus, thermodynamic principles provide the background needed to understand the calculated values for \( \Delta G_{\text{hydr.}}^{\circ} \) and \( \Delta H_{\text{hydr.}}^{\circ} \) and relate them to the more conceptually challenging \( \Delta S_{\text{hydr.}}^{\circ} \) values.

Entropy Change

Entropy change (\( \Delta S \) is a measure of the disorder or randomness in a system. An increase in entropy corresponds to an increase in disorder, while a decrease indicates a more ordered state. In chemical processes, entropy can play a significant role in determining the spontaneity of a reaction or a physical change.

During the hydration of alkali metal ions, as seen in the exercise, the entropy change \( \Delta S_{\text{hydr.}}^{\circ} \) is a way to quantify the change in order when ions in the gas phase become surrounded by water molecules in the aqueous phase. Smaller ions, which have a stronger charge density, can more significantly structure water molecules in their hydration shells, resulting in a larger decrease in entropy. This trend is highlighted in the solution provided, which demonstrates that larger ions like Cs+ contribute to a smaller entropy change upon hydration because the water molecules around them remain relatively more disordered compared to those around smaller ions like Li+.