Thermodynamics: Entropy, Gibbs Free Energy, and Spontaneity (Ch.19.3–19.6)

Thermodynamics: Entropy, Gibbs Free Energy, and Spontaneity

Entropy at the Molecular Level

Entropy (S) is a measure of disorder or randomness in a system. At the molecular level, entropy is related to the number of possible arrangements (microstates) of molecules in a system. Statistical thermodynamics connects macroscopic properties to microscopic behaviors.

• Microstate: A specific arrangement of the positions and energies of all molecules in a system.

• Boltzmann Equation: The relationship between entropy and microstates is given by $S = k \ln W$, where k is the Boltzmann constant and W is the number of microstates.

• Entropy increases as the number of microstates increases.

Molecular Motions: Molecules exhibit translational, vibrational, and rotational motions, each contributing to the number of microstates and thus the entropy.

• Translational: Movement from one place to another.

• Vibrational: Periodic motion of atoms within a molecule.

• Rotational: Rotation about an axis.

Factors Affecting Entropy

The number of microstates and thus entropy increases with:

• Increasing volume (more possible positions)

• Increasing temperature (greater distribution of molecular speeds)

• Increasing number of atoms/molecules (more degrees of freedom)

Entropy increases with the freedom of motion: $S_{\text{gas}} > S_{\text{liquid}} > S_{\text{solid}}$.

• Processes that increase entropy include formation of gases from solids or liquids, formation of liquids or solutions from solids, and increase in the number of gas molecules during a reaction.

The Third Law of Thermodynamics

The third law states that the entropy of a pure crystalline substance at absolute zero (0 K) is zero. At this temperature, the substance has only one microstate.

• At 0 K, $S = k \ln 1 = 0$.

Standard Molar Entropy and Entropy Changes in Reactions

Standard molar entropy ($S^\circ$) is the entropy of one mole of a substance under standard conditions. Entropy changes in reactions can be calculated similarly to enthalpy changes:

• $\Delta S^\circ = \sum n S^\circ(\text{products}) - \sum m S^\circ(\text{reactants})$

• Standard molar entropy is not zero for elements.

• Gases have higher entropy than liquids, which have higher entropy than solids.

• Entropy values generally increase with increasing molar mass and number of atoms in the formula.

Gibbs Free Energy ($\Delta G$)

Gibbs free energy is a thermodynamic state function that predicts the spontaneity of a reaction. It combines enthalpy and entropy:

• $\Delta G = \Delta H - T \Delta S$

• If $\Delta G < 0$, the reaction is spontaneous.

• If $\Delta G > 0$, the reaction is non-spontaneous.

• If $\Delta G = 0$, the system is at equilibrium.

Standard free energy changes can be calculated as:

• $\Delta G^\circ = \sum n \Delta G_f^\circ(\text{products}) - \sum m \Delta G_f^\circ(\text{reactants})$

• $\Delta G_f^\circ$ for elements in their standard state is zero.

Temperature Dependence of Free Energy

The sign and magnitude of $\Delta G$ depend on both $\Delta H$ and $\Delta S$, as well as temperature:

• $\Delta G = \Delta H - T \Delta S$

• Spontaneity can change with temperature, especially for reactions where $\Delta H$ and $\Delta S$ have opposite signs.

Key Concepts and Examples

• Spontaneous Process: Occurs without external intervention; $\Delta G < 0$.

• Irreversible Process: Cannot return to original state without external work.

• Reversible Process: Can return to original state; $\Delta S_{\text{univ}} = 0$.

• Entropy Change ($\Delta S$): Positive when disorder increases (e.g., solid to gas).

• Enthalpy Change ($\Delta H$): Positive for endothermic, negative for exothermic processes.

Example: Boiling water ($\text{H}_2\text{O}(l) \rightarrow \text{H}_2\text{O}(g)$) has $\Delta S > 0$ and $\Delta H > 0$.

Example: Condensation ($\text{H}_2\text{O}(g) \rightarrow \text{H}_2\text{O}(l)$) has $\Delta S < 0$ and $\Delta H < 0$.

Sample Calculations and Practice

• Calculate $\Delta S$ for vaporization of methanol using provided data:

• $\Delta S = S^\circ(\text{CH}_3\text{OH}(g)) - S^\circ(\text{CH}_3\text{OH}(l))$

• Boiling point temperature can be found by setting $\Delta G = 0$:

• $T_{\text{boiling}} = \frac{\Delta H_{\text{vap}}}{\Delta S_{\text{vap}}}$

Summary Table: Entropy and Spontaneity

Process	ΔS	ΔH	ΔG	Spontaneity
Solid → Liquid	+	+	Depends on T	Spontaneous at high T
Liquid → Gas	+	+	Depends on T	Spontaneous at high T
Gas → Liquid	-	-	Depends on T	Spontaneous at low T
Formation of gas from solid	+	Varies	Varies	Varies

Practice Questions and Applications

• Predict the sign of $\Delta S$ for sublimation, dissolution, and condensation processes.

• Calculate standard entropy changes for reactions using tabulated values.

• Determine spontaneity based on $\Delta G$ and temperature.

Additional info:

• Entropy is a central concept in predicting the direction of chemical reactions.

• Gibbs free energy links enthalpy and entropy to determine spontaneity.

• Standard molar entropy and free energy values are essential for quantitative calculations in thermochemistry.