Thermodynamics: Entropy, Free Energy, and Equilibrium Study Notes

Chapter 15: Thermodynamics - Entropy, Free Energy, and Equilibrium

Fundamental Concepts of Spontaneity

• Definition of Spontaneous Process: A process that does occur under a specific set of conditions.
• Definition of Nonspontaneous Process: A process that does not occur under a specific set of conditions.
• Spontaneity vs. Kinetics: Spontaneity is NOT the same as speed. Kinetics deals with the rate of a reaction, while thermodynamics deals with whether the reaction can occur at all.
• Comparative Examples of Spontaneity:     * Spontaneous Processes:         * Ice melting at room temperature.         * A ball rolling downhill.         * Iron rusting at room temperature.     * Nonspontaneous Processes:         * Water freezing at room temperature.         * A ball rolling uphill.         * Conversion of rust back to iron metal at room temperature.
• Enthalpy as an Incomplete Predictor: Spontaneity involves more than just enthalpy changes ($\Delta H$). Several spontaneous processes are endothermic (require energy):     * Dissolving $\text{NaI}(s)$ in water is endothermic (\Delta H > 0) but spontaneous.     * Ice melting above $0 \, ^\circ \text{C}$ is endothermic (\Delta H > 0) but spontaneous.     * $H_2O$ evaporating is endothermic (\Delta H > 0) but spontaneous.
• Determining Factor: In the cases above, the substances move toward a state that is more "random" or characterized by more "disorder."

Entropy ($S$) and Microstates

• Definition of Entropy ($S$): A measure of how spread out or dispersed the system’s energy is. It is often correlated with the disorder or randomness of a system.     * Increasing disorder corresponds to an increase in entropy (\Delta S > 0).     * Order and disorder can be visualized through probabilities. For example, there is only one way to have cards in a specific order, but many ways to have cards not in order. More options result in greater entropy.
• Statistical Definitions:     * Entropy is frequently related to "chaos," but this is an oversimplification.     * True definition refers to Microstates ($W$): The specific arrangements of particles and energy distributions.     * The distribution with the largest number of energetically equivalent microstates is the most probable because it is the most random and energy is most dispersed.     * Nature moves toward a greater number of microstates because more possibilities are more favorable.
• Boltzmann Constant and Mathematical Definition:     * In 1868, Ludwig Boltzmann provided a quantitative definition of entropy:     * $S = k \ln(W)$     * $k$ = Boltzmann constant ($1.38 \times 10^{-23} \, J \, K^{-1}$).     * $W$ = Number of energetically equivalent microstates.
• Entropy Practice: The Four Gas Particle Thought Experiment:     * Scenario: 4 gas particles (or balls) in two flasks (Left and Right).     * Arrangement possibilities:         * All 4 balls on the RIGHT side: 1 option (1 microstate).         * All 4 balls on the LEFT side: 1 option (1 microstate).         * 3 balls on the right side: 4 options (4 microstates).         * Most Probable Arrangement: 2 balls on each side (Greatest number of microstates).
• Entropy as a State Function:     * $\Delta S = S_{final} - S_{initial}$     * A process that increases the number of microstates moves the system to a more probable state and is therefore favorable (\Delta S > 0).

Standard Entropy and Trends

• Standard Entropy ($S^\circ$): The absolute entropy of a substance at $1 \, \text{atm}$ (typically at $25 \, ^\circ \text{C}$).     * Units: $J \, mol^{-1} \, K^{-1}$ (Note: Enthalpies are typically provided in $kJ \, mol^{-1}$).     * Standard entropies are always positive, even for elements in their standard states.
• Trends in Standard Entropy:     1. Phases: $S^\circ$ for gas phase is greater than liquid or solid for the same substance (e.g., I_2(g) > I_2(s)).         * General System Trend: S^\circ_{vapor} > S^\circ_{liquid} > S^\circ_{solid}.         * Solution Trend: S^\circ_{aqueous} > S^\circ_{pure}.     2. Structural Complexity: For species with similar molar masses and the same phase, more complex structures have greater $S^\circ$ because more types of motion ("wiggles") are possible.         * O_3(g) > F_2(g)         * C_2H_6(g) > CH_4(g)     3. Allotropes: Different forms of the same atom. More ordered forms (where atoms have less mobility) have lower $S^\circ$.         * Diamond has a lower $S^\circ$ than graphite.         * Other allotropes follow the structural complexity rule (e.g., $O_2$ vs $O_3$; cyclic vs linear Phosphorus/Sulfur).     4. Monatomic Species: For heavier atoms, $S^\circ$ is greater (e.g., Helium has a lower $S^\circ$ than Neon).     5. Physical Conditions:         * Temperature: $S^\circ$ at higher temp > $S^\circ$ at lower temp.         * Moles of Gas: $S^\circ$ for more moles of gas > $S^\circ$ for fewer moles of gas.         * Volume: $S^\circ$ for larger volume of gas > $S^\circ$ for smaller volume of gas.

Calculating Entropy Changes and Thermodynamics Laws

• Entropy Changes in a System ($\Delta S^\circ_{rxn}$): Calculated using the table of standard values.     * $\Delta S^\circ_{rxn} = \sum n S^\circ_{products} - \sum m S^\circ_{reactants}$     * Where $n$ and $m$ are coefficients from the balanced reaction.
• Comparison to Enthalpy Calculation:     * $\Delta H^\circ_{rxn} = \sum n \Delta H_f^\circ(products) - \sum m \Delta H_f^\circ(reactants)$     * Note: $\Delta H_f^\circ$ for an element in its most stable form (e.g., $O_2(g)$, $C(graphite)$) is $0$ by convention; this is NOT true for absolute entropy.
• Third Law of Thermodynamics: The entropy of a perfect crystalline substance at $0 \, K$ is zero ($S = k \ln(1) = 0$).     * This allows for the calculation of absolute entropies ($S^\circ$) using $0 \, K$ as the reference point.
• Second Law of Thermodynamics: The universe is composed of the System (the reaction) and the Surroundings (everything else).     * Spontaneous process: Entropy of the universe increases (\Delta S_{universe} > 0).     * Equilibrium process: Entropy of the universe remains unchanged ($\Delta S_{universe} = 0$). An equilibrium process can be induced to occur by adding/removing energy at the equilibrium point (e.g., melting ice exactly at $0 \, ^\circ \text{C}$).     * $\Delta S_{universe} = \Delta S_{system} + \Delta S_{surroundings}$     * Note: $\Delta S_{system}$ can be negative as long as $\Delta S_{surroundings}$ is sufficiently positive to make the sum greater than zero.

Entropy Changes in the Surroundings

• Relation to Enthalpy: $\Delta S_{surroundings} \propto -\Delta H_{system}$. An exothermic process ($-\Delta H$) corresponds to a positive entropy change in the surroundings.
• Relation to Temperature: $\Delta S_{surroundings} \propto T^{-1}$. The change is inversely proportional to temperature; heat transfer has a larger effect on entropy at lower temperatures.
• Atkins Equation: Combined expression:     * $\Delta S_{surroundings} = -\frac{\Delta H_{system}}{T}$
• Metaphor for Temperature Dependence: Small energy transfer to a high-temp system (disorganized) is like sneezing in a busy street—barely noticed. Small energy transfer to a low-temp system (highly organized) is like sneezing in a quiet library—highly disruptive.
• Spontaneity Practice Problem:     * Given: $\Delta S_{system} = -187.5 \, J \, K^{-1} \, mol^{-1}$; $\Delta H_{system} = -35.8 \, kJ \, mol^{-1}$ at $25 \, ^\circ \text{C}$ ($298 \, K$).     * $\Delta S_{surroundings} = -\frac{-35800 \, J \, mol^{-1}}{298 \, K} = 120.0 \, J \, mol^{-1} \, K^{-1}$.     * $\Delta S_{universe} = -187.5 + 120.0 = -67.5 \, J \, K^{-1} \, mol^{-1}$.     * Since \Delta S_{universe} < 0, the reaction is non-spontaneous.

Gibbs Free Energy ($G$)

• Definition: A thermodynamic function defined solely based on the system that predicts spontaneity.
• Derivation from Second Law:     * \Delta S_{universe} = \Delta S_{system} + \left[-\frac{\Delta H_{system}}{T}\right] > 0     * T \Delta S_{universe} = T \Delta S_{system} - \Delta H_{system} > 0     * Multiplying by -1: -T \Delta S_{universe} = \Delta H_{system} - T \Delta S_{system} < 0     * Resulting function: $\Delta G = \Delta H - T \Delta S$
• Predicting Spontaneity using $\Delta G$:     * $\Delta G < 0$: Spontaneous in the forward direction.     * $\Delta G > 0$: Non-spontaneous in the forward direction.     * $\Delta G = 0$: The system is at equilibrium.
• Summary Table/Cases for $\Delta G = \Delta H - T \Delta S$:     * Case 1: $\Delta H < 0, \Delta S > 0$: $\Delta G$ is always negative. Spontaneous at all temperatures. Example: Building collapsing.     * Case 2: $\Delta H > 0, \Delta S < 0$: $\Delta G$ is always positive. Spontaneous never (non-spontaneous at all temperatures). Example: Bricks spontaneously forming a building.     * Case 3: $\Delta H > 0, \Delta S > 0$: \Delta G < 0 only at High Temperatures ($T \Delta S$ must be larger than $\Delta H$). Example: Ice melting, water evaporating.     * Case 4: $\Delta H < 0, \Delta S < 0$: \Delta G < 0 only at Low Temperatures ($T \Delta S$ must be smaller in magnitude than $\Delta H$). Example: Water freezing, condensation.

Thermodynamics of Phase Changes and Standard Conditions

• Determining Temperature of Spontaneity: To find the threshold temperature, set the equilibrium condition $\Delta G = 0$.     * $0 = \Delta H - T \Delta S \implies T = \frac{\Delta H}{\Delta S}$
• Entropy Change for Phase Change:     * $\Delta S = \frac{\Delta H}{T}$     * For Melting: $\Delta S_{fusion} = \frac{\Delta H_{fus}}{T_{mp}}$     * For Boiling: $\Delta S_{vaporization} = \frac{\Delta H_{vap}}{T_{bp}}$
• Standard Free-Energy Change ($\Delta G^\circ$): $\Delta G$ under standard state conditions ($1 \, \text{atm}$, typ. $25 \, ^\circ \text{C}$, $1 \, M$ concentration for solutions).     * $\Delta G_{rxn}^\circ = \sum n \Delta G_f^\circ(products) - \sum m \Delta G_f^\circ(reactants)$     * Practice: $2 \text{KClO}_3(s) \rightarrow 2 \text{KCl}(s) + 3 \text{O}_2(g)$     * $\Delta G^\circ = [2(-408.3 \, kJ \, mol^{-1}) + 3(0)] - [2(-289.9 \, kJ \, mol^{-1})] = -236.8 \, kJ \, mol^{-1}$ (Spontaneous).
• Calculating $\Delta G^∘$ at Non-standard Temperatures:     * $\Delta H$ and $\Delta S$ are relatively insensitive to temperature changes.     * $\Delta G_T^∘ \approx \Delta H_{298}^∘ - T \Delta S_{298}^∘$     * Validity: Assumes no phase changes occur in the temperature range. For aqueous species, restricted to approx. $1 \, ^\circ \text{C}$ to $99 \, ^\circ \text{C}$.

Free Energy and Equilibrium ($K$)

• Standard ($\Delta G^∘$) vs. Actual ($\Delta G$):     * $\Delta G^∘$: Standard textbook value; indicates if products ($-$) or reactants ($+$) are favored at equilibrium.     * $\Delta G$: What is actually happening in the reaction; indicates spontaneity and the amount of work that can be done.
• Spontaneity and Reaction Quotient ($Q$):     * If K > Q ($\ln(Q/K) < 0$): Reaction proceeds RIGHT; $\Delta G < 0$.     * If $K < Q$ ($\ln(Q/K) > 0$): Reaction proceeds LEFT; \Delta G > 0.     * If $K = Q$ ($\ln(Q/K) = 0$): Equilibrium; $\Delta G = 0$.
• Thermodynamic Equilibrium Equations:     * $\Delta G = \Delta G^∘ + RT \ln(Q)$     * At equilibrium ($\Delta G = 0, Q = K$):     * $\Delta G^∘ = -RT \ln(K)$     * $R = 8.314 \, J \, mol^{-1} \, K^{-1}$
• Relationship between $\Delta G^∘$ and $K$:     * Large negative $\Delta G^∘$: Equilibrium goes to completion (large $K$).     * Large positive $\Delta G^∘$: Essentially no forward reaction (small $K$).     * $\Delta G^∘ = 0$: neither reactants nor products favored ($K = 1$).     * Table of Magnitude (approximate) at $298 \, K$:         * $\Delta G^∘ = 200 \, kJ \implies K = 9 \times 10^{-36}$         * $\Delta G^∘ = 50 \, kJ \implies K = 2 \times 10^{-9}$         * $\Delta G^∘ = 0 \, kJ \implies K = 1$         * $\Delta G^∘ = -50 \, kJ \implies K = 6 \times 10^{8}$         * $\Delta G^∘ = -200 \, kJ \implies K = 1 \times 10^{35}$
• Temperature Effects on Equilibrium: Since $\Delta G^∘$ changes with temperature, so does $K$. This can be calculated as $\Delta G_T^∘ = -RT \ln(K_T)$.

Summary and Visualization

• Free Energy Diagram:     * Plots Free Energy ($G$) vs. Reaction Coordinate.     * $\Delta G^∘ = G^∘<em>{final} - G^∘</em>{initial}$.     * The lowest point on the curve represents equilibrium.     * The slope of the curve at any point is $\Delta G$.     * $\Delta G^∘$ tells us the position of equilibrium (magnitude of $K$). The bigger the absolute value, the further to the left or right the equilibrium lies.     * $\Delta G$ tells us the direction the reaction must move to reach equilibrium (approach from either side depending on $Q$ vs $K$).