Entropy, Spontaneity, and Gibbs Free Energy Study Guide

Fundamentals of Entropy

• Entropy ($S$): This is defined as the amount of molecular randomness, or "disorder," in a system.

• State Function: Like enthalpy, entropy is a state function, meaning it depends only on the current state of the system, not the path taken to reach it.

• Calculating Change in Entropy ($\Delta S$): The change is determined by the final and initial states:     * $\Delta S = S_{final} - S_{initial}$

• Entropy Value Magnitude: The more entropy a substance possesses, the more positive its entropy value is.

• Units: Entropy is typically measured in Joules per Kelvin ($J/K$).

• 3rd Law of Thermodynamics: A perfectly ordered crystalline substance at $0\,K$ has zero entropy.

Qualitative Predictions of Change in Entropy

Phase Changes (for a given substance)

• Solid State: Crystalline solids are generally very orderly, exhibiting less randomness and lower entropy.

• Liquid State: The liquid phase has more entropy than the solid phase.

• Gas State: The gas phase has a great deal more entropy than the liquid phase. This is because molecules can move independently in a much larger volume, causing significantly more randomness.

• Entropy Sign and Processes:     * Melting (Solid to Liquid): \Delta S > 0 (Entropy increases due to more randomness).     * Freezing (Liquid to Solid): \Delta S < 0 (Entropy decreases due to less randomness).     * Vaporization (Liquid to Gas): \Delta S > 0 (Entropy increases significantly).     * Condensation (Gas to Liquid): \Delta S < 0 (Entropy decreases).

Temperature Changes (for a given substance and phase)

• Direct Relationship: The higher the temperature, the greater the entropy.

• Kinetic Energy: Higher temperature corresponds to greater average kinetic energy. The increased randomness of motion at higher kinetic energies corresponds to greater entropy.

• Entropy vs. Temperature Diagram:     * A perfectly ordered, crystalline solid starts at $0$ entropy at $0\,K$.     * Discontinuous Jumps: The diagram shows vertical lines at phase transition points where significant changes in entropy occur without a change in temperature.     * Melting Point ($mp$): There is a discontinuous jump in entropy at the melting point.     * Boiling Point ($bp$): There is a discontinuous jump in entropy at the boiling point.

Volume and Pressure Changes

• Volume Increase / Pressure Decrease: Increasing the volume or decreasing the pressure of a gas sample increases its entropy. A larger volume allows for greater randomness of molecular distribution.

Dissolving Solids

• Competing Factors for Ionic Solids:     * Solid's Perspective: Randomness increases on dissolution as the fixed lattice breaks apart.     * Solvent's Perspective: Randomness decreases on dissolution because water molecules become arranged into organized shells around hydrated ions.

• Charge Influence: The orderliness of hydrated ions increases with the increasing charge of the ions ($M^{2+}$ vs $M^+$) because water molecules are held more tightly by higher charges.

• Entropy Trends in Dissolution:     * For $M^+X^-$ salts (low charge), the overall entropy generally increases (\Delta S > 0) upon dissolution.     * For $M^{2+}X^{2-}$ (or higher charged) salts, the overall entropy generally decreases (\Delta S < 0) upon dissolution.

Reactions Involving Gases

• Gas Mole Comparison: For reactions involving gases (which may also involve solids or liquids), the side of the reaction with the higher number of moles of gas has the higher entropy. While solids and liquids contribute entropy, their contribution is relatively small compared to gases.

• Examples for Analysis:     1. $2H_2(g) + O_2(g) \rightarrow 2H_2O(g)$     2. $SiO_2(s) + 2C(s) + 2Cl_2(g) \rightarrow SiCl_4(g) + 2CO(g)$     3. $HgS(s) + O_2(g) \rightarrow Hg(l) + SO_2(g)$

Mixing

• Mixing without Reaction: Mixing two different gases results in greater entropy for the system. Opening a stopcock between two containers of different gases results in a positive $\Delta S$.

Quantitative Calculation of Entropy

Standard Molar Entropy ($S^\circ$)

• Definition: The entropy of one mole of a pure substance at $1\,atm$ pressure and a specified temperature (usually $25\,^{\circ}C$).

• Absolute Values: Unlike enthalpy of formation values (which can be zero for elements), all entropy values are positive.

• Phase Comparison Caveat: The standard molar entropy of a liquid is not necessarily greater than that of a solid when comparing two different substances.

Table of Standard Molar Entropies ($25\,^{\circ}C$)

Calculating Standard Entropy of Reaction ($\Delta S^\circ$)

• The following equation is used:     * $\Delta S^\circ = \sum n S^\circ(\text{products}) - \sum m S^\circ(\text{reactants})$

• Practice Calculations (Calculate the standard entropy of reaction for):     * $2NH_3(g) \rightarrow N_2(g) + 3H_2(g)$     * $2NO_2(g) \rightarrow N_2O_4(g)$     * $CH_4(g) + 2O_2(g) \rightarrow CO_2(g) + 2H_2O(l)$

Spontaneity and the Laws of Thermodynamics

Spontaneous Processes

• Spontaneous process: A process that proceeds on its own without a continuous external influence. There is a natural direction to events in the universe (macroscale and chemical).

• Nonspontaneous process: The reverse of any spontaneous process. This requires the continuous application of an outside force, such as electricity.

• Equilibrium Direction: Every chemical reaction naturally proceeds toward its equilibrium mixture. A spontaneous reaction moves a mixture toward equilibrium, whereas a nonspontaneous one moves it away from equilibrium.

Reaction Quotient ($Q$) and Spontaneity

• If Q < K: The reaction proceeds to the right (net production of product) spontaneously.

• If Q > K: The reaction proceeds to the left (net production of reactant) spontaneously.

• If $Q = K$: The reaction is at equilibrium (no net change).

• Defining Spontaneity: Spontaneity is always defined based on how the reaction is written (reactants on left to products on right). A reaction is only called "spontaneous" if it proceeds left-to-right on its own.

Scenario Analysis ($A(g) \rightleftharpoons B(g)$)

• Given equilibrium mixture: $400$ molecules of $A$, $600$ molecules of $B$.     * Scenario 1: Initial mixture: $200\,A$, $800\,B$. Result: Reaction proceeds left ($B \rightarrow A$) to reach equilibrium.     * Scenario 2: Initial mixture: $500\,A$, $500\,B$. Result: Reaction proceeds right ($A \rightarrow B$) to reach equilibrium.     * Scenario 3: Initial mixture: $400\,A$, $600\,B$. Result: At equilibrium; no net change ($Q = K$).

2nd Law of Thermodynamics

• Verbatim Statement: In any spontaneous event, the entropy of the universe (the total of the system and the surroundings) must increase.

• Fundamental Equation:     * $\Delta S_{Total} = \Delta S_{System} + \Delta S_{Surroundings}$     * $\Delta S_{Total}$ must be positive for a spontaneous event.

• Components:     * $\Delta S_{System} = \sum S^\circ(\text{products}) - \sum S^\circ(\text{reactants})$     * $\Delta S_{Surroundings} = -\frac{\Delta H}{T}$     * $\Delta H = \sum \Delta H_f^\circ(\text{products}) - \sum \Delta H_f^\circ(\text{reactants})$

Gibbs Free Energy ($\Delta G$)

• Chemists focus primarily on the system (the reaction). The 2nd Law is restated using "Change in Free Energy" ($\Delta G$):     * $\Delta G = \Delta H - T\Delta S$

• Spontaneity Criteria:     * If \Delta G < 0 (negative): The reaction is spontaneous as written.     * If \Delta G > 0 (positive): The reaction is nonspontaneous as written.

Relationships between Spontaneity, $\Delta H$, and $\Delta S$

• Entropy Contribution:     * An increase in system entropy ($+\Delta S$) contributes to spontaneity.     * A decrease in system entropy ($-\Delta S$) detracts from spontaneity.

• Enthalpy Contribution:     * Exothermic processes (release of heat, $-\Delta H$) contribute to spontaneity.     * Endothermic processes (intake of heat, $+\Delta H$) detract from spontaneity.

• Ultimately, the value of $\Delta G$ (incorporating both factors) determines whether a reaction is spontaneous and quantifies its degree of spontaneity.