Entropy, Free Energy, and Equilibrium - Flashcards

Spontaneous Physical and Chemical Processes

• A spontaneous process is one that occurs naturally under a specific set of conditions without external intervention.

• Examples of spontaneous processes include:

  • A waterfall running downhill due to gravity.

  • A lump of sugar dissolving in a cup of coffee.

  • At $1\,atm$, water freezing below $0\,^{\circ}C$ and ice melting above $0\,^{\circ}C$.

  • Heat flowing from a hotter object to a colder object.

  • A gas expanding into an evacuated bulb (vacuum).

  • Iron forming rust when exposed to oxygen ($O_2$) and water ($H_2O$).

• If a process is spontaneous in one direction, the reverse process under the same conditions is nonspontaneous.

Enthalpy and Spontaneity

• A decrease in enthalpy (\Delta H < 0) often accompanies spontaneous reactions, but it is not a sole requirement for spontaneity.

• Exothermic spontaneous reactions:

  • Combustion of methane: $CH_4(g) + 2O_2(g) \rightarrow CO_2(g) + 2H_2O(l)$ where $\Delta H^0 = -890.4\,kJ$.

  • Acid-base neutralization: $H^+(aq) + OH^-(aq) \rightarrow H_2O(l)$ where $\Delta H^0 = -56.2\,kJ$.

• Endothermic spontaneous reactions:

  • Melting of ice: $H_2O(s) \rightarrow H_2O(l)$ where $\Delta H^0 = 6.01\,kJ$.

  • Dissolution of ammonium nitrate in water: $NH_4NO_3(s) \xrightarrow{H_2O} NH_4^+(aq) + NO_3^-(aq)$ where $\Delta H^0 = 25\,kJ$.

Entropy ($S$) and Microstates

• Definition: Entropy ($S$) is a measure of the randomness or disorder of a system.

• Change in entropy is calculated as: $\Delta S = S_f - S_i$.

• If the change results in increased randomness (S_f > S_i), then \Delta S > 0.

• State of Matter Order: For any given substance, the solid state is more ordered than the liquid state, and the liquid state is significantly more ordered than the gas state.

  • Order of entropy: S_{solid} < S_{liquid} << S_{gas}.

  • Example: $H_2O(s) \rightarrow H_2O(l)$ results in \Delta S > 0.

• Boltzmann Equation: The entropy is related to the number of microstates ($W$), which are the ways the molecules of a system can be arranged while keeping the same total energy.

  • $S = k \ln(W)$

  • Where $k$ is the Boltzmann constant.

  • Change in entropy based on microstates: $\Delta S = k \ln\left(\frac{W_f}{W_i}\right)$.

  • If W_f > W_i, then \Delta S > 0. If W_f < W_i, then \Delta S < 0.

Processes Increasing Entropy (\Delta S > 0)

• Phase transistions: Solid to Liquid or Liquid to Vapor.

• Dissolution: Solute + Solvent to Solution.

• Temperature changes: A system at a higher temperature ($T_2$) has higher entropy than at a lower temperature ($T_1$) where T_2 > T_1.

• Qualitative predictions of entropy changes:

  • Condensing water vapor: Randomness decreases, so entropy decreases (\Delta S < 0).

  • Forming sucrose crystals from a supersaturated solution: Randomness decreases, so \Delta S < 0.

  • Heating hydrogen gas from $60\,^{\circ}C$ to $80\,^{\circ}C$: Randomness increases, so \Delta S > 0.

  • Subliming dry ice ($CO_2(s) \rightarrow CO_2(g)$): Randomness increases, so \Delta S > 0.

Thermodynamics Laws and State Functions

• State Functions: Properties determined solely by the state of the system regardless of the path taken. Examples: Potential energy, Enthalpy ($H$), Pressure ($P$), Volume ($V$), Temperature ($T$), and Entropy ($S$).

• First Law of Thermodynamics: Energy can be converted from one form to another but cannot be created or destroyed.

• Second Law of Thermodynamics: The entropy of the universe increases in a spontaneous process and remains unchanged in an equilibrium process.

  • Spontaneous process: \Delta S_{univ} = \Delta S_{sys} + \Delta S_{surr} > 0.

  • Equilibrium process: $\Delta S_{univ} = \Delta S_{sys} + \Delta S_{surr} = 0$.

• Third Law of Thermodynamics: The entropy of a perfect crystalline substance is zero ($S = 0$) at the absolute zero of temperature ($0\,K$).

  • At $0\,K$, $W = 1$, thus $S = k \ln(1) = 0$.

Entropy Changes in the System ($\Delta S_{sys}$)

• The standard entropy of reaction ($\Delta S^0_{rxn}$) is measured at $1\,atm$ and $25\,^{\circ}C$.

• Equation: $\Delta S^0_{rxn} = \sum n S^0(products) - \sum m S^0(reactants)$.

• Calculation Example (Combustion of CO):

  • $2CO(g) + O_2(g) \rightarrow 2CO_2(g)$

  • Values: $S^0(CO) = 197.9\,J/K \cdot mol$, $S^0(O_2) = 205.0\,J/K \cdot mol$, $S^0(CO_2) = 213.6\,J/K \cdot mol$.

  • $\Delta S^0_{rxn} = [2 \times 213.6] - [2 \times 197.9 + 205.0] = -173.6\,J/K \cdot mol$.

• Gaseous effects on System Entropy:

  • If reaction produces more gas molecules: \Delta S^0 > 0.

  • If gas molecules diminish: \Delta S^0 < 0.

  • No net change in gas molecules: $\Delta S^0$ is a small number (can be positive or negative).

  • Example: $2Zn(s) + O_2(g) \rightarrow 2ZnO(s)$ results in a negative $\Delta S$ because the number of gas molecules decreases.

Entropy Changes in the Surroundings ($\Delta S_{surr}$)

• Exothermic Process: Heat is transferred from system to surroundings. Surroundings entropy increases (\Delta S_{surr} > 0).

• Endothermic Process: Heat is absorbed from surroundings to system. Surroundings entropy decreases (\Delta S_{surr} < 0).

Gibbs Free Energy ($G$)

• For processes at constant temperature: $\Delta G = \Delta H_{sys} - T \Delta S_{sys}$.

• Spontaneity Criteria:

  • \Delta G < 0: Reaction is spontaneous in the forward direction.

  • \Delta G > 0: Reaction is nonspontaneous as written (spontaneous in the reverse direction).

  • $\Delta G = 0$: The reaction is at equilibrium.

• Standard Free-energy of Reaction ($\Delta G^0_{rxn}$): The change for a reaction under standard-state conditions.

  • $\Delta G^0_{rxn} = \sum n \Delta G^0_f(products) - \sum m \Delta G^0_f(reactants)$.

  • Standard free energy of formation ($\Delta G^0_f$) is for 1 mole of compound from elements in stable forms.

  • $\Delta G^0_f$ for any element in its stable form is zero.

Temperature and Spontaneity

• Calculation Example (Benzene Combustion at $25\,^{\circ}C$):

  • $2C_6H_6(l) + 15O_2(g) \rightarrow 12CO_2(g) + 6H_2O(l)$

  • $\Delta G^0_{rxn} = [12 \times -394.4 + 6 \times -237.2] - [2 \times 124.5] = -6405\,kJ$.

  • The reaction is spontaneous since \Delta G^0 < 0.

• Factors affecting the sign of $\Delta G$ ($\Delta G = \Delta H - T \Delta S$):

  • $\Delta H (+), \Delta S (+)$: Spontaneous at high temperatures. Nonspontaneous at low temperatures. Example: $2HgO(s) \rightarrow 2Hg(l) + O_2(g)$.

  • $\Delta H (+), \Delta S (-)$: $\Delta G$ is always positive. Nonspontaneous at all temperatures. Example: $3O_2(g) \rightarrow 2O_3(g)$.

  • $\Delta H (-), \Delta S (+)$: $\Delta G$ is always negative. Spontaneous at all temperatures. Example: $2H_2O_2(aq) \rightarrow 2H_2O(l) + O_2(g)$.

  • $\Delta H (-), \Delta S (-)$: Spontaneous at low temperatures. Reverse reaction spontaneous at high temperatures. Example: $NH_3(g) + HCl(g) \rightarrow NH_4Cl(s)$.

• Temperature Case Study ($CaCO_3$):

  • $CaCO_3(s) \rightarrow CaO(s) + CO_2(g)$

  • $\Delta H^0 = 177.8\,kJ$, $\Delta S^0 = 160.5\,J/K$.

  • At $25\,^{\circ}C$, $\Delta G^0 = 130.0\,kJ$ (nonspontaneous).

  • At $835\,^{\circ}C$, $\Delta G^0 = 0$ (reaches equilibrium).

Gibbs Free Energy, Phase Transitions, and Efficiency

• Phase Transitions: At equilibrium transitions (like boiling), $\Delta G^0 = 0$.

  • Therefore, $0 = \Delta H^0 - T \Delta S^0 \implies \Delta S = \frac{\Delta H}{T}$.

  • For water vaporization: $\Delta S = \frac{40.79\,kJ}{373\,K} = 109\,J/K$.

• Heat Engines: Efficiency of a heat engine is defined by the temperatures of the hot reservoir ($T_h$) and cold reservoir ($T_c$).

  • $Efficiency = \frac{T_h - T_c}{T_h} \times 100\%$.

Free Energy and Chemical Equilibrium

• Relation for non-standard conditions: $\Delta G = \Delta G^0 + RT \ln(Q)$.

  • $R = 8.314\,J/K \cdot mol$.

  • $T$ is absolute temperature in Kelvin.

  • $Q$ is the reaction quotient.

• At equilibrium: $\Delta G = 0$ and $Q = K$.

  • Equation: $\Delta G^0 = -RT \ln(K)$.

• Relationship between $\Delta G^0$ and $K$:

  • If K > 1, $\ln(K)$ is positive, $\Delta G^0$ is negative: Products are favored.

  • If $K = 1$, $\ln(K) = 0$, $\Delta G^0 = 0$: Products and reactants favored equally.

  • If K < 1, $\ln(K)$ is negative, $\Delta G^0$ is positive: Reactants are favored.

Biochemical and Physical Applications

• Metabolic Coupling: Reactions with positive $\Delta G^0$ can be driven by coupling them with the hydrolysis of Adenosine Triphosphate (ATP).

  • Alanine + Glycine $\rightarrow$ Alanylglycine ($\Delta G^0 = +29\,kJ$, K < 1).

  • Coupled with ATP hydrolysis: ATP + $H_2O$ + Alanine + Glycine $\rightarrow$ ADP + $H_3PO_4$ + Alanylglycine ($\Delta G^0 = -2\,kJ$, K > 1).

• Rubber Band Thermodynamics:

  • Relaxed state represents High Entropy ($S$).

  • Stretched state represents Low Entropy ($S$).

  • Thermodynamic relation: $T \Delta S = \Delta H - \Delta G$.