HK

Entropy, Free Energy, and Equilibrium

Spontaneous Processes

  • Spontaneous processes occur without external intervention.
  • Examples:
    • Waterfall running downhill.
    • Sugar dissolving in coffee.
    • Water freezing below 0°C and ice melting above 0°C at 1 atm.
    • Heat flowing from a hotter to a colder object.
    • Gas expanding into an evacuated bulb.
    • Iron rusting when exposed to oxygen and water.

Enthalpy and Spontaneity

  • A decrease in enthalpy does not guarantee a spontaneous reaction.
  • Examples of spontaneous reactions with varying enthalpy changes (\Delta H^0):
    • CH4 (g) + 2O2 (g) \rightarrow CO2 (g) + 2H2O (l), \Delta H^0 = -890.4 \text{ kJ}
    • H^+ (aq) + OH^- (aq) \rightarrow H_2O (l), \Delta H^0 = -56.2 \text{ kJ}
    • H2O (s) \rightarrow H2O (l), \Delta H^0 = 6.01 \text{ kJ}
    • NH4NO3 (s) \rightarrow NH4^+ (aq) + NO3^- (aq), \Delta H^0 = 25 \text{ kJ}

Entropy (S)

  • Entropy is a measure of randomness or disorder in a system.
  • Order to disorder: Increase in Entropy (S).
  • Change in entropy: \Delta S = Sf - Si
    • If randomness increases (Sf > Si), then \Delta S > 0
  • For a substance, entropy increases from solid to liquid to gas: S{\text{solid}} < S{\text{liquid}} << S_{\text{gas}}
  • Example: H2O (s) \rightarrow H2O (l), \Delta S > 0

Microstates and Entropy

  • W = number of microstates
  • S = k \ln W where k is Boltzmann's constant.
  • \Delta S = k \ln \frac{Wf}{Wi}
    • If Wf > Wi, then \Delta S > 0
    • If Wf < Wi, then \Delta S < 0

Processes Increasing Entropy

  • Phase transitions from solid to liquid to gas increase entropy.
  • Dissolving a solute in a solvent increases entropy.
  • Heating a system increases entropy.

Entropy Changes in Processes

  • (a) Condensing water vapor: Decreases randomness, \Delta S < 0
  • (b) Forming sucrose crystals: Decreases randomness, \Delta S < 0
  • (c) Heating hydrogen gas from 60°C to 80°C: Increases randomness, \Delta S > 0
  • (d) Subliming dry ice: Increases randomness, \Delta S > 0

State Functions

  • State functions depend only on the current state of the system, not the path taken to reach that state.
  • Examples: energy, enthalpy, pressure, volume, temperature, entropy

Laws of Thermodynamics

  • First Law: Energy is conserved.
  • Second Law: The entropy of the universe increases in a spontaneous process.
    • Spontaneous process: \Delta S{\text{univ}} = \Delta S{\text{sys}} + \Delta S_{\text{surr}} > 0
    • Equilibrium process: \Delta S{\text{univ}} = \Delta S{\text{sys}} + \Delta S_{\text{surr}} = 0

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

  • For a reaction aA + bB \rightarrow cC + dD
  • \Delta S_{\text{rxn}}^0 = [ cS^0(C) + dS^0(D) ] - [ aS^0(A) + bS^0(B) ]
  • \Delta S_{\text{rxn}}^0 = \sum nS^0(\text{products}) - \sum mS^0(\text{reactants})
  • The standard entropy of reaction (\Delta S_{\text{rxn}}^0) is the entropy change for a reaction carried out at 1 atm and 25°C.
  • Example:
    • 2CO (g) + O2 (g) \rightarrow 2CO2 (g)
    • S^0(CO) = 197.9 \text{ J/K•mol}
    • S^0(O_2) = 205.0 \text{ J/K•mol}
    • S^0(CO_2) = 213.6 \text{ J/K•mol}
    • \Delta S{\text{rxn}}^0 = 2 \times S^0(CO2) – [2 \times S^0(CO) + S^0 (O_2)]
    • \Delta S_{\text{rxn}}^0 = 427.2 – [395.8 + 205.0] = -173.6 \text{ J/K•mol}

Entropy Changes and Gases

  • If a reaction produces more gas molecules than it consumes, \Delta S^0 > 0
  • If the total number of gas molecules diminishes, \Delta S^0 < 0
  • If there is no net change in the total number of gas molecules, \Delta S^0 may be positive or negative but will be a small number.
  • Example: 2Zn (s) + O_2 (g) \rightarrow 2ZnO (s)
    • The total number of gas molecules decreases, so \Delta S is negative.

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

  • Exothermic process: Heat released by the system increases the entropy of the surroundings (\Delta S_{surr} > 0).
  • Endothermic process: Heat absorbed by the system decreases the entropy of the surroundings (\Delta S_{surr} < 0).

Third Law of Thermodynamics

  • The entropy of a perfect crystalline substance is zero at absolute zero (0 K).
  • S = k \ln W and at absolute zero, W = 1, so S = 0

Gibbs Free Energy (G)

  • For a constant-temperature process: \Delta G = \Delta H{sys} - T\Delta S{sys}
  • \Delta G < 0: The reaction is spontaneous in the forward direction.
  • \Delta G > 0: The reaction is nonspontaneous as written (spontaneous in the reverse direction).
  • \Delta G = 0: The reaction is at equilibrium.

Standard Free-Energy Change (\Delta G^0)

  • For a reaction aA + bB \rightarrow cC + dD
  • \Delta G{\text{rxn}}^0 = [ c\Delta Gf^0(C) + d\Delta Gf^0(D) ] - [ a\Delta Gf^0(A) + b\Delta G_f^0(B) ]
  • \Delta G{\text{rxn}}^0 = \sum n\Delta Gf^0(\text{products}) - \sum m\Delta G_f^0(\text{reactants})
  • The standard free-energy of reaction (\Delta G_{\text{rxn}}^0) is the free-energy change for a reaction under standard-state conditions.
  • The standard free energy of formation (\Delta G_f^0) is the free-energy change when 1 mole of the compound is formed from its elements in their standard states.
  • \Delta G_f^0 of any element in its stable form is zero.
  • Example:
    • 2C6H6 (l) + 15O2 (g) \rightarrow 12CO2 (g) + 6H_2O (l)
    • \Delta G{\text{rxn}}^0 = [ 12\Delta Gf^0(CO2) + 6\Delta Gf^0(H2O) ] - [ 2\Delta Gf^0(C6H6) ]
    • \Delta G_{\text{rxn}}^0 = [ 12 \times -394.4 + 6 \times -237.2 ] – [ 2 \times 124.5 ] = -6405 \text{ kJ}
    • Since \Delta G^0 = -6405 \text{ kJ} < 0, the reaction is spontaneous at 25°C.

Factors Affecting the Sign of \Delta G

  • \Delta G = \Delta H - T\Delta S
  • \Delta H negative, \Delta S positive: \Delta G is always negative (spontaneous at all temperatures).
  • \Delta H positive, \Delta S negative: \Delta G is always positive (nonspontaneous at all temperatures).
  • \Delta H negative, \Delta S negative: Spontaneous at low temperatures.
  • \Delta H positive, \Delta S positive: Spontaneous at high temperatures.

Temperature and Spontaneity

  • Example: CaCO3 (s) \rightarrow CaO (s) + CO2 (g)
    • \Delta H^0 = 177.8 \text{ kJ}
    • \Delta S^0 = 160.5 \text{ J/K}
    • \Delta G^0 = \Delta H^0 – T\Delta S^0
    • At 25°C, \Delta G^0 = 130.0 \text{ kJ}
    • \Delta G^0 = 0 at 835°C

Gibbs Free Energy and Phase Transitions

  • Example: H2O (l) \rightarrow H2O (g)
  • \Delta G^0 = 0 = \Delta H^0 – T\Delta S^0
  • \Delta S = \frac{\Delta H}{T} = \frac{40.79 \text{ kJ}}{373 \text{ K}} = 109 \text{ J/K}

Efficiency of Heat Engines

  • \text{Efficiency} = \frac{Th - Tc}{Th} \times 100\%, where Th is the temperature of the hot reservoir and T_c is the temperature of the cold reservoir.

Gibbs Free Energy and Chemical Equilibrium

  • \Delta G = \Delta G^0 + RT \ln Q, where
    • R is the gas constant (8.314 J/K•mol)
    • T is the absolute temperature (K)
    • Q is the reaction quotient
  • At equilibrium, \Delta G = 0 and Q = K
  • 0 = \Delta G^0 + RT \ln K
  • \Delta G^0 = - RT \ln K

Relation Between \Delta G^0 and K

  • If K > 1, then \ln K is positive and \Delta G^0 is negative (products are favored).
  • If K = 1, then \ln K = 0 and \Delta G^0 = 0 (products and reactants are equally favored).
  • If K < 1, then \ln K is negative and \Delta G^0 is positive (reactants are favored).

Coupled Reactions

  • Reactions can be coupled to drive non-spontaneous reactions.
  • Example:
    • Alanine + Glycine \rightarrow Alanylglycine, \Delta G^0 = +29 \text{ kJ}, K < 1
    • ATP + H2O + \text{Alanine} + \text{Glycine} \rightarrow ADP + H3PO_4 + \text{Alanylglycine}, \Delta G^0 = -2 \text{ kJ}, K > 1

ATP and ADP

  • ATP (Adenosine triphosphate) and ADP (Adenosine diphosphate) are important energy carriers in biological systems.

Thermodynamics of a Rubber Band

  • Stretched rubber band has low entropy.
  • Relaxed rubber band has high entropy.