Free Energy & Thermodynamics Notes

Thermodynamics & The First Law

• The energy of the universe is constant.
• $\Delta E = w + q$
  • $w = -P\Delta V$ (+ if work done to system, - if work done by system).
  • Constant pressure: $q_P = \Delta E - w = \Delta E + P\Delta V = \Delta H$
  • q (+) if heat enters the system (endothermic), q (-) if heat leaves the system (exothermic).

Spontaneous Reactions

• Proceeds on its own without continuous external influence.
• Non-spontaneous reactions require continuous external influence (energy).
• Combustion is a spontaneous, exothermic reaction.
• Cold pack reactions are spontaneous and endothermic.
• Enthalpy change $\Delta H$ alone doesn't indicate spontaneity.

Gibbs Free Energy and Spontaneity

• Spontaneity is determined by the free energy change of the system.
• Thermodynamically favorable: \Delta G < 0 (negative).
• Thermodynamically unfavorable: \Delta G > 0 (positive).
• $\Delta G = \Delta H - T\Delta S$
  • $\Delta G$: Gibbs Free Energy change (kJ/mol).
  • $\Delta H$: Enthalpy change (kJ/mol).
  • $\Delta S$: Entropy change (J/K).
  • T: Temperature (K).

Entropy

• Measure of disorder or energy dispersal in a system (J/K or J/(mol·K)).
• $S = k<em>B \cdot \ln W$ ($k</em>B$ is Boltzmann constant, W is # of energetically equivalent arrangements).
• Third Law: Entropy of a perfectly ordered crystalline substance at 0 K is zero.
• Absolute entropy (S) increases with increasing temperature.
• Standard Molar Entropy ($S^\circ$): Absolute entropy of one mole of a pure substance at 1 atm and a specified temperature (usually 25 °C).
• In general: $S^\circ$ (gas) > $S^\circ$ (liquid) > $S^\circ$ (solid).

Entropy Change

• \Delta S > 0: Entropy increases (greater energy dispersal).
• \Delta S < 0: Entropy decreases (less energy dispersal).
• $\Delta S = S<em>{\text{final}} - S</em>{\text{initial}}$
• Entropy increases with temperature, volume, and number of independent particles.
• Phase changes: fusion/melting, vaporization, and sublimation increase entropy (\Delta S > 0); freezing, condensation, and deposition decrease entropy (\Delta S < 0).

Standard Entropy of Reaction

• $\Delta S^\circ_{rxn} = [cS^\circ(C) + dS^\circ(D)] − [aS^\circ(A) + bS^\circ(B)]$ for the reaction: aA + bB → cC + dD

The Laws of Thermodynamics

• 1st Law: Energy of the universe is constant.
• 3rd Law: Entropy of a perfectly ordered crystalline substance at 0 K is zero.
• 2nd Law: For a spontaneous process, the total entropy of the universe always increases (\Delta S_{universe} > 0).
• \Delta S{universe} = \Delta S{system} + \Delta S_{surroundings} > 0

Entropy of Surroundings

• $\Delta S<em>{surr} \propto -q</em>{sys} \propto \Delta H_{rx}$
• Exothermic: $\Delta H<em>{rx} < 0$, \Delta S{surroundings} > 0
• Endothermic: $\Delta H<em>{rx} > 0$, \Delta S{surroundings} < 0

Gibbs Free Energy and the Universe

• $-T\Delta S_{universe} = \Delta G = \Delta H - T\Delta S$
• If \Delta S_{universe} > 0, then \Delta G < 0 (thermodynamically favorable/spontaneous).
• If $\Delta S_{universe} < 0$, then $\Delta G > 0$ (thermodynamically unfavorable/non-spontaneous).

Gibbs Free Energy and Equilibrium

• \Delta G < 0: Reaction is thermodynamically favorable.
• \Delta G > 0: Reaction is thermodynamically unfavorable.
• $\Delta G = 0$: Reaction is at equilibrium.
• $\Delta G + T\Delta S = \Delta H$

Spontaneity and Temperature

• $\Delta G = \Delta H - T\Delta S$
• $\Delta H < 0$ and $\Delta S > 0$: $\Delta G$ always negative (spontaneous at all temperatures).
• \Delta H > 0 and \Delta S < 0: $\Delta G$ always positive (non-spontaneous at all temperatures).
• \Delta H < 0 and \Delta S < 0: $\Delta G$ temperature-dependent (spontaneous at low temperatures).
• \Delta H > 0 and \Delta S > 0: $\Delta G$ temperature-dependent (spontaneous at high temperatures).

Calculating Standard Free Energy Changes ($\Delta G^\circ_{rx}$)

• $\Delta G^\circ<em>{rx} = \Delta H^\circ</em>{rx} - T\Delta S^\circ_{rx}$
• $\Delta H^\circ<em>{rx} = \sum n</em>p \Delta H^\circ<em>f (products) - \sum n</em>r \Delta H^\circ_f (reactants)$
• $\Delta S^\circ<em>{rx} = \sum n</em>p S^\circ(products) - \sum n_r S^\circ(reactants)$

Using Tabulated Values of $\Delta G^\circ_f$

• $\Delta G^\circ_f$: Standard free energy of formation of 1 mole of a compound from its elements in their standard states.
• For an element in its most stable form at 25 °C, $\Delta H^\circ<em>f = 0$ and $\Delta G^\circ</em>f = 0$
• A substance with a negative $\Delta G^\circ_f$ is thermodynamically stable.
• A substance with a positive $\Delta G^\circ_f$ is thermodynamically unstable.

Hess’s Law

• If $\Delta G^\circ<em>{rx}$ is known for a series of stepwise reactions, $\Delta G^\circ</em>{rx}$ can be determined.

Non-Standard Conditions

• $\Delta G<em>{rx} = \Delta G^\circ</em>{rx} + RT \cdot \ln Q$
  • R: gas constant 8.314 J/(mol·K)
  • T: absolute temperature, K