Free Energy, Nonstandard States, and Equilibrium in Chemical Reactions

Free Energy Changes for Nonstandard States

From Standard Free Energy (ΔG°) to Actual Free Energy (ΔG)

In real chemical systems, conditions often differ from the standard state (1 M concentration, 1 atm pressure). The Gibbs free energy change under these nonstandard conditions is given by:

• Key Equation: $\Delta G = \Delta G^\circ + RT \ln Q$

• R: Universal gas constant, 8.314 J/(mol·K)

• T: Temperature in Kelvin

• Q: Reaction quotient, reflecting current concentrations/pressures

Physical Interpretation:

• When Q = 1 (standard conditions): $\ln Q = 0 \Rightarrow \Delta G = \Delta G^\circ$

• When Q < K: $\Delta G < 0$ (forward reaction is spontaneous)

• When Q > K: $\Delta G > 0$ (reverse reaction is spontaneous)

• When Q = K: $\Delta G = 0$ (system at equilibrium)

The $RT \ln Q$ term corrects the standard free energy for the actual conditions of the system.

Free Energy and Equilibrium: Relating ΔG° to K

Derivation and Interpretation of ΔG° = −RT ln K

At equilibrium, the reaction quotient Q equals the equilibrium constant K, and the free energy change ΔG is zero. Substituting these values into the key equation yields:

• $0 = \Delta G^\circ + RT \ln K$

• Therefore, $\Delta G^\circ = -RT \ln K$

• Or, rearranged: $K = e^{-\Delta G^\circ/RT}$

Implications:

• Small changes in ΔG° cause large changes in K due to the logarithmic relationship.

• ΔG° < 0: K > 1 (products favored at equilibrium)

• ΔG° > 0: K < 1 (reactants favored at equilibrium)

• ΔG° = 0: K = 1 (reactants and products equally favored)

Summary Table: Relationship Between ΔG° and K

The following table summarizes how the sign and magnitude of ΔG° relate to the equilibrium constant K and the position of equilibrium:

ΔG°rxn	K value	Equilibrium Position	Reaction at Standard Conditions
ΔG° << 0 (large negative)	K >> 1	Products strongly favored	Spontaneous forward
ΔG° < 0 (moderately negative)	K > 1	Products favored	Spontaneous forward
ΔG° = 0	K = 1	Reactants = products	At equilibrium
ΔG° > 0 (moderately positive)	K < 1	Reactants favored	Spontaneous reverse
ΔG° >> 0 (large positive)	K << 1	Reactants strongly favored	Spontaneous reverse

Additional info: Small changes in ΔG° produce large changes in K due to the exponential relationship.

Example: Calculating K from ΔG°rxn

Consider the reaction: N2O4(g) ⇌ 2 NO2(g). Given the following standard free energies of formation:

Calculate ΔG°rxn:

Calculate K at 298 K:

Result: K = 0.32, indicating that products are slightly disfavored at standard conditions.

Coupling Reactions: Driving Nonspontaneous Processes

Thermodynamic Coupling in Chemistry and Biology

Some reactions with ΔG > 0 (nonspontaneous) can be made to occur by coupling them with a spontaneous reaction (ΔG < 0) so that the overall ΔG is negative. This principle is fundamental in both industrial and biological processes.

• Industrial Example: Extraction of iron from iron ore (Fe2O3) is nonspontaneous, but becomes spontaneous when coupled with the combustion of carbon.

• Biological Example: ATP hydrolysis (ΔG ≈ −30.5 kJ/mol) is coupled to many cellular reactions to make them spontaneous overall.

Key Equations in Thermodynamics and Equilibrium

• Boltzmann Equation (Entropy): $S = k \ln W$

• Entropy Change (Reversible Process): $\Delta S = \frac{q_{\text{rev}}}{T}$

• Entropy of Surroundings: $\Delta S_{\text{surr}} = -\frac{\Delta H_{\text{sys}}}{T}$

• Gibbs Free Energy: $\Delta G = \Delta H - T \Delta S$

• Standard Entropy Change: $\Delta S^\circ_{\text{rxn}} = \sum n_p S^\circ(\text{products}) - \sum n_r S^\circ(\text{reactants})$

• Standard Free Energy Change: $\Delta G^\circ_{\text{rxn}} = \Delta H^\circ_{\text{rxn}} - T \Delta S^\circ_{\text{rxn}} = \sum \Delta G^\circ_f(\text{products}) - \sum \Delta G^\circ_f(\text{reactants})$

• Free Energy at Nonstandard Conditions: $\Delta G = \Delta G^\circ + RT \ln Q$

• Relationship to Equilibrium Constant: $\Delta G^\circ = -RT \ln K$

Comprehensive Review: Final Takeaways

1. ΔG = ΔG° + RT ln Q: Corrects standard free energy for actual conditions.

2. ΔG° = −RT ln K: Master equation linking thermodynamics and equilibrium.

3. Large negative ΔG° → K ≫ 1 (products favored); large positive ΔG° → K ≪ 1 (reactants favored).

4. Coupling reactions: Nonspontaneous processes can be driven by coupling with spontaneous ones.

5. All spontaneous processes are driven by ΔS_universe > 0; ΔG = −TΔS_universe is the practical criterion.

6. Key equations: S = k ln W; ΔS = q/T; ΔS_surr = −ΔH/T; ΔG = ΔH−TΔS; ΔS°_rxn and ΔG°_rxn formulas; ΔG = ΔG° + RT ln Q; ΔG° = −RT ln K.