Gibbs Free Energy Equations

Why This Matters

Gibbs free energy is the ultimate thermodynamic scorecard—it tells you whether a reaction will actually happen on its own or if you need to put energy in to make it go. In General Chemistry II, you're being tested on your ability to connect enthalpy, entropy, and temperature into a single predictive framework. Every equilibrium problem, every electrochemistry question, and every discussion of phase changes ultimately comes back to $\Delta G$.

These equations aren't isolated formulas to memorize in a vacuum. They represent different ways of asking the same fundamental question: is this process thermodynamically favorable? Whether you're calculating equilibrium constants, predicting cell voltages, or determining if a reaction becomes spontaneous at high temperatures, you need to understand what each equation reveals about the system. Don't just memorize the math—know which equation to reach for and why it applies to the situation at hand.

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Predicting Spontaneity: The Core Equation

The foundation of Gibbs free energy lies in balancing two competing thermodynamic drives: the tendency to minimize energy (enthalpy) and the tendency to maximize disorder (entropy).

The Fundamental Gibbs Equation

$\Delta G = \Delta H - T\Delta S$

• Negative $\Delta G$ indicates a spontaneous process—this is the single most important criterion for spontaneity at constant T and P
• $\Delta H$ (enthalpy change) captures heat flow; exothermic reactions ($\Delta H < 0$) favor spontaneity
• $T\Delta S$ (entropy term) increases with temperature; reactions with positive $\Delta S$ become more spontaneous as T rises

Gibbs Free Energy and Maximum Work

$\Delta G = w_{max}$ (at constant T and P)

• $\Delta G$ equals the maximum non-expansion work obtainable from a reversible process—this connects abstract thermodynamics to practical applications
• Negative $\Delta G$ means the system can perform work on surroundings; positive $\Delta G$ means work must be done on the system
• Reversibility requirement is key—real processes always yield less work due to irreversibilities and friction

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Connecting to Equilibrium: Standard and Non-Standard Conditions

These equations bridge thermodynamics and equilibrium, showing that $\Delta G$ determines where a reaction "wants" to go, while K tells you where it ends up.

Standard Free Energy and Equilibrium Constant

$\Delta G° = -RT \ln K$

• $\Delta G°$ is the standard free energy change—measured with all species at 1 M (solutions) or 1 atm (gases) at 298 K
• Large K (>1) corresponds to negative $\Delta G°$, meaning products are favored at equilibrium
• The relationship is logarithmic—a change of ~5.7 kJ/mol in $\Delta G°$ shifts K by a factor of 10 at 298 K

Non-Standard Conditions: The Reaction Quotient

$\Delta G = \Delta G° + RT \ln Q$

• Q (reaction quotient) represents the current ratio of products to reactants—not the equilibrium ratio
• When Q < K, $\Delta G < 0$ and the reaction proceeds forward spontaneously toward equilibrium
• When Q = K, $\Delta G = 0$ and the system has reached equilibrium—no net driving force remains

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Electrochemistry Connection

Electrochemical cells convert chemical energy directly to electrical work, and the cell potential is just another way of expressing Gibbs free energy.

Gibbs Free Energy and Cell Potential

$\Delta G = -nFE$

• n = moles of electrons transferred in the balanced redox reaction; F = Faraday's constant (96,485 C/mol)
• Positive E (cell potential) gives negative $\Delta G$—spontaneous galvanic cells have positive voltages
• This equation lets you interconvert between thermodynamic spontaneity and measurable voltage—essential for electrochemistry problems

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Phase Transitions and Equilibrium Conditions

At phase boundaries and equilibrium points, the system has minimized its Gibbs free energy, and competing phases or reactions are perfectly balanced.

Equilibrium Criterion

$\Delta G = 0$ (at equilibrium)

• Minimum Gibbs free energy defines the equilibrium state—the system has no thermodynamic driving force to change
• Phase transitions (melting, boiling, sublimation) occur when $\Delta G = 0$ between phases at specific T and P
• Forward and reverse rates are equal—this connects thermodynamic equilibrium to kinetic equilibrium

Temperature Dependence: Gibbs-Helmholtz Equation

$\left(\frac{\partial(\Delta G/T)}{\partial T}\right)_P = -\frac{\Delta H}{T^2}$

• Relates temperature dependence of $\Delta G$ to enthalpy—useful for predicting how spontaneity changes with T
• Exothermic reactions ($\Delta H < 0$) become less favorable as temperature increases
• Enables calculation of $\Delta G$ at new temperatures when you know $\Delta H$ and $\Delta G$ at a reference temperature

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Multicomponent Systems and Mixtures

For systems with multiple species, Gibbs free energy depends on both the amounts and the chemical potentials of each component.

Chemical Potential and Total Free Energy

$G = \sum_i \mu_i n_i$

• Chemical potential ($\mu_i$) is the partial molar Gibbs free energy—how much G changes when you add one mole of component i
• Each component contributes to total G proportionally to its amount ($n_i$) and its "thermodynamic influence" ($\mu_i$)
• At equilibrium, chemical potential of each species is equal in all phases—this drives phase distribution

Gibbs Free Energy of Mixing (Ideal Solutions)

$\Delta G_{mix} = RT(x_1 \ln x_1 + x_2 \ln x_2)$

• Always negative for ideal solutions because mole fractions are less than 1, making logarithms negative—mixing is always spontaneous
• $x_1$ and $x_2$ are mole fractions—this equation assumes no enthalpy change upon mixing ($\Delta H_{mix} = 0$)
• Driven entirely by entropy—the increase in disorder from mixing overcomes any other factors in ideal systems

Gibbs-Duhem Equation

$\sum_i x_i \, d\mu_i = 0$

• Chemical potentials are interdependent—you cannot change one component's $\mu$ without affecting others
• Constrains the system—if you know how $\mu$ changes for most components, you can calculate the rest
• Essential for non-ideal solution thermodynamics and activity coefficient calculations

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Quick Reference Table

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Self-Check Questions

1. A reaction has $\Delta H > 0$ and $\Delta S > 0$. Under what temperature conditions will this reaction be spontaneous, and which equation would you use to determine the crossover temperature?

2. If $\Delta G° = -30$ kJ/mol for a reaction, is K greater than, less than, or equal to 1? What if you're told that current concentrations give Q > K—is the reaction spontaneous in the forward direction?

3. Compare and contrast $\Delta G°$ and $\Delta G$. Why is it critical to distinguish between them when solving equilibrium problems with non-standard concentrations?

4. An electrochemical cell has a positive standard cell potential ($E° > 0$). What does this tell you about $\Delta G°$ and K for the cell reaction? Write the equation that connects all three quantities.

5. Why is $\Delta G_{mix}$ always negative for ideal solutions, even though there's no enthalpy change? Which term in the fundamental Gibbs equation ($\Delta G = \Delta H - T\Delta S$) is responsible?