Free Energy and Equilibrium - AP Chemistry Study Guide

Getting Started

Every chemical reaction is a tug-of-war between reactants and products. How can we predict which side will "win" when the pulling stops and the system reaches equilibrium? This chapter connects the macroscopic world of chemical equilibrium—the final mixture of substances—to the molecular-level driving forces of energy and entropy, which are captured by a single powerful quantity: Gibbs free energy.

What You Should Be able to Do

By the end of this section, you should be able to:

• Calculate the standard free energy change (ΔG°) for a reaction using its equilibrium constant (K), and vice versa.

• Predict whether reactants or products will be favored at equilibrium based on the sign of ΔG°.

• Qualitatively describe how the magnitude of ΔG° relates to the magnitude of K.

• Explain what the term "thermodynamically favored" signifies in the context of a chemical equilibrium.

Key Concepts & Analysis

We can understand the relationship between free energy and equilibrium as a process of cause and effect, where the inherent energy properties of a reaction dictate the final state of the chemical system.

Inputs & Preconditions

The primary input for determining the position of an equilibrium is the standard Gibbs free energy change (ΔG°). This value represents the change in free energy when a reaction occurs under standard conditions (typically 298 K, 1 atm for gases, 1 M for solutes). It is a measure of the maximum amount of non-expansion work that can be extracted from a reaction. A negative ΔG° indicates a process that can occur spontaneously under these conditions. The other key inputs are the absolute temperature (T) in Kelvin and the ideal gas constant (R), which is 8.314 J/(mol·K).

Key Steps / The Mathematical Relationship

The connection between the thermodynamic input (ΔG°) and the equilibrium output (K) is defined by a fundamental equation:

ΔG° = −RT ln K

Here, ln K is the natural logarithm of the equilibrium constant. This equation allows us to perform a quantitative analysis:

1. From K to ΔG°: If you know the equilibrium constant for a reaction at a specific temperature, you can calculate the standard free energy change. A K value greater than 1 will result in a negative ΔG°, while a K value less than 1 will result in a positive ΔG°.

2. From ΔG° to K: The equation can be rearranged to solve for the equilibrium constant. By dividing by -RT and then taking the exponential of both sides, we get:

   K = e^−ΔG°/RT

   This form is incredibly powerful, as it allows us to predict the extent of a reaction (its K value) using only thermodynamic data (like ΔH° and ΔS°, which are used to find ΔG°).

Example Calculation: Consider a reaction at 298 K with a ΔG° of -10.0 kJ/mol. First, convert ΔG° to J/mol (-10,000 J/mol) to match the units of R.

K = e^{−(−10000 J/mol) / (8.314 J/mol·K * 298 K)}

K = e^{10000 / 2477.6}

K = e^4.036

K ≈ 56.6

The negative ΔG° correctly predicts a K value significantly greater than 1.

Outputs & Effects

The output of this relationship is the equilibrium constant (K), which describes the state of the system once it has reached equilibrium. The sign and magnitude of ΔG° directly determine the value of K and, consequently, the composition of the equilibrium mixture.

Controls & Limiting Factors

The primary control in this relationship is temperature (T). Because T appears directly in the equation, it modulates the connection between ΔG° and K. For a given ΔG°, a higher temperature will make the exponent (−ΔG°/RT) smaller in magnitude, pushing K closer to 1. This is why the equilibrium position of a reaction can often be manipulated by changing the temperature. A reaction that is not favored at room temperature might become favored at a higher temperature, or vice versa, depending on the signs of ΔH° and ΔS°.

Key Models & Representations

This flowchart illustrates the decision-making process for relating ΔG° to the characteristics of a chemical equilibrium.

graph TD

    A[Start with the value of ΔG°] --> B{What is the sign of ΔG°?};

    B --> C[ΔG° < 0 <br/>(Negative)];

    B --> D[ΔG° > 0 <br/>(Positive)];

    B --> E[ΔG° ≈ 0];


    C --> F["ln(K) must be positive"];

    F --> G["K > 1"];

    G --> H["Products are favored at equilibrium"];


    D --> I["ln(K) must be negative"];

    I --> J["K < 1"];

    J --> K["Reactants are favored at equilibrium"];


    E --> L["ln(K) ≈ 0"];

    L --> M["K ≈ 1"];

    M --> N["Significant amounts of both reactants and products exist at equilibrium"];

Key Terms, Quantities, & Concepts

• Gibbs Free Energy (G): A thermodynamic quantity representing the energy in a system available to do useful work at constant temperature and pressure.

• Standard Gibbs Free Energy Change (ΔG°): The change in Gibbs free energy for a process when all reactants and products are in their standard states (1 M, 1 atm, 298 K). Its sign indicates thermodynamic favorability under these specific conditions.

• Equilibrium Constant (K): The ratio of product concentrations to reactant concentrations at equilibrium. A large K indicates that the reaction proceeds far toward the products.

• Thermodynamically Favored: A process with a negative ΔG°. This means that the process can proceed toward products without an external input of energy and that the equilibrium state will favor products (K > 1).

• Standard Conditions: A set of reference conditions (298 K, 1 atm, 1 M solutions) used for comparing thermodynamic data between different reactions.

• Ideal Gas Constant (R): A fundamental physical constant. In thermodynamic calculations, the value 8.314 J/(mol·K) is used to ensure unit consistency with energy values.

• Natural Logarithm (ln): The mathematical function that is the inverse of the exponential function (e^x). It is central to the relationship between ΔG° and K.

Skill Snapshots

• Causation:

  1. A negative standard free energy change (ΔG° < 0) causes the equilibrium constant (K) to be greater than 1.

  2. A very large positive ΔG° causes K to be extremely small (K << 1), meaning the forward reaction is highly unfavored.

  3. The presence of the temperature term (T) in the equation ΔG° = -RT ln K causes the equilibrium constant to be temperature-dependent.

• Comparison:

  1. A reaction with ΔG° < 0 favors products, whereas a reaction with ΔG° > 0 favors reactants at equilibrium.

  2. ΔG° describes the free energy change under standard conditions, while the non-standard free energy change (ΔG) describes the energy change under any conditions and is equal to zero at the point of equilibrium.

  3. Thermodynamics (ΔG°) predicts the final equilibrium position of a reaction, while kinetics describes the speed or rate at which that equilibrium is approached.

• Change and Continuity Over Time:

  • Baseline: A system is at equilibrium when ΔG = 0 and the reaction quotient Q equals the equilibrium constant K.

  • Change 1: For a reaction with ΔG° < 0 that starts with only reactants, the system will spontaneously change by converting reactants to products until the equilibrium state is reached.

  • Change 2: If the temperature of an exothermic reaction is increased, the value of ΔG° becomes less negative, changing the value of K to be smaller and shifting the equilibrium position toward the reactants.

  • Continuity: Throughout any reversible reaction, the relationship defined by the equation ΔG° = -RT ln K remains valid for that specific temperature.

Common Misconceptions & Clarifications

1. Misconception: A reaction with a negative ΔG° goes to 100% completion.

   Clarification: A negative ΔG° means products are favored at equilibrium (K > 1), not that reactants are completely consumed. All reversible reactions establish an equilibrium that includes both reactants and products.

2. Misconception: A thermodynamically favored reaction (ΔG° < 0) is always a fast reaction.

   Clarification: Thermodynamics and kinetics are separate. ΔG° tells us where the equilibrium lies, not how fast we get there. The conversion of diamond to graphite is highly favored (ΔG° is very negative) but is so slow it is unobservable on a human timescale.

3. Misconception: If ΔG° is positive, the reaction is impossible.

   Clarification: A positive ΔG° means that under standard conditions, the reaction favors reactants (K < 1). The forward reaction still occurs to a small extent to establish equilibrium. Furthermore, changing the conditions (e.g., temperature or concentrations) can make the reaction proceed.

4. Misconception: The value of K is always the same for a given reaction.

   Clarification: The equilibrium constant K is only constant at a specific temperature. Because temperature is part of the equation ΔG° = -RT ln K, changing T will change K.

One-Paragraph Summary

The standard Gibbs free energy change, ΔG°, serves as the definitive link between thermodynamics and chemical equilibrium. Its sign and magnitude provide a clear prediction of a reaction's outcome: a negative ΔG° corresponds to an equilibrium constant K greater than 1, indicating that products are favored, while a positive ΔG° corresponds to K less than 1, where reactants are favored. The equation ΔG° = −RT ln K provides the quantitative tool to convert between the potential energy of a system and the final composition of its equilibrium mixture. This principle is fundamental for predicting the extent of chemical reactions and for manipulating reaction conditions to maximize the yield of desired products.