ChemistryChemical Energetics and EquilibriaA-Level

Relationship between Gibbs Free Energy and Equilibrium Constant

This equation establishes the thermodynamic link between the standard Gibbs free energy change and the equilibrium constant of a chemical reaction.

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Δ G^{\circ} = - R T ln K

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Core idea

Overview

The equation shows that the spontaneity of a reaction under standard conditions is directly related to the position of its equilibrium. A negative Gibbs free energy value results in an equilibrium constant greater than one, indicating that the products are favored at equilibrium. Conversely, a positive Gibbs free energy value implies that the reactants are favored.

When to use: Use this equation when you need to calculate the equilibrium constant from standard thermodynamic data or vice versa.

Why it matters: It allows chemists to predict the extent of a chemical reaction without having to measure concentrations experimentally at equilibrium.

Symbols

Variables

$Δ$ G^ $\circ$ = Standard Gibbs Free Energy change (J/mol), R = Gas Constant (J/mol·K), T = Temperature (K), K = Equilibrium Constant

Δ G^{\circ}

Standard Gibbs Free Energy change (J/mol)

unit

R

Gas Constant (J/mol·K)

s^{-} 1

T

Temperature (K)

K

K

Equilibrium Constant

dimensionless

Walkthrough

Derivation

Derivation of Relationship between Gibbs Free Energy and Equilibrium Constant

This derivation relates the chemical potential of reactants and products to the state of chemical equilibrium using the thermodynamic definition of Gibbs free energy.

The system is at constant temperature and pressure.
The reactants and products behave as ideal gases or ideal solutes.
The system is at equilibrium, where the total Gibbs free energy is at a minimum.

General Gibbs Free Energy Equation

This fundamental equation describes the change in Gibbs free energy for a reaction under non-standard conditions, where Q is the reaction quotient.

Δ G = Δ G^{°} + R T ln Q

Note: Remember that Q is calculated using the activities of products divided by reactants.

Application of Equilibrium Conditions

At chemical equilibrium, the net change in Gibbs free energy is zero (ΔG = 0) and the reaction quotient Q becomes equal to the equilibrium constant K.

0 = Δ G^{°} + R T ln K

Note: This is a common exam derivation point; ensure you define K as the equilibrium constant.

Rearrangement to Final Form

By subtracting RT ln K from both sides, we isolate the standard Gibbs free energy change, showing its direct dependence on the equilibrium constant.

Δ G^{°} = - R T ln K

Note: A negative value for ΔG° indicates that K > 1, meaning the reaction favors the products at equilibrium.

Result

Δ G^{°} = - R T ln K

Source: A-Level Chemistry (OCR/AQA/Edexcel Specification - Thermodynamics section)

Free formulas

Rearrangements

Solve for deltaG

Make deltaG the subject

Δ G^{\circ} = - R T ln K

Deterministic rearrangement generated from calculator baseLaTeX for deltaG.

Difficulty: 2/5

Solve for $R$

Make R the subject

R = - \frac{Δ G ^{\circ}}{T ln K}

Deterministic rearrangement generated from calculator baseLaTeX for R.

Difficulty: 2/5

Solve for $T$

Make T the subject

T = - \frac{Δ G ^{\circ}}{R ln K}

Deterministic rearrangement generated from calculator baseLaTeX for T.

Difficulty: 2/5

Solve for $K$

Make K the subject

K = e^{- \frac{Δ G ^{\circ}}{R T}}

Deterministic rearrangement generated from calculator baseLaTeX for K.

Difficulty: 2/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Why it behaves this way

Intuition

Think of this as a 'potential energy landscape' where the system acts like a ball rolling toward the lowest point. The equilibrium constant K is the 'depth' of the valley—the deeper the valley (larger K), the more the ball is trapped there. ΔG measures the height difference between the starting state and the bottom of the valley. The negative sign ensures that a spontaneous reaction (negative ΔG) always moves toward a state of higher stability (larger K).

Δ G

Standard Gibbs Free Energy Change

The 'net energy profit' or 'chemical potential' available to do work as a reaction proceeds to equilibrium.

R

Ideal Gas Constant

The scaling factor that translates thermal energy into the energy units used for chemical reactions.

T

Absolute Temperature

A measure of the thermal 'noise' or agitation that influences how effectively the system can find and settle into its equilibrium state.

ln K

Natural Logarithm of the Equilibrium Constant

The relative 'occupancy' of products versus reactants; it represents how strongly the system 'prefers' the product side.

Signs and relationships

Negative sign (-): It acts as a 'direction inverter': if K is large (products favored), ln K is positive, resulting in a negative ΔG, which signifies a spontaneous, downhill reaction.

One free problem

Practice Problem

Calculate the equilibrium constant K for a reaction at 298 K where ΔG° = -5.70 kJ/mol. (R = 8.314 J/mol·K)

Standard Gibbs Free Energy change (J/mol)-5700 unit

Temperature (K)298 K

Solve for: $K$

Hint: Rearrange to K = e^(-ΔG / RT). Ensure ΔG is in Joules.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

In industrial ammonia production (Haber Process), this equation helps engineers determine the maximum theoretical yield of ammonia at specific temperatures.

Study smarter

Tips

Ensure the temperature is always in Kelvin.
Check that the gas constant R is in units of J/(mol·K) if ΔG is in J/mol.
Remember that ΔG must be in Joules, not kiloJoules, to match the units of R.

Avoid these traps

Common Mistakes

Forgetting to convert kiloJoules (kJ) to Joules (J) when performing calculations.
Using the wrong temperature units (Celsius instead of Kelvin).
Confusing the standard Gibbs free energy (ΔG°) with non-standard Gibbs free energy (ΔG).

Common questions

Frequently Asked Questions

This derivation relates the chemical potential of reactants and products to the state of chemical equilibrium using the thermodynamic definition of Gibbs free energy.

Use this equation when you need to calculate the equilibrium constant from standard thermodynamic data or vice versa.

It allows chemists to predict the extent of a chemical reaction without having to measure concentrations experimentally at equilibrium.

Forgetting to convert kiloJoules (kJ) to Joules (J) when performing calculations. Using the wrong temperature units (Celsius instead of Kelvin). Confusing the standard Gibbs free energy (ΔG°) with non-standard Gibbs free energy (ΔG).

In industrial ammonia production (Haber Process), this equation helps engineers determine the maximum theoretical yield of ammonia at specific temperatures.

Ensure the temperature is always in Kelvin. Check that the gas constant R is in units of J/(mol·K) if ΔG is in J/mol. Remember that ΔG must be in Joules, not kiloJoules, to match the units of R.

References

Sources

Atkins, P., & de Paula, J. (2014). Physical Chemistry (10th ed.). Oxford University Press.
Zumdahl, S. S., & Zumdahl, S. A. (2017). Chemistry (10th ed.). Cengage Learning.
Atkins, P., & de Paula, J. (2014). Physical Chemistry, 10th Edition.
Royal Society of Chemistry, A-Level Chemistry Resources.
A-Level Chemistry (OCR/AQA/Edexcel Specification - Thermodynamics section)

Relationship between Gibbs Free Energy and Equilibrium Constant

Overview

Variables

Derivation

General Gibbs Free Energy Equation

Application of Equilibrium Conditions

Rearrangement to Final Form

Rearrangements

Intuition

Practice Problem

Real-World Context

Tips

Common Mistakes

Related Formulas

Gibbs Free Energy Equation

Frequently Asked Questions

Sources