Standard Gibbs free energy | Equation, Derivation and Rearrangements

Core idea

Overview

This fundamental thermodynamic equation relates the standard Gibbs free energy change (ΔG°) to the equilibrium constant (K) of a chemical reaction. It provides a bridge between energetics and the final ratio of products to reactants at a specific temperature.

When to use: Apply this equation when calculating the extent of a reaction at equilibrium or finding the spontaneity of a process under standard conditions. It is specifically for systems at a constant temperature where standard state values (1 atm or 1 M) are provided.

Why it matters: It allows scientists to predict how temperature changes will shift equilibrium positions in industrial synthesis, like the Haber process. It also helps biochemists understand the energetics of enzyme-catalyzed reactions in the human body.

Symbols

Variables

R = Gas Constant, T = Temperature, K = Equilibrium Constant, $Δ$ G^ $\circ$ = Standard Gibbs Energy

R

Gas Constant

J/molK

T

Temperature

K

K

Equilibrium Constant

Variable

Δ G^{\circ}

Standard Gibbs Energy

J/mol

Walkthrough

Derivation

Formula: Standard Gibbs Free Energy and Equilibrium

Relates standard Gibbs free energy change to equilibrium constant, linking thermodynamics and equilibrium.

Standard conditions apply (e.g., 100 kPa, 298 K, 1 mol dm^{-3} where relevant).
K is defined consistently for the balanced equation written.

1

State the Relationship:

If K>1 then $ln$ K>0 so $Δ$ $G^{⊖}$ <0, meaning products are favoured under standard conditions.

Δ G^{⊖} = - R T ln K

Result

Δ G^{⊖} = - R T ln K

Source: AQA A-Level Chemistry — Thermodynamics

Free formulas

Rearrangements

Solve for $K$

Make K the subject

K = e^{- \frac{Δ G ^{\circ}}{R T}}

Start from the Standard Gibbs free energy equation. To make K the subject, isolate the natural logarithm by dividing by $- R T$ , then apply the inverse exponential function ( $e^{x}$ ) to both sides.

Difficulty: 2/5

Solve for $T$

Make T the subject

T = - \frac{Δ G ^{\circ}}{R ln K}

To make T the subject, start with the Standard Gibbs free energy equation and divide both sides by the terms multiplying T.

Difficulty: 2/5

Solve for $R$

Make R the subject

R = - \frac{Δ G ^{\circ}}{T ln K}

To make R the subject of the Standard Gibbs free energy equation, divide both sides by the terms multiplying R (-T ln K) and then simplify the expression by moving the negative sign to the front.

Difficulty: 2/5

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

Visual intuition

Graph

The graph follows a logarithmic curve where the Standard Gibbs Energy decreases as the Equilibrium Constant increases, approaching a vertical asymptote at zero. For a chemistry student, this shape shows that a very small Equilibrium Constant corresponds to a large positive Standard Gibbs Energy, while a large Equilibrium Constant indicates a more negative value. The most important feature of this curve is the inverse relationship between the variables, meaning that as the system moves toward a more spontaneous state, the equilibrium position shifts significantly toward the products.

Graph type: logarithmic

Why it behaves this way

Intuition

This equation links the inherent energetic 'drive' of a reaction (ΔG°) to the relative amounts of reactants and products present when the system reaches its lowest energy state (K) at a given temperature.

Δ G °

The change in Gibbs free energy for a reaction when all reactants and products are in their standard states.

A measure of the maximum useful (non-PV) work a reaction can perform under standard conditions, indicating its spontaneity.

R

The ideal gas constant.

A fundamental constant that scales temperature into energy units, linking thermal energy to other forms of energy.

T

Absolute temperature in Kelvin.

Represents the thermal energy available in the system; higher T means more thermal energy.

K

The equilibrium constant for the reaction.

Quantifies the extent to which a reaction proceeds to products at equilibrium; a large K means products are favored.

ln

Natural logarithm.

Converts the multiplicative ratio of K into a linear energy scale, allowing it to be directly related to ΔG°.

Signs and relationships

-RTlnK: The negative sign ensures consistency with the definition of spontaneity: if K > 1 (products favored), lnK is positive, making ΔG° negative (spontaneous).

Free study cues

Insight

Canonical usage

The standard Gibbs free energy change (ΔG°) is typically expressed in joules per mole (J/mol) or kilojoules per mole (kJ/mol), with the ideal gas constant (R) in J/(mol·K), and temperature (T) in Kelvin (K).

Common confusion

A common error is using temperature in Celsius instead of Kelvin, or failing to ensure consistent energy units between ΔG° (often in kJ/mol) and R (typically in J/(mol·K)) without proper conversion.

Dimension note

The equilibrium constant (K) is a ratio of activities or effective concentrations/pressures at equilibrium, making it inherently dimensionless. The natural logarithm (ln K) is also dimensionless.

Unit systems

$Δ G °$ J/mol - Often reported in kJ/mol, requiring conversion if R is in J/(mol·K).

$R$ J/(mol·K) - The ideal gas constant. Ensure its energy unit is consistent with ΔG°.

$T$ K - Must be absolute temperature in Kelvin.

$K$ dimensionless - The equilibrium constant is a ratio of activities or effective concentrations/pressures, making it dimensionless.

One free problem

Practice Problem

Calculate the standard Gibbs free energy change (ΔG°) for a reaction at 298.15 K that has an equilibrium constant (K) of 2.0 × 10⁴.

Gas Constant8.314 J/molK

Temperature298.15 K

Equilibrium Constant20000

Solve for: $G 0$

Hint: Use the natural logarithm (ln) of the equilibrium constant and ensure the result is in Joules per mole.

The full worked solution stays in the interactive walkthrough.

Where it shows up

Real-World Context

In k from tabulated Δ G values, Standard Gibbs free energy is used to calculate Standard Gibbs Energy from Gas Constant, Temperature, and Equilibrium Constant. The result matters because it helps connect measured amounts to reaction yield, concentration, energy change, rate, or equilibrium.

Study smarter

Tips

Ensure the unit of energy in ΔG° (often kJ) matches the unit in the gas constant R (J/mol·K).
A large K value (> 1) results in a negative ΔG°, indicating the reaction is spontaneous in the forward direction.
Always use absolute temperature in Kelvin (K = °C + 273.15).

Avoid these traps

Common Mistakes

Using log10 instead of ln.
Forgetting the negative sign.

Keep going

Related Formulas

Common questions

Frequently Asked Questions

Relates standard Gibbs free energy change to equilibrium constant, linking thermodynamics and equilibrium.

Apply this equation when calculating the extent of a reaction at equilibrium or finding the spontaneity of a process under standard conditions. It is specifically for systems at a constant temperature where standard state values (1 atm or 1 M) are provided.

It allows scientists to predict how temperature changes will shift equilibrium positions in industrial synthesis, like the Haber process. It also helps biochemists understand the energetics of enzyme-catalyzed reactions in the human body.

Using log10 instead of ln. Forgetting the negative sign.

In k from tabulated Δ G values, Standard Gibbs free energy is used to calculate Standard Gibbs Energy from Gas Constant, Temperature, and Equilibrium Constant. The result matters because it helps connect measured amounts to reaction yield, concentration, energy change, rate, or equilibrium.

Ensure the unit of energy in ΔG° (often kJ) matches the unit in the gas constant R (J/mol·K). A large K value (> 1) results in a negative ΔG°, indicating the reaction is spontaneous in the forward direction. Always use absolute temperature in Kelvin (K = °C + 273.15).

References

Sources

Atkins' Physical Chemistry
Callen, H. B. (1985). Thermodynamics and an Introduction to Thermostatistics.
Wikipedia: Gibbs free energy
Wikipedia: Equilibrium constant
NIST CODATA
IUPAC Gold Book
Atkins, P. W.; de Paula, J. Atkins' Physical Chemistry. 11th ed. Oxford University Press, 2018.
Callen, H. B. Thermodynamics and an Introduction to Thermostatistics. 2nd ed. John Wiley & Sons, 1985.