Gibbs Free Energy Formula: Definition, Equations, Example

Gibbs Free Energy Formula: Gibbs free energy, commonly represented as G or ΔG, serves as a thermodynamic parameter that quantifies the utmost reversible work attainable within a system operating at a consistent temperature and pressure. Its primary utility lies in evaluating the spontaneity of chemical reactions under these specific conditions. The formula for Gibbs free energy is as follows:

ΔG = ΔH – TΔS

Where:

• ΔG represents the alteration in Gibbs free energy.
• ΔH stands for the modification in enthalpy, which reflects the heat content of the system.
• T represents the temperature in units of Kelvin.
• ΔS signifies the Change in entropy, a quantification of the system’s disorder or randomness.

Here’s a brief derivation of the Gibbs free energy formula:

1. Start with the definition of the change in Gibbs free energy:

ΔG = G(final state) – G(initial state)

1. In a system under constant temperature and pressure, the alteration in enthalpy is connected to the heat transferred between the system and its surroundings:
2. ΔH = q (heat transfer) at constant 
3. The change in entropy can be related to the heat transfer as well:

ΔS = q (reversible) / T

1. Substitute the expressions for ΔH and ΔS into the definition of ΔG:

ΔG = [G(final state) – G(initial state)] – T[q (reversible) / T]

Simplify the equation:

ΔG = ΔH – TΔS

Also Check – Partial Pressure Formula

Gibbs Free Energy Formula Example

Example: Is the formation of water from its elements, hydrogen and oxygen, spontaneous at 298 K and 1 atm?

The chemically balanced equation for this particular reaction is:

2 H₂(g) + O₂(g) → 2 H₂O(g)

Given that ΔH° (standard enthalpy change) for this reaction is -483.7 kJ/mol, and ΔS° (standard entropy change) is +188.8 J/(mol·K), we can calculate ΔG at 298 K:

ΔG = ΔH – TΔS

ΔG = (-483.7 kJ/mol) – (298 K) * (0.1888 kJ/mol·K)

ΔG = -483.7 kJ/mol – 56.3264 kJ/mol

ΔG ≈ -539.03 kJ/mol

Since ΔG is negative, the formation of water from its elements is spontaneous under the given conditions of 298 K and 1 atm. This means the reaction can occur without requiring external work and will proceed in the forward direction.

Also Check – Net Ionic Formula

Relationship Between Free Energy and Equilibrium Constant

The Gibbs free energy (ΔG) and the equilibrium constant (K) are linked through the equation known as the Gibbs-Helmholtz equation, as follows:

ΔG° = -RT ln(K)

Where:

• ΔG° is the standard Gibbs free energy change for a chemical reaction at standard conditions (usually 298 K and 1 atm).
• R is the gas constant (8.314 J/(mol·K) or 0.008314 kJ/(mol·K)).
• T corresponds to the temperature measured in Kelvin.
• K is the equilibrium constant for the reaction.

This equation establishes a measurable connection between the alteration in Gibbs free energy and the equilibrium constant for a chemical reaction.

If ΔG° is negative:

This indicates that the reaction is spontaneous in the forward direction under standard conditions.

The equilibrium constant (K) for the reaction is greater than 1, meaning that the products are favored at equilibrium.

If ΔG° is positive:

This indicates that the reaction is non-spontaneous in the forward direction under standard conditions.

The equilibrium constant (K) for the reaction is less than 1, meaning that the reactants are favored at equilibrium.

If ΔG° is zero:

This suggests that the reaction is in equilibrium when evaluated under standard conditions. Under these conditions, the equilibrium constant (K) takes on a value of 1, indicating that the concentrations of both reactants and products are equal at equilibrium. It’s worth emphasizing that ΔG can also be computed for non-standard conditions using the following formula:

ΔG = ΔG° + RT ln(Q)

Where:

• ΔG is the Gibbs free energy change at non-standard conditions.
• ΔG° is the standard Gibbs free energy change.
• R is the gas constant.
• T is the absolute temperature in Kelvin.
• Q is the reaction quotient, which is similar to K but is calculated using concentrations or pressures at non-standard conditions.

By comparing ΔG to ΔG°, we can determine whether a reaction will proceed spontaneously or non-spontaneously under non-standard conditions, and you can calculate the equilibrium constant (K) at those conditions as well.

Also Check – Internal Energy Formula

Relationship Between Gibbs Free Energy and EMF of a Cell

The relationship between the Gibbs free energy (ΔG) and the electromotive force (EMF) of a cell is described by the Nernst equation, which relates the EMF (also known as cell potential or voltage) of an electrochemical cell to the standard Gibbs free energy change (ΔG°) for the cell reaction. The Nernst equation is as follows:

E = E° – (RT/nF) * ln(Q)

Where:

• E represents the cell potential (EMF) under non-standard conditions.
• E° signifies the standard cell potential (EMF) measured under standard conditions, typically at 298 K and 1 atm.
• R stands for the gas constant, which has two common values: 8.314 J/(mol·K) or 0.008314 kJ/(mol·K).
• T stands for the absolute temperature, specifically measured in Kelvin.
• n indicates the quantity of moles of electrons exchanged during the balanced cell reaction.
• F represents the Faraday constant, approximately equivalent to 96485 C/mol.
• Q is the reaction quotient, calculated as the ratio of product to reactant concentrations (or activities), with each term raised to the power of their respective stoichiometric coefficients as specified in the balanced cell reaction.

The Nernst equation allows you to calculate the cell potential (EMF) of an electrochemical cell at non-standard conditions (different concentrations of reactants and products) when you know the standard cell potential (E°) and the reaction quotient (Q). Here’s how the relationship between ΔG and EMF is understood through the Nernst equation:

If the cell potential (E) is positive:

This indicates that the cell reaction is spontaneous in the direction written under the given conditions.

The Gibbs free energy change (ΔG) for the reaction will have a negative value.

If the cell potential (E) is negative:

This indicates that the cell reaction is nonspontaneous in the direction written under the given conditions.

The Gibbs free energy change (ΔG) for the reaction will be of a positive value.

If the cell potential (E) is zero:

This indicates that the cell reaction is at equilibrium under the given conditions.

The Gibbs free energy change (ΔG) for the reaction will be zero.

In summary, the Nernst equation connects the EMF (cell potential) of an electrochemical cell to the Gibbs free energy change for the cell reaction. A positive EMF corresponds to a spontaneous reaction with a negative ΔG, while a negative EMF corresponds to a non-spontaneous reaction with a positive ΔG. The Nernst equation allows you to quantitatively relate these electrochemical and thermodynamic properties.