Gibbs-Helmholtz Equation

J. Willard Gibbs employed the term “free energy” to predict the direction of spontaneity. Free energy (G) is defined as the amount of energy accessible for doing productive work under constant temperature and pressure.

H denotes the system’s enthalpy, S signifies the system’s entropy, and T represents the system’s temperature measured in Kelvin.

Gibbs free energy changes may be expressed as:

Under conditions of constant temperature and constant pressure, ΔT = 0 and ΔP = 0.

Consequently, ΔG = ΔU + PΔV – TΔS

Given that ΔH = ΔU + PΔV

The equation ΔG = ΔH – TΔS is referred to as the Gibbs-Helmholtz equation.

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Gibbs and Helmholtz formulated two equations, together referred to as the Gibbs-Helmholtz equation. One equation may be articulated in terms of variations in free energy and enthalpy, while another can be articulated in terms of variations in internal energy and work function.

The Gibbs free energy (G) is represented by the equation:

For an isothermal process:

Subtracting equation (2) from equation (3), we obtain:

Where ΔG represents the change in the system’s free energy, ΔH represents the change in the system’s enthalpy, and ΔS denotes the change in the system’s entropy.

Substituting H = U + PV into equation (1) yields:

By fully differentiating this equation, we obtain:

However, it follows that dS = dq_revT = (dU + PdV)T (according to the first law of thermodynamics).

By substituting this value into equation (6), we obtain:

If pressure remains constant, dP = 0. So, that the above equation reduces to:

∴ for the initial state: ( ∂G₁∂T )_P = –S₁ (8)

∴ for the final state: ( ∂G₂∂T )_P = –S₂ (9)

Subtracting equation (8) from (9), we get:

It can also be expressed as:

By substituting the value of –ΔS into equation (4), we obtain:

This equation is referred to as the Gibbs-Helmholtz equation, relating free energy and enthalpy change under constant pressure, applicable to all processes.

This equation is employed to calculate the enthalpy change ΔH for a chemical process when the values of free energy changes at two distinct temperatures are known.

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An equivalent equation pertaining to work function A may be derived using the equation:

For a reaction at constant volume, the relevant equation is:

This equation is referred to as the Gibbs-Helmholtz equation concerning internal energy and work function at constant volume.