J. Willard Gibbs employed the term “free energy” to predict the direction of spontaneity.
Free energy (G) is defined as the amount of energy available for doing productive work under constant temperature and pressure.
G = H – TS
H denotes the system’s enthalpy, S signifies the system’s entropy, and T represents the system’s temperature measured in Kelvin.
The Gibbs free energy is expressed as G = H – TS
Enthalpy is defined as H = U + PV.
Consequently, G = U + PV – TS
Gibbs free energy changes may be expressed as
ΔG = ΔU + Δ(PV) – Δ(TS)
ΔG = ΔU + PΔV + VΔP – TΔS – SΔT
Under conditions of constant temperature and constant pressure, ΔT = 0 and ΔP = 0. Consequently, ΔG = ΔU + PΔV – TΔS.
Given that ΔH = ΔU + PΔV
ΔG = ΔH – TΔS
The equation ΔG=ΔH–TΔS is referred to as the Gibbs-Helmholtz equation.
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.
G = H – TS …………………. (1)
For an isothermal process,
G₁ = H₁ – T S₁ for the initial state ………………. (2)
And G₂ = H₂ – T S₂ for the final state ………………… (3)
Subtracting equation (2) from equation (3), we obtain
G₂ – G₁ = (H₂ – T S₂) – (H₁ – T S₁)
Or G₂ – G₁ = (H₂ – H₁) – T (S₂ – S₁)
Or ΔG = ΔH – T ΔS …………………………. (4)
Where ΔG represents the change in the system’s free energy.
ΔH represents the change in the system’s enthalpy, whereas ΔS denotes the change in the system’s entropy.
Substituting H=U + PV into equation (1) yields
G=U + PV-TS ………………… (5)
By fully differentiating this equation, we obtain
dG = dU + PdV + VdP – TdS – SdT. ………………… (6)
However, it follows that dS = =
(according to the first law of thermodynamics).
Or TdS = dU + PdV
By substituting this value into equation (6), we obtain
dG = VdP – SdT.
If pressure remains constant, dP = 0
So, that the above equation reduces to
(dG) P = –(SdT) P
or
()P = – S …………………. (7)
∴ for the initial state
()P = – ���� …………………. (8)
∴ for the final state
()P = – ���� ………………… (9)
Subtracting equation (8) from (9), we get
()P – (
)P = -S2 – (-����)
= – ���� + ���� = – (���� – ����)
= – ΔS
It can also be expressed as
[]P= – ΔS
[]p = – ΔS
By substituting the value of –ΔS into equation (4), we obtain
ΔG = ΔH – T []ₚ
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.
An equivalent equation pertaining to work function A may be derived using the equation
A = U – TS
For a reaction at constant volume, the relevant equation is
ΔA = ΔU + T []V
This equation is referred to as the Gibbs-Helmholtz equation concerning internal energy and work function at constant volume.
