The Gibbs-Helmholtz equation is used to determine the change in the Gibbs energy of a system as a function of temperature at a fixed pressure. It serves as a vital foundational tool in thermodynamics by encompassing the core essence of both the First and Second Laws of Thermodynamics. Consequently, almost any thermodynamic relation regarding system equilibrium and state variables can be systematically deduced from this single formulation.
Derivation of the Gibbs-Helmholtz Equation
When a system undergoes a reversible change, the fundamental thermodynamic variation in free energy ($G$) with temperature ($T$) and pressure ($P$) is given by the state expression:
At constant pressure, the change in pressure is zero ($dP = 0$). Therefore, Equation 1 simplifies to:
$$dG = -SdT$$Expressing this as a partial derivative with respect to temperature at constant pressure ($P$):
$$\left(\frac{\partial G}{\partial T}\right)_P = -S$$For a finite thermodynamic change from an initial state (1) to a final state (2):
$$-\Delta S = -\left(S_2 - S_1\right)$$ $$-\Delta S = \left(\frac{\partial G_2}{\partial T}\right)_P - \left(\frac{\partial G_1}{\partial T}\right)_P$$ $$\Delta S = -\left[\frac{\partial(\Delta G)}{\partial T}\right]_P \quad \text{--- (Equation 2)}$$We already know the fundamental relationship for Gibbs free energy:
$$\Delta G = \Delta H - T\Delta S \quad \text{--- (Equation 3)}$$Substituting the precise value of $\Delta S$ from Equation 2 into Equation 3 yields the standard form of the Gibbs-Helmholtz Equation:
$$\Delta G = \Delta H + T\left[\frac{\partial(\Delta G)}{\partial T}\right]_P$$The above equation is applicable to all closed processes taking place at constant pressure.