Thermodynamic Equilibrium: Gibbs-Helmholtz

Analyze spontaneity and thermodynamic equilibrium with the Gibbs-Helmholtz equation: ΔG° = ΔH° − TΔS°, equilibrium constant Keq, and inversion temperature

Parameters

Standard enthalpy

Enthalpy change of the reaction under standard conditions

-500500

Standard entropy

Entropy change of the reaction under standard conditions

-500500

Temperature

System temperature

13000

Results

Standard Gibbs free energy

—

kJ/mol

Equilibrium constant

—

Inversion temperature

—

Temperature dependence of ΔG°

Calculating…

Fundamentals & Explanation

Gibbs-Helmholtz Equation and Equilibrium

The standard Gibbs free energy (ΔG°) is the thermodynamic potential that determines the spontaneity of a reaction at constant temperature and pressure. Its temperature dependence is governed by the Gibbs-Helmholtz equation (integrated assuming constant heat capacities). The thermodynamic equilibrium constant (K_eq) is directly linked to this potential:

Free Energy

\Delta G^\circ = \Delta H^\circ - T\Delta S^\circ

Equilibrium Const.

K_{eq} = \exp\left(-\frac{\Delta G^\circ}{RT}\right)

The simulator rigorously evaluates both functions. If K_eq exceeds the 64-bit precision limits (exponent greater than 700 or less than -700), it is visually restricted, but a warning for extreme thermal asymmetry is triggered.

Linearization: van't Hoff Equation

By substituting ΔG° into the K_eq expression and applying the natural logarithm, the linear form of the van't Hoff isochore is obtained. This expression is fundamental because it demonstrates that plotting ln(K_eq) vs 1/T yields a straight line, whose slope allows experimental isolation of the enthalpy:

\ln K_{eq} = -\frac{\Delta H^\circ}{R}\frac{1}{T} + \frac{\Delta S^\circ}{R}

The slope is

m = -\Delta H^\circ / R

and the y-intercept is

b = \Delta S^\circ / R

. This is the guiding principle behind the simulator's third chart, which does not suffer from the exponential visual distortions of the direct graph.

Thermodynamic Quadrants and Inversion T

The inversion temperature occurs when

\Delta G^\circ = 0

, which implies

T_{inv} = \Delta H^\circ / \Delta S^\circ

. This temperature marks the exact crossover between spontaneity and non-spontaneity. Depending on the enthalpic and entropic signs, thermodynamics classifies reactions into four universal regimes:

ΔH°	ΔS°	Thermodynamic Behavior & Example
< 0	> 0	Always spontaneous (ΔG° < 0 for all T). No physical T_inv. Ex: Exothermic combustion generating gases (glucose, methane).
> 0	< 0	Never spontaneous (ΔG° > 0 for all T). No physical T_inv. Ex: Decomposition of liquid water into gaseous H₂ and O₂ at room T.
< 0	< 0	Spontaneous at low T (T < T_inv). Enthalpically driven. Ex: Freezing of water (H₂O liq → sol) below 0°C.
> 0	> 0	Spontaneous at high T (T > T_inv). Entropically driven. Ex: Endothermic thermal decomposition of limestone (CaCO₃).

Physical Limits of the Model

Ellingham Approximation (ΔCp ≈ 0)

The simulator assumes that ΔH° and ΔS° are truly independent of temperature. In thermodynamic reality, both vary with T according to Kirchhoff's Law and the integral of (ΔC_p/T)dT. This linear approximation is very robust for moderate intervals and is widely used in Ellingham Diagrams for metallurgy, but loses severe accuracy at extremely high temperatures or if the substance undergoes critical phase transitions.

Standard vs Actual State (Q)

The simulator exclusively calculates standardspontaneity (ΔG°). However, the actual in-situ spontaneity (ΔG) also depends on the instantaneous reaction quotient: ΔG = ΔG° + RT ln(Q). A reaction with ΔG° > 0 can, in industrial practice, be driven to be spontaneous if the concentrations of the reactants are forced to be very high (achieving a Q ≪ 1).

Constant / Ref.	Official Value	Rigorous Source
R (Ideal Gas)	8.314462618... J/(mol·K)	CODATA 2018 (NIST)
Standard T (Thermo)	298.15 K (25 °C)	IUPAC Green Book
Standard P	10⁵ Pa (1 bar)	IUPAC 1982

Academic References

[1] Atkins, P. & de Paula, J. (2014). Physical Chemistry (10th ed.). Oxford University Press. Chapters 3 and 5 (Thermodynamic Fundamentals of the Gibbs-Helmholtz Equation and Equilibrium Regimes).
[2] IUPAC. (2007). Quantities, Units and Symbols in Physical Chemistry (3rd ed., "Green Book"). RSC Publishing. Sec. 2.11 (Thermodynamic Conventions and Standard Pressure).
[3] Denbigh, K. (1981). The Principles of Chemical Equilibrium (4th ed.). Cambridge University Press. (For exact derivations of the van't Hoff isochore).

Thermodynamic Equilibrium: Gibbs-Helmholtz

Parameters

Results

Temperature dependence of ΔG°

Fundamentals & Explanation

Gibbs-Helmholtz Equation and Equilibrium

Linearization: van't Hoff Equation

Thermodynamic Quadrants and Inversion T

Physical Limits of the Model

Ellingham Approximation (ΔCp ≈ 0)

Standard vs Actual State (Q)

Academic References

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Gibbs-Helmholtz Equation and Equilibrium

Linearization: van't Hoff Equation

Thermodynamic Quadrants and Inversion T

Physical Limits of the Model

Ellingham Approximation (ΔCp ≈ 0)

Standard vs Actual State (Q)

Academic References