Temperature's Hidden Power: The Gibbs-Helmholtz Equation

Temperature appears in ΔG = ΔH - TΔS as a simple multiplier. But its influence is deeper than that.

The Gibbs-Helmholtz equation reveals how temperature changes shift the free energy landscape. It shows why reactions that are non-spontaneous at one temperature become spontaneous at another. And it connects the temperature derivative of free energy directly to enthalpy—a measurable quantity.

This equation is thermodynamics doing calculus.

The Equation

∂(ΔG/T)/∂T = -ΔH/T²

Or equivalently:

∂(ΔG)/∂T = -ΔS

The first form is more useful because it relates the temperature dependence of ΔG/T directly to enthalpy. The second form shows that entropy is the "slope" of free energy versus temperature.

These equations let you: - Predict how ΔG changes with temperature - Calculate ΔH from ΔG measurements at different temperatures - Understand why equilibrium constants shift with temperature

Deriving the Result

Start from ΔG = ΔH - TΔS.

Take the temperature derivative (at constant pressure):

∂(ΔG)/∂T = ∂(ΔH)/∂T - ∂(TΔS)/∂T

The first term: ∂(ΔH)/∂T = ΔCp (heat capacity change at constant pressure)

The second term: ∂(TΔS)/∂T = ΔS + T∂(ΔS)/∂T

This gets complicated. But at constant ΔH and ΔS (a useful approximation for small temperature ranges):

∂(ΔG)/∂T = -ΔS

If ΔS > 0: ΔG decreases with temperature (entropy-driven reactions favor high T) If ΔS < 0: ΔG increases with temperature (enthalpy-driven reactions favor low T)

The pebble: Entropy is the slope of free energy versus temperature. Positive slope (ΔS < 0) means ΔG increases with T. Negative slope (ΔS > 0) means ΔG decreases with T.

Why ΔG/T Matters

The form with ΔG/T is special because it connects to the equilibrium constant.

From ΔG° = -RT ln K:

ln K = -ΔG°/RT

So ΔG°/T = -R ln K.

The Gibbs-Helmholtz equation becomes:

∂(ln K)/∂T = ΔH°/RT²

This is the van't Hoff equation. It describes how equilibrium constants change with temperature.

If ΔH° > 0 (endothermic): K increases with T (equilibrium shifts toward products) If ΔH° < 0 (exothermic): K decreases with T (equilibrium shifts toward reactants)

The pebble: Heat the system, and equilibrium shifts toward whichever direction absorbs heat. This is Le Chatelier's principle, derived from Gibbs-Helmholtz.

Integrating the van't Hoff Equation

Integrate the van't Hoff equation:

∫d(ln K) = ∫(ΔH°/RT²)dT

Assuming ΔH° is constant over the temperature range:

ln(K₂/K₁) = (ΔH°/R)(1/T₁ - 1/T₂)

This lets you calculate K at a new temperature from K at a known temperature, if you know ΔH°.

This equation is widely used: - Predict equilibrium shifts with temperature - Determine ΔH° from K measurements at two temperatures - Design industrial processes at optimal temperatures

Phase Transition Temperatures

At a phase transition (melting, boiling), the two phases are in equilibrium: ΔG = 0.

The Gibbs-Helmholtz equation explains why phase transitions occur at specific temperatures.

For melting ice at 1 atm: - Below 0°C: ΔG(solid→liquid) > 0 (ice is stable) - At 0°C: ΔG = 0 (ice and water coexist) - Above 0°C: ΔG < 0 (liquid is stable)

The melting point is where the free energy curves of solid and liquid cross. The Gibbs-Helmholtz equation determines the slopes of these curves (via entropy), hence where they cross (via enthalpy/entropy balance).

Clausius-Clapeyron Equation

For phase equilibria, the Gibbs-Helmholtz equation leads to the Clausius-Clapeyron equation:

dP/dT = ΔH_transition / (TΔV_transition)

This describes how the equilibrium pressure changes with temperature—i.e., how boiling points change with pressure.

Water boils at 100°C at 1 atm. At lower pressure (high altitude), it boils at lower temperature. At higher pressure (pressure cooker), it boils at higher temperature.

The Clausius-Clapeyron equation quantifies this from enthalpy and volume changes.

Temperature Dependence of ΔG: Examples

Example 1: Decomposition of calcium carbonate CaCO₃ → Caite + CO₂

ΔH° = +178 kJ/mol (endothermic) ΔS° = +161 J/(mol·K) (entropy increases, gas produced)

At 298 K: ΔG° = 178 - 298(0.161) = +130 kJ/mol (non-spontaneous) At 1000 K: ΔG° = 178 - 1000(0.161) = +17 kJ/mol (weakly non-spontaneous) At 1200 K: ΔG° = 178 - 1200(0.161) = -15 kJ/mol (spontaneous)

The crossover is around 1100 K (827°C). This is why lime production requires high temperatures.

Example 2: Synthesis of ammonia N₂ + 3H₂ → 2NH₃

ΔH° = -92 kJ/mol (exothermic) ΔS° = -199 J/(mol·K) (entropy decreases, fewer moles of gas)

Low T: Enthalpy dominates, ΔG < 0, spontaneous High T: Entropy term grows, ΔG becomes less negative, eventually positive

But low T means slow kinetics. The Haber process compromises: moderate temperature with catalyst.

Graphical Analysis

Plot ΔG vs T: - Intercept at T = 0: ΔH (extrapolated) - Slope: -ΔS

For a reaction that becomes spontaneous at high T (Case 4: ΔH > 0, ΔS > 0): - Line starts above zero (ΔH > 0) - Slopes downward (-ΔS < 0 because ΔS > 0) - Crosses zero at T = ΔH/ΔS

For a reaction that becomes non-spontaneous at high T (Case 3: ΔH < 0, ΔS < 0): - Line starts below zero (ΔH < 0) - Slopes upward (-ΔS > 0 because ΔS < 0) - Crosses zero at T = ΔH/ΔS

The graph makes temperature effects intuitive.

Heat Capacity Corrections

For accurate work over wide temperature ranges, ΔH and ΔS aren't constant.

ΔH(T) = ΔH(T_ref) + ∫ΔCp dT

ΔS(T) = ΔS(T_ref) + ∫(ΔCp/T) dT

Where ΔCp is the heat capacity difference between products and reactants.

This matters for: - Reactions at extreme temperatures (combustion, metallurgy) - Precision thermodynamic calculations - Long extrapolations from reference conditions

For rough estimates over modest ranges, constant ΔH and ΔS work fine.

Applications in Biology

Protein stability depends on temperature. The Gibbs-Helmholtz equation explains:

Cold denaturation: Proteins can unfold at low temperature as well as high temperature. The ΔG vs T curve for folding can cross zero twice.

Enzyme activity: Enzymes have optimal temperatures. Below optimal: kinetics are slow. Above optimal: denaturation reduces functional protein.

Membrane phase transitions: Lipid bilayers transition from gel to fluid phase at specific temperatures, determined by lipid composition and Gibbs-Helmholtz analysis.

Connection to Entropy

The equation ∂(ΔG)/∂T = -ΔS has a deep meaning:

Entropy measures how free energy responds to temperature.

High entropy systems are temperature-sensitive: their free energy drops rapidly with heating.

Low entropy systems are temperature-insensitive: their free energy changes slowly with temperature.

This makes sense: entropy is about thermal disorder, so high-entropy systems are more affected by thermal changes.

Summary

The Gibbs-Helmholtz equation: - Shows temperature is not just a multiplier but a controller of spontaneity - Derives the van't Hoff equation for equilibrium constant shifts - Explains phase transition temperatures - Connects entropy to the temperature derivative of free energy - Enables predictions across temperature ranges

Temperature is thermodynamics' most versatile lever. The Gibbs-Helmholtz equation is the manual for pulling it.

Further Reading

- Atkins, P. & de Paula, J. (2014). Physical Chemistry. Oxford University Press. - Denbigh, K. (1981). The Principles of Chemical Equilibrium. Cambridge University Press.

This is Part 5 of the Gibbs Free Energy series. Next: "Chemical Equilibrium: Where ΔG Equals Zero"