Gibbs Free Energy

Equivalent forms

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

dG = -S dT + V dP

One sign change on ΔG decides every chemical reaction, every phase transition, every biological process — the universal arrow of spontaneity at constant T,P.

Symbol	Name	SI unit	Dimension	Range
$ΔG$	Change in Gibbs energyoutput Gibbs free energy change of process	J/mol	M·L²·T⁻²·N⁻¹	-100000 – 100000
$ΔH$	Change in enthalpy Enthalpy change of process	J/mol	M·L²·T⁻²·N⁻¹	-100000 – 100000
$T$	Temperature Absolute temperature	K	Θ	100 – 1000
$ΔS$	Change in entropy Entropy change of process	J/(mol·K)	M·L²·T⁻²·N⁻¹·Θ⁻¹	-500 – 500

Unit systems

Symbol	SI	CGS	Imperial
$ΔG$	J/mol	erg/mol	BTU/mol
$ΔH$	J/mol	erg/mol	BTU/mol
$T$	K	K	°R
$ΔS$	J/(mol·K)	erg/(mol·K)	BTU/(mol·°R)

Where it holds

Valid for closed systems at constant T and P. For variable T,P or open systems use chemical potentials and the full Gibbs differential.

Dimensional analysis

$[\Delta G] = [\Delta H] - [T][\Delta S] = (J/mol) - (K)(J/(mol\cdot K)) = J/mol \checkmark$

Discovery

Josiah Willard Gibbs · 1873

Gibbs introduced this potential in his foundational treatise 'On the Equilibrium of Heterogeneous Substances', creating modern chemical thermodynamics single-handedly.

Try this

Why does ice melt at room temperature but not in a freezer?

For water-to-ice at 250 K: ΔH = -6010 J/mol, ΔS = -22.0 J/(mol·K). Compute ΔG and decide direction.

Research status: stable

Real-world applications

Battery EMF: $\Delta G = -nFE$
ATP hydrolysis powers all biology $(\Delta G \approx -30\,\mathrm{kJ/mol})$
Metallurgy: Ellingham diagrams predict reducibility of oxides
Drug binding affinity: $K_d = \exp (\Delta G/RT)$

Common misconceptions

$\Delta G$ < 0 means spontaneous, NOT fast — kinetics can make spontaneous reactions arbitrarily slow
G is a state function — $\Delta G$ depends only on endpoints, not path
'Free energy' doesn't mean free of cost; it's the max non-PV work obtainable

Experimental verification

Equilibrium constants

K = \exp (-\Delta G^{\circ}/RT)

match measured composition for thousands of reactions to 1% precision. Phase equilibria (e.g., water at 273.15 K, 1 atm) satisfy

\Delta G = 0

exactly.

Derivation

From the second law: for spontaneous process at constant T,P, dS_universe

\ge 0

.

Splitting into system + surroundings: dS_sys + dS_surr

\ge 0

.

Surroundings absorb heat dQ_surr

= -dH_sys

, so dS_surr

= -dH_sys/T

.

Multiplying through by -T:

dH_sys - T dS_sys \le 0

, i.e.,

dG \le 0

.

This defines

G = H - TS

as the potential whose decrease drives spontaneity at fixed T,P.

Limiting cases

T \to 0

⟶

\Delta G \to \Delta H

Enthalpy dominates at low T — exothermic reactions favored.

T \to \infty

⟶

\Delta G \to -T\cdot \Delta S

Entropy dominates at high T — disordering reactions favored.

\Delta H = T\Delta S

⟶

\Delta G = 0

Phase transition or equilibrium condition.

What if…

What

if \Delta H

and \Delta S

are both positive?

Spontaneous only above $T = \Delta H/\Delta S$ — endothermic but entropy-driven (e.g., $NH_{4}NO_{3}$ dissolution).

What

if \Delta H

<

0\,\mathrm{and} \Delta S

> 0?

Always spontaneous regardless of T — both driving forces align.

What

if \Delta G = 0

?

System is at equilibrium — at the phase transition temperature or reaction balance point.

1

Ice formation at 250 K

Given ·

ΔH:: -6010
T:: 250
ΔS:: -22

Find ·

\Delta G

Steps

Plug into $\Delta G = \Delta H - T\Delta S$
$\Delta G = -6010 - (250)(-22.0)$
$\Delta G = -6010 + 5500 = -510\,\mathrm{J/mol}$ (negative → freezing is spontaneous below $0^{\circ}C)$

Result ·

\Delta G = -6010 - 250\times (-22.0) = -6010 + 5500 = -510\,\mathrm{J/mol}

→ spontaneous

2

Same process at 300 K

Given ·

ΔH:: -6010
T:: 300
ΔS:: -22

Find ·

\Delta G

Steps

$\Delta G = -6010 - (300)(-22.0)$
$\Delta G = -6010 + 6600 = +590\,\mathrm{J/mol}$
Positive $\Delta G \to ice$ melts spontaneously instead at 300 K

Result ·

\Delta G = -6010 - 300\times (-22.0) = -6010 + 6600 = +590\,\mathrm{J/mol} \to non

-spontaneous

Helmholtz Free Energy→Entropy Change→First Law of Thermodynamics→