Science Wiki

Gibbs free energy (also referred to as Gibbs energy, the Gibbs function, or free enthalpy, denoted by the symbol G) is a thermodynamic potential that measures the maximum amount of reversible non-expansion work that can be extracted from a closed thermodynamic system at constant temperature and pressure. Formulated in the 1870s by American physicist and mathematician Josiah Willard Gibbs, this state function integrates the first law of thermodynamics (conservation of internal energy and enthalpy) with the second law of thermodynamics (the universal tendency toward increased total entropy) into a single Master equation defined exclusively in terms of system variables:

G = H - TS = U + PV - TS

where H is enthalpy, T is absolute temperature (in Kelvin), S is entropy, U is internal energy, P is pressure, and V is volume. Across physical chemistry, chemical engineering, geochemistry, and bioenergetics, the change in Gibbs free energy (ΔG) serves as the definitive thermodynamic criterion for predicting whether a physical phase transition or chemical reaction can proceed spontaneously under isothermal and isobaric conditions, and determines the exact composition of a system at chemical equilibrium.

Clausius Inequality and System State Variables

According to the Second Law of Thermodynamics, any spontaneous process occurring within an isolated universe must result in a net increase in total universal entropy (dS_univ ≥ 0). When analyzing a laboratory reaction vessel or a living cell, the universe is partitioned into the system of interest (sys) and its surrounding thermal and barometric reservoir (surr):

dS_univ = dS_sys + dS_surr ≥ 0

If the system is held at constant temperature (dT = 0) and constant pressure (dP = 0), and interacts with its surroundings solely through the transfer of heat (q) and mechanical pressure-volume expansion work (w_PV = -_P__extdV), the heat absorbed by the system equals its change in enthalpy (dq_P = dH_sys). Because the heat gained by the system is lost reversibly by the isothermal surroundings (dq_surr = -_dq__sys = -_dH__sys), the entropy change of the surroundings is given by:

dS_surr = dq_surr / T = -_dH__sys / T

Substituting this relation into the universal entropy inequality yields the Clausius inequality expressed entirely in state variables of the system:

dS_sys - (dH_sys / T) ≥ 0

Multiplying both sides of the inequality by -_T_ (which reverses the direction of the inequality sign because absolute temperature T > 0) defines the differential of the Gibbs free energy (dG = dH - TdS at constant T):

dG_{T,_P_} = dH_sys - TdS_sys = -_TdS__univ ≤ 0

For a macroscopic transformation occurring at constant temperature and pressure, integrating this expression yields the finite Gibbs free energy equation:

ΔG = ΔH - TΔS

Exergonic, Endergonic, and Equilibrium Regimes

Because ΔG = -_TΔS__univ at constant T and P, the algebraic sign of ΔG dictates the thermodynamic directionality of any chemical or physical process:

• ΔG < 0 (Exergonic): The transformation increases universal entropy (ΔS_univ > 0) and is thermodynamically spontaneous (favorable) in the forward direction, capable of performing useful work as it approaches equilibrium.
• ΔG > 0 (Endergonic): The transformation would decrease universal entropy if executed in the forward direction and is therefore thermodynamically non-spontaneous without the input of external work; the reverse reaction is spontaneous.
• ΔG = 0 (Equilibrium): The system has reached its minimum Gibbs free energy state (maximum universal entropy); no net macroscopic driving force exists in either direction.

The interplay between the enthalpic term (ΔH, reflecting covalent bond energies and intermolecular forces) and the entropic term (-_TΔS_, reflecting the statistical dispersal of energy across molecular microstates) establishes four distinct thermodynamic regimes:

Double Legendre Transformation of Internal Energy

In classical chemical thermodynamics, the fundamental equation of internal energy for an open, multicomponent system is a function of extensive natural variables: _U_(S, V, {_N__i}), where N_i represents the molar amount of chemical species i:

dU = TdS - PdV + Σ_μ__idN_i

Because entropy (S) and volume (V) are difficult to hold constant or control directly in experimental settings, Gibbs free energy is mathematically constructed via a double Legendre transformation of U that replaces the extensive variables S and V with their conjugate intensive variables, temperature (T = (_∂U_/_∂S_)_{V,_N_}) and pressure (-_P_ = (_∂U_/_∂V_)_{S,_N_}):

G = U - _T_(∂U / ∂S)_{V,_N_} - _V_(∂U / ∂V)_{S,_N_} = U - TS + PV

Taking the total differential of G = U + PV - TS yields:

dG = dU + PdV + VdP - TdS - SdT

Substituting the fundamental differential for dU into this expression cancels the TdS and PdV terms, producing the fundamental differential equation for Gibbs free energy in terms of its natural variables _G_(T, P, {_N__i}):

dG = -_SdT_ + VdP + Σ_μ__idN_i

Partial Derivatives and Maxwell Relations

Comparing the fundamental differential with the exact total differential of _G_(T, P, {_N__i}) identifies the first partial derivatives of the Gibbs potential:

• Temperature Dependence: (∂G / ∂T)_{P,{_N_i}} = -_S_ (Because absolute entropy S is always positive according to the third law of thermodynamics, G always decreases monotonically as temperature increases at constant pressure, with a steeper negative slope for gases than for liquids or solids).
• Pressure Dependence: (∂G / ∂P)_{T,{_N_i}} = V (Because molar volume V is positive, G always increases as pressure increases at constant temperature).
• Compositional Dependence (Chemical potential): μ_i = (∂G / ∂N_i)_{T,_P_,_N__j_≠_i_} (Thus, the chemical potential μ_i is precisely the partial molar Gibbs free energy, Ḡ_i, of component i).

Applying Clairaut's theorem on the equality of mixed second partial derivatives of state functions (∂²G / ∂T∂P = ∂²G / ∂P∂T) directly generates the corresponding Maxwell relation:

(∂S / ∂P)_{T,{_N_i}} = -(∂V / ∂T)_{P,{_N_i}} = -_Vα_

where α is the isobaric thermal expansion coefficient.

Euler's Theorem, the Gibbs-Duhem Equation, and Gibbs-Helmholtz Relation

Because G is a first-order homogeneous function with respect to the extensive molar amounts {_N__i} at fixed intensive T and P, Euler's homogeneous function theorem dictates that the total Gibbs free energy of any multicomponent mixture is the sum of the chemical potentials weighted by their molar amounts:

G = Σ_N__iμ_i

Differentiating this Euler relation (dG = Σ_N__idμ_i + Σ_μ__idN_i) and setting it equal to the fundamental differential dG yields the Gibbs-Duhem equation, which proves that the intensive properties (T, P, and {_μ__i}) of a thermodynamic phase cannot vary independently:

SdT - VdP + Σ_N__idμ_i = 0

To evaluate how _G_/_T_ varies with temperature at constant pressure, applying the quotient rule alongside G = H - TS and (_∂G_/_∂T_)_P = -_S_ yields the Gibbs-Helmholtz equation:

(_∂_(G / T) / ∂T)_P = (_T_(∂G / ∂T)_P - G) / T² = (-_TS_ - (H - TS)) / T² = -_H_ / T²

Equivalently, expressed in terms of inverse temperature (1/_T_) for a chemical reaction (ΔG):

(_∂_(ΔG / T) / _∂_(1 / T))_P = ΔH

Thermodynamic Activity and the Reaction Isotherm

For a pure ideal gas at constant temperature T, integrating dG = VdP = (_nRT_/_P_)_dP_ from the standard pressure _P_° (1 bar = 100 kPa) to an arbitrary pressure P yields the molar chemical potential:

μ = _μ_° + RT · ln(P / _P_°)

For real gases, non-ideal liquid solutions, and electrolyte mixtures, intermolecular attractions and steric repulsions cause deviations from ideal behavior. To preserve the mathematical structure of the chemical potential across all phases, G. N. Lewis introduced the dimensionless thermodynamic activity (a_i), defined for a dissolved solute i as:

a_i = γ_i · ([C_i] / _C_°)

where [C_i] is the molarity or molality, _C_° is the standard reference concentration (1 M or 1 mol/kg), and γ_i is the dimensionless activity coefficient (which approaches 1 as concentration approaches infinite dilution, modeled in ionic solutions by Debye-Hückel theory). The chemical potential of any species i in any phase is then written rigorously as:

μ_i = μ_i° + RT · ln(a_i)

Consider a generalized reversible chemical reaction written with signed stoichiometric coefficients ν_i (where ν_i > 0 for products and ν_i < 0 for reactants):

0 ↔ Σ_ν__iA_i

As the reaction progresses by an infinitesimal extent of reaction dξ (defined such that dN_i = ν_idξ), the change in the system's Gibbs free energy at constant T and P is:

dG_{T,_P_} = (Σ_ν__iμ_i)_dξ_

The molar reaction Gibbs free energy (Δ_rG, commonly abbreviated ΔG) is therefore the partial derivative (slope) of the total Gibbs energy plotted against the extent of reaction ξ:

Δ_rG = (∂G / ∂ξ)_{T,_P_} = Σ_ν__iμ_i

Substituting μ_i = μ_i° + RT · ln(a_i) into this sum yields the reaction isotherm:

ΔG = Σ_ν__iμ_i° + RT · Σln(a_i^ν_i) = _ΔG_° + RT · ln(Π_a__i^ν_i)

Defining the mass-action product of activities at any arbitrary composition as the reaction quotient (Q = Π_a__i^ν_i) produces the central equation of chemical thermodynamics:

ΔG = _ΔG_° + RT · ln(Q)

where _ΔG_° = Σ_ν__iΔ_f_G_°_i is the standard Gibbs free energy change when all reactants and products are present at unit activity (a_i = 1, so Q = 1 and ln(Q) = 0).

Equilibrium Constant (K_eq) and the van 't Hoff Equation

When the reaction mixture reaches chemical equilibrium, the Gibbs free energy of the system reaches its minimum with respect to ξ, meaning the slope vanishes (Δ_rG = (_∂G_/_∂ξ_)_{T,_P_} = 0) and the reaction quotient assumes its time-invariant equilibrium value, the thermodynamic equilibrium constant (Q = K_eq):

0 = _ΔG_° + RT · ln(K_eq)

Rearranging this identity reveals that the standard free energy change _ΔG_° uniquely fixes the equilibrium constant at a given temperature:

_ΔG_° = -_RT_ · ln(K_eq) ↔ K_eq = e^{-_ΔG_° / (RT)}

Substituting _ΔG_° = -_RT_ · ln(K_eq) back into the reaction isotherm expresses the instantaneous thermodynamic driving force directly as the displacement of the current reaction quotient Q from the equilibrium constant K_eq:

ΔG = RT · ln(Q / K_eq)

Thus, when Q < K_eq, ln(_Q_/_K__eq) < 0 and the forward reaction proceeds spontaneously (ΔG < 0); when Q > K_eq, ΔG > 0 and the reverse reaction proceeds spontaneously. Furthermore, combining _ΔG_° / T = -_R_ · ln(K_eq) with the Gibbs-Helmholtz equation yields the van 't Hoff equation, which governs how the equilibrium constant shifts with temperature:

_d_(ln K_eq) / dT = _ΔH_° / (RT²)

Maximum Non-Expansion Work and the Nernst Equation

To prove that ΔG represents the maximum useful non-expansion work extractable from a process, consider the First Law of Thermodynamics where total work (dw) is separated into mechanical pressure-volume work (-_PdV_) and useful non-expansion work (dw_{non-_PV_}, such as electrical work in a galvanic battery or mechanical contraction in myosin motor proteins):

dU = dq - PdV + dw_{non-_PV_}

By the Second Law inequality (dq ≤ TdS), substituting dU into the differential dG = dU + PdV + VdP - TdS - SdT at constant T (dT = 0) and constant P (dP = 0) yields:

dG_{T,_P_} = (dq - PdV + dw_{non-_PV_}) + PdV - TdS = (dq - TdS) + dw_{non-_PV_} ≤ dw_{non-_PV_}

When the transformation is executed completely reversibly (such that dq_rev = TdS and no energy is dissipated as irreversible frictional heat), the equality holds:

ΔG = w_{max,non-_PV_}

In electrochemistry, moving n moles of electrons across a reversible electrical potential difference (the electromotive force or cell potential, E_cell) performs electrical work equal to w_elec = -_nFE__cell, where F is the Faraday constant (96,485.33 C/mol). Equating this electrical work to ΔG connects thermodynamics directly to electrochemistry:

ΔG = -_nFE__cell and _ΔG_° = -_nFE_°_cell

Substituting these identities into the van 't Hoff reaction isotherm (ΔG = _ΔG_° + RT · ln Q) and dividing through by -_nF_ derives the electrochemical Nernst equation:

E_cell = _E_°_cell - (RT / (nF)) · ln(Q)

Transformed Biochemical Conventions (ΔG°' at pH 7.0)

In physical chemistry, the thermodynamic standard state (denoted by the superscript °, as in _ΔG_°) mandates that all dissolved solutes, including hydrogen ions (H⁺), be present at an activity of 1 M, which corresponds to an strongly acidic pH of 0. Because living cells operate within buffered aqueous cytosol near neutral pH (where proteins would denature at pH 0), biochemistry and bioenergetics adopt a transformed biochemical standard state, denoted by a prime symbol (_ΔG_°' and K_eq):

• Hydrogen Ion Concentration: The activity of H⁺ is held constant at pH 7.0 ([H⁺] = 10^-7 M). For any reaction that releases m moles of H⁺ as a product, the biochemical standard free energy is related to the physical chemical standard state by _ΔG_°' = _ΔG_° + mRT · ln(10^-7) ≈ _ΔG_° - _m_(39.96 kJ/mol) at 298.15 K.
• Water Activity: Because liquid water (H₂O) serves as the ubiquitous solvent at a nearly invariant concentration of ~55.5 M, its activity is set to unity (a_H2O = 1) and incorporated directly into K_eq rather than entered as ~55.5 in the mass-action quotient.
• Magnesium and Ionic Strength: Because intracellular phosphorylated metabolites (such as adenosine triphosphate, ATP, and adenosine diphosphate, ADP) exist as equilibrium mixtures of protonated species and magnesium coordination complexes (primarily MgATP^2- and MgADP^-), biochemical conventions define the total analytical concentration of all ionization and metal-chelated states of a compound as a single pseudo-isomer group, evaluated at a specified free [Mg²⁺] (typically 1 mM) and physiological ionic strength (I = 0.10 to 0.25 M).

ATP Hydrolysis and Intracellular Phosphorylation Potential (ΔG_p)

Living organisms maintain a low-entropy, non-equilibrium steady state by coupling endergonic anabolic reactions (ΔG > 0, such as DNA synthesis, ribosomal translation, and active transport) to strongly exergonic catabolic pathways via a shared chemical intermediate, primarily adenosine triphosphate (ATP). Because Gibbs free energy is an additive state function (per Hess's law), an endergonic reaction (A → B, ΔG₁ > 0) becomes thermodynamically favorable when enzymatically coupled via a shared phosphorylated intermediate to a reaction with a larger negative free energy (C → D, ΔG₂ < 0) such that the net sum satisfies ΔG_net = ΔG₁ + ΔG₂ < 0.

The terminal phosphoanhydride bond hydrolysis of ATP is written in biochemical convention as:

ATP + H₂O ↔ ADP + P_i

At 298.15 K (25 °C) and pH 7.0, the biochemical standard free energy of ATP hydrolysis is _ΔG_°' = -30.5 kJ/mol (-7.3 kcal/mol). This negative _ΔG_°' arises from three physical-organic factors that stabilize the products (ADP^3- and inorganic orthophosphate, HPO₄^2-) relative to the reactant (ATP^4-):

1. Electrostatic Charge Relief: At pH 7.0, the polyphosphate tail of ATP carries roughly four closely spaced negative charges that strongly repel one another; hydrolytic cleavage separates these like charges, lowering electrostatic potential energy.
2. Resonance Stabilization: In the intact phosphoanhydride linkage (P-O-P), the bridging oxygen atom's lone-pair electrons cannot simultaneously delocalize into both adjacent phosphorus atoms without competing. Once cleaved, the liberated orthophosphate ion (HPO₄^2-) distributes its negative charge symmetrically across multiple equivalent resonance structures of lower electronic energy.
3. Solvation and Ionization Entropy: The separated ADP^3- and HPO₄^2- ions form stronger hydrogen-bonded hydration shells with surrounding water molecules than intact ATP^4-, and the immediate deprotonation of the released H₂PO₄^- at pH 7.0 pulls the reaction forward.

Crucially, cellular bioenergetics is governed by the actual intracellular free energy (the phosphorylation potential, ΔG_p), not _ΔG_°':

ΔG_p = _ΔG_°' + RT · ln( ([ADP][P_i]) / [ATP] )

Because mitochondria and chloroplasts continuously drive oxidative phosphorylation and photophosphorylation to hold the mass-action ratio ([ADP][P_i])/[ATP] several orders of magnitude below its equilibrium value (maintaining cytosolic [ATP] ≈ 2 to 8 mM, [ADP] ≈ 0.01 to 0.2 mM, and [P_i] ≈ 1 to 5 mM), the actual ΔG_p inside a living cell ranges from -50 to -65 kJ/mol. If a cell's mass-action ratio ever relaxes to K_eq (such that ΔG_p = 0), all metabolic driving force ceases and the cell is dead.

Eyring Transition State Theory and ΔG^‡

A negative Gibbs free energy change (ΔG < 0) establishes that a reaction is thermodynamically favorable, but provides zero information regarding the kinetic rate at which the reaction occurs. For example, while the hydrolysis of ATP in aqueous solution is strongly exergonic (_ΔG_°' = -30.5 kJ/mol), ATP is kinetically stable (metastable) in water because the negatively charged polyphosphate tail electrostatically repels attacking nucleophilic water molecules.

In Eyring transition state theory, the rate constant (k) of an elementary reaction step is governed not by the net reaction free energy ΔG, but by the Gibbs free energy of activation (ΔG^‡ = ΔH^‡ - TΔS^‡), which represents the free energy barrier between the ground-state reactants and the high-energy transition state complex:

k = κ · (k_BT / h) · e^{-_ΔG_‡ / (RT)}

where κ is the transmission coefficient, k_B is the Boltzmann constant, and h is the Planck constant. Biological enzymes and abiotic catalysts accelerate chemical reactions by orders of magnitude solely by stabilizing the transition state and lowering ΔG^‡; because G is a path-independent state function, a catalyst cannot alter ΔG, _ΔG_°', or the equilibrium constant K_eq.

What is the difference between ΔG, ΔG°, and ΔG°' in biochemistry?

_ΔG_° is the physical-chemistry standard free-energy change when all reactants and products—including H⁺—are at unit activity (1 M, or pH 0). _ΔG_°' is the transformed biochemical standard free-energy change defined at physiological pH 7.0 ([H⁺] = 10⁻⁷ M) and 1 mM Mg²⁺. Both _ΔG_° and _ΔG_°' are constants for a given reaction at fixed temperature, whereas ΔG is the instantaneous free-energy change at actual non-standard cellular concentrations (ΔG = _ΔG_°' + RT ln Q).

How can a glycolytic reaction with a positive standard free energy (ΔG°' > 0) proceed forward in living cells?

In glycolysis, aldolase cleaves fructose-1,6-bisphosphate into dihydroxyacetone phosphate (DHAP) and glyceraldehyde-3-phosphate (GAP) with a strongly endergonic standard free energy of _ΔG_°' = +23.8 kJ/mol. However, downstream enzymes rapidly consume GAP and DHAP, keeping the intracellular reaction quotient Q = [DHAP][GAP]/[F-1,6-BP] around ~10⁻⁴ M; consequently, RT ln Q is strongly negative (~−25 kJ/mol), rendering the actual intracellular ΔG slightly negative (−1.2 kJ/mol).

Why does Gibbs free energy determine spontaneity only at constant temperature and pressure?

The equivalence dG = −_TdS__univ ≤ 0 holds strictly because at constant pressure (dP = 0), heat exchanged with the surroundings equals the enthalpy change (dq_P = dH), and at constant temperature (dT = 0), _d_(TS) = TdS. Under constant volume and constant temperature (dV = 0, dT = 0) where no PV work occurs (dq_V = dU), spontaneity is governed instead by the Helmholtz free energy (A = U − TS, dA_T,V ≤ 0).