In physical organic chemistry, a free-energy relationship or Gibbs energy relation relates the logarithm of a reaction rate constant or equilibrium constant for one series of chemical reactions with the logarithm of the rate or equilibrium constant for a related series of reactions.^[1] Free energy relationships establish the extent at which bond formation and breakage happen in the transition state of a reaction, and in combination with kinetic isotope experiments a reaction mechanism can be determined. Free energy relationships are often used to calculate equilibrium constants since they are experimentally difficult to determine.^[2]

The most common form of free-energy relationships are linear free-energy relationships (LFER). The Brønsted catalysis equation describes the relationship between the ionization constant of a series of catalysts and the reaction rate constant for a reaction on which the catalyst operates. The Hammett equation predicts the equilibrium constant or reaction rate of a reaction from a substituent constant and a reaction type constant. The Edwards equation relates the nucleophilic power to polarisability and basicity. The Marcus equation is an example of a quadratic free-energy relationship (QFER).^{[citation needed]}

IUPAC has suggested that this name should be replaced by linear Gibbs energy relation, but at present there is little sign of acceptance of this change.^[1] The area of physical organic chemistry which deals with such relations is commonly referred to as 'linear free-energy relationships'.

A typical LFER relation for predicting the equilibrium concentration of a compound or solute in the vapor phase to a condensed (or solvent) phase can be defined as follows (following M.H. Abraham and co-workers):^[3][4]

${\displaystyle \log \mathrm {SP} =c+e\mathrm {E} +s\mathrm {S} +a\mathrm {A} +b\mathrm {B} +l\mathrm {L} }$

where SP is some free-energy related property, such as an adsorption or absorption constant, log K, anesthetic potency, etc. The lowercase letters (e, s, a, b, l) are system constants describing the contribution of the aerosol phase to the sorption process.^[5] The capital letters (E, S, A, B, L) are solute descriptors representing the complementary properties of the compounds. Specifically,

• L is the gas–liquid partition constant on n-hexadecane at 298 K;
• E = the excess molar refraction (E = 0 for n-alkanes).
• S = the ability of a solute to stabilize a neighbouring dipole by virtue of its capacity for orientation and induction interactions;
• A = the solute's effective hydrogen bond acidity; and
• B = the solute's effective hydrogen-bond basicity.

The complementary system constants are identified as

• l = the contribution from cavity formation and dispersion interactions;
• e = the contribution from interactions with solute n-electrons and pi electrons;
• s = the contribution from dipole-type interactions;
• a = the contribution from hydrogen-bond basicity (because a basic sorbent will interact with an acidic solute); and
• b = the contribution from hydrogen-bond acidity to the transfer of the solute from air to the aerosol phase.

Similarly, the correlation of solvent–solvent partition coefficients as log SP, is given by

${\displaystyle \log \mathrm {SP} =c+e\mathrm {E} +s\mathrm {S} +a\mathrm {A} +b\mathrm {B} +v\mathrm {V} }$

where V is McGowan's characteristic molecular volume in cubic centimeters per mole divided by 100.

• Brønsted catalysis equation
• Hammett equation
• Taft equation
• Swain–Lupton equation
• Grunwald–Winstein equation
• Yukawa–Tsuno equation
• Edwards equation
• Marcus equation
• Bell–Evans–Polanyi principle
• Quantitative structure–activity relationship

• IUPAC, Compendium of Chemical Terminology, 2nd ed. (the "Gold Book") (1997). Online corrected version: (2006–) "linear free-energy relation". doi:10.1351/goldbook.L03551