Linear free-energy relationships (LFERs) exist when the relative influence of changing substituents on one reactant is similar to the effect on another reactant, and include linear Hammett plots, Swain–Scott plots, and Brønsted plots. LFERs are not always found to hold, and to see when one can expect them to, we examine the relationship between the free-energy differences for the two reactions under comparison. The extent to which the free energy of the new reaction is changed, via a change in substituent, is proportional to the extent to which the reference reaction was changed by the same substitution. A ratio of the free-energy differences is the reaction quotient or constant .
The above equation may be rewritten as the difference () in free-energy changes ():
Substituting the Gibbs free-energy equation () into the equation above yields a form that makes clear the requirements for LFERs to hold.
One should expect LFERs to hold if one of three conditions are met:
#s are coincidentally the same for both the new reaction under study and the reference reaction, and the s are linearly proportional for the two reactions being compared.
#s are coincidentally the same for both the new reaction under study and the reference reaction, and the s are linearly proportional for the two reactions being compared.
#s and s are linearly related to each other for both the reference reaction and the new reaction.
The third condition describes the enthalpy–entropy effect and is the condition most commonly met.