The Bell–Evans–Polanyi model was developed independently by Ronald Percy Bell and by Meredith Gwynne Evans and Michael Polanyi to explain the apparent linear relationship between activation energy and free energy in acid disassociation, as described in the Brønsted catalysis equation, which was the original linear free-energy relationship published in 1924.
Consider the reaction
The system is assumed to have two degrees of freedom: r, the distance between atoms A and B, and r, the distance between atoms B and C. The distance between A and C is assumed to be fixed such that
As the A—B bond stretches, the energy of the system increases up to the activation energy associated with the transition state, whereupon the bond breaks. The energy then decreases as the B—C bond is formed. Evans and Polanyi approximated the two energy functions between reactants, the transition state, and the products by two straight lines (with slopes m and m respectively) that intersect at the transition state.
For the AB molecule, the energy is given as a function of bond distance r:
At the transition state, r = r and E = E. Therefore, we can write that
which rearranges to give
For the BC molecule, a similar expression of energy as a function of r is given by
The overall enthalpy change ΔH of the system can thus be expressed as
Plugging equation  into equation  and rearranging gives the following:
The constants in equation  can be condensed into the common form of the Evans–Polanyi equation given above.