Bell–Evans–Polanyi principle

The Bell–Evans–Polanyi model was developed independently by Ronald Percy Bell[1] and by Meredith Gwynne Evans and Michael Polanyi[2] 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.[3]

Consider the reaction

${\displaystyle AB+C\to A+BC.}$

The system is assumed to have two degrees of freedom: r_AB, the distance between atoms A and B, and r_BC, the distance between atoms B and C. The distance between A and C is assumed to be fixed such that

${\displaystyle r=r_{\text{AB}}=const-r_{\text{BC}}}$

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 m₁ and m₂ respectively) that intersect at the transition state.

For the AB molecule, the energy is given as a function of bond distance r:

${\displaystyle E_{\text{AB}}(r)=m_{1}(r-r_{1}).}$

(1)

At the transition state, r = r^‡ and E = E_a. Therefore, we can write that

${\displaystyle E_{\text{a}}=m_{1}(r^{\ddagger }-r_{1}),}$

(2)

which rearranges to give

${\displaystyle r^{\ddagger }={\frac {E_{\text{a}}}{m_{1}}}+r_{1}.}$

(3)

For the BC molecule, a similar expression of energy as a function of r is given by

${\displaystyle E_{\text{BC}}(r)=m_{2}(r-r^{\ddagger })+E_{\text{a}}.}$

(4)

The overall enthalpy change ΔH of the system can thus be expressed as

${\displaystyle \Delta H=m_{2}(r_{2}-r^{\ddagger })+E_{\text{a}}.}$

(5)

Plugging equation (3) into equation (5) and rearranging gives the following:

${\displaystyle E_{\text{a}}={\frac {m_{1}}{m_{1}-m_{2}}}[\Delta H-m_{2}(r_{2}-r_{1})].}$

(6)

The constants in equation (6) can be condensed into the common form of the Evans–Polanyi equation given above.

• Free-energy relationship
• Brønsted catalysis equation

• Carey, Francis, A.; Sundberg, Richard, J. (2007). Advanced Organic Chemistry (Part A: Structure and Mechanisms) (5th ed.). New York: Springer. ISBN 978-0-387-44897-8. OCLC 154040953.
• Dill, Ken A.; Bromberg, Sarina (2011). Molecular Driving Forces: Statistical Thermodynamics in Biology, Chemistry, Physics, and Nanoscience (2nd ed.). London: Garland Science. ISBN 978-0-8153-4430-8. OCLC 660161826.
• Vinu, R.; Broadbelt, Linda J. (2012). "Unraveling Reaction Pathways and Specifying Reaction Kinetics for Complex Systems". Annual Review of Chemical and Biomolecular Engineering. 3 (1): 29–54. doi:10.1146/annurev-chembioeng-062011-081108. ISSN 1947-5438. PMID 22468596.