# What does “reaction coordinate” mean? What does it mean when a reaction has TWO reaction coordinates?

I'm familiar with the concept of a reaction coordinate from high school chemistry as some generic conformational parameter that all of the intermediate states of a chemical reaction lie upon. Recently, though, I've seen some presentations where chemists presented plots of the free energy with respect to two reaction coordinates, and my understanding is that the plot was of a measurement (I could be wrong, though).

What exactly does the reaction coordinate mean? What does it mean to have more than one reaction coordinate for the same reaction? Is the free energy as a function of the reaction coordinate a measurable quantity?

• I suggest to you the book "Computational Chemistry", especially the second chapter, in which you will find an interesting discussion on the potential energy surface, which is the concept underlying plots such as the ones you're referring to. – CHM May 19 '12 at 23:14
• @CHM: I'm familiar with the potential energy surface, but I was curious as to how one could generate a potential energy surface with respect to two reaction coordinates when the precise definition of those coordinates is unknown. – Dan May 23 '12 at 20:09
• Well, you don't seem to be too familiar with the concept of PES. What you're seeing in Ashu's graph is a slice of a potential energy surface, so that only two dimensions are present. The "reaction coordinates" really are the geometry of atoms in the system, when they follow this particular path. Also, that book will probably be in your university's library, you really should read it. – CHM May 24 '12 at 1:40
• @CHM: I have read it. I understand that a "normal" reaction coordinate plot is a slice of the potential energy surface. What I don't understand is how one could measure or calculate the PES as a function of two reaction coordinates without knowing what either reaction coordinate represents in terms of the spatial configuration of the molecules. I was under the impression that a given reaction could only have one reaction coordinate. – Dan May 24 '12 at 1:59
• Hi Dan, I wonder if you might help us launch Materials Modeling Stack Exchange, it would be great to have you as one of the users in the private beta! – user1271772 Feb 10 at 1:19

The reaction coordinate is the progress of a reaction from reactants to products with various intermediates and transition states in between. It is an abstraction. It has no relation to time. Rather it is the progress of bond-forming and bond-breaking reaction steps. The free energy change of partially formed and partially broken bonds cannot be measured. The free energies of transition states (local maxima) cannot be directly measured. These states are transient and cannot be directly observed. For a simple reaction with few mechanistic steps, the activation energy (transition state energy) can be calculated from kinetics measurements. The free energy of intermediates can often be determined directly or indirectly if they are long-lived. Most of these diagrams are assembled from a few measured data points and either extrapolation or theoretical computation.

If two reaction coordinate axes are presented, this often means that there are two separate pathways the reaction can take, and each pathway is presented on one of the two reaction coordinate axes. Perhaps the reaction involves two different steps, but those steps need not happen in the same order every time. As a generic example, consider nucleophilic substitution reactions in organic chemistry: A generic nucleophile ($\ce{Nu-}$) reacts with an organic molecule R-X, where X is a leaving group, and R is the rest of the molecule (which is unchanged or nearly unchanged). The overall reaction would be

$$\ce{Nu- + R-X -> Nu-R + X-}$$

This process involves two steps: forming the bond between Nu and R, and breaking the bond between R and X. These two steps can happen in three different orders: 1) the R-X bond breaks first and the R-Nu bond forms second (this is the dissociative or $\mathrm{S_{N}1}$ mechanism); 2) The R-Nu bond forms first and the R-X bond breaks second (this is the associative or $\mathrm{S_{N}Ac}$ mechanism); and 3) The R-X bond breaks and the R-Nu bond forms at the same time (this is the intermediate or $\mathrm{S_{N}2}$ mechanism). All three of these cases occur in reactions that are covered in any undergraduate organic chemistry text. These three processes would look like the following. The items in parentheses are transition states with partially formed/broken bonds ($\ce{\bond{....}}$).

Dissociative:

• Step 1: $\ce{Nu- + R-X -> (Nu- + R\bond{....}X) -> Nu- + R+ + X-}$
• Step 2: $\ce{Nu- + R+ + X- -> (Nu\bond{....}R + X-) -> Nu-R + X-}$

Associative:

• Step 1: $\ce{Nu- + R-X -> (Nu\bond{....}R-X) -> [Nu-R-X]-}$
• Step 2: $\ce{[Nu-R-X]- -> (Nu-R\bond{....}X) -> Nu-R + X-}$

Intermediate:

• Only step: $\ce{Nu- + R-X -> (Nu\bond{...}R\bond{....}X) -> Nu-R + X-}$

A reaction coordinate energy diagram for a substitution process could then be graphed over two different reaction coordinates: breaking the R-X bond and forming the R-Nu bond. For ease of two dimensional representation, these are often presented as contour diagrams. Various trajectories could be taken through this space based on what real species R, Nu, and X represent. For example, if R+ is a stabilized cation, then the dissociative pathway becomes more likely. These plots are called More O'Ferrall-Jencks plots. This link will take you to the wikipedia article, which is not a very good article. The references in that article to the papers by R. A. More O'Ferrall and W. P. Jencks are probably better places to go to find out more. Most advanced undergraduate or graduate level textbooks of physical organic chemistry, particularly Advanced Organic Chemistry, Part A: Structure and Mechanisms by Carey and Sundberg, will cover this topic, at least as it relates to organic chemistry.