Statistical Physics, Second Revised and Enlarged Edition

Mixed systems and chemical reactions 151

AB

Configuration parameter,η

Energy

01

fi i

A+BA+B

ΔV

ΔW

Fig. 13. 1 Schematic graph of the energetics of a chemical reaction. The valley atη=0represents separate
Aand B atoms. The valley atη=1represents the AB molecule.

theform ofthefree energyplottedagainst a ‘configuration parameter’,ηsay.We
have definedη= 0 to correspond to the system when it consists of A + B, andη= 1
when the systemis AB. We note that theABvalley atη= 1 islower than theA+B
valleyatη=0, thedifferenceinheightbeingeffectivelythe energyWdiscussed
above. However for the reaction to occur, the pair ofA+ B atoms must somehow deal
withthebarrier ofheightVwhichexistsinthe transition process atintermediate
values ofη.
There is a very pretty analogy here. Consider Fig. 13.1 as a description of a mass
ofwaterinthe Earth’sgravitationalfield,i.e.it canbethoughtofas agraphof
gravitational potential energy(essentiallyheight above sea level) against distance.
The lowest energy state is in the deep valley atη=1. The gravitational potential
energyofthe water whichstartsinthehighvalleyatη= 0 ishigher, so that energy
Wis available if the transition can be made (ask anyhydroelectric engineer). But in
order to effect this transition, one has to do one of three things: (i) tunnel through the
hill,afrequent ployforhydroelectric power, or (ii)if itexists,findsome other way
around which is not shown in the section shown in the figure (perhaps theyare the
same valley curved around in another dimension?), or (iii) pump or carry the water
over thehill, requiringatbest theloan ofan amount ofenergyV.
The same thinking helps us with the chemical reaction. What can make the reaction
go? There are three possible routes. Route (i)involves tunnelling throughthebarrier.
Although foramicroscopicsystem thisisinprinciple possiblebyquantum tunnelling
through the barrier, for most ordinary chemical reactions this has such a low proba-
bility thatitis entirely negligible. Route (ii)involves evading thebarrier, andthisis
awayofcheatingcommonlyusedinchemistry.Itistheprincipleofusinga catalyst