Essentials of Computational Chemistry

380 10 THERMODYNAMIC PROPERTIES
where x is the concentration of E (and half the concentration of D) in units of partial pressure
at equilibrium, and the initial partial pressures of reactants A, B, and C appear as constants
in the denominator on the l.h.s. of Eq. (10.58). Note that the sign of the free energy change
for Eq. (10.56) is only predictive of the side to which the equilibrium shifts when all species
are initially present at their unit standard-state concentrations. All other situations require
explicit evaluation of equations like Eq. (10.58) in order to determine the final concentrations
predicted at equilibrium.
One variation on this theme that should be borne in mind when analyzing actual chemical
situations is that certain species in the real system may be ‘buffered’. That is, their
concentrations may be held constant by external means. A good example of this occurs
in condensed phases, where solvent molecules may play explicit roles in chemical equilibria
but the concentration of the free solvent is so much larger than that for any other species that
it may be considered to be effectively constant. Modeling solvation phenomena in general is
covered in detail in the next two chapters, but it is instructive to consider here a particular
case as it relates to computing equilibria. Consider such a reaction as
3(AžS) ⇀↽ Bž2S + S (10.59)
That is, three monosolvates of A are in equilibrium with a disolvate of trimeric B (i.e.,
B = A3) and a liberated solvent molecule. A rather typical protocol for evaluating the ratio
of monomer to trimer in solution would be the following: (i) compute the gas-phase free
energies of A·S, B·2S, and S at the appropriate temperature and a partial pressure of 1 atm
(the default in most electronic structure programs), (ii) add to these gas-phase free energies
the appropriate solvation free energies (usually computed assuming no change in standardstate
concentration, as described in Chapters 11 and 12), and (iii) convert the free-energy
change on going from reactants to products to standard-state units of 1 M concentration
following the protocol of Eq. (10.55) because this is the more conventional standard state in
solution. Having accomplished this, we would then be able to write
(x/3)[S]0
(x0,AžS − x) 3 = e−Go′ /RT
(10.60)
where x is the moles of A monosolvate converted at equilibrium to x/3 moles of trimeric
B disolvate and [S]0 is the concentration of the solvent (determined from its density and
molecular weight). To cement this example with actual values, imagine the solvent to be
water ([S]0 = 55.56 M) and, for a particular choice of A and B, the free energy change
in solution (i.e., for the ‘o ′ ’ standard state) to be −3.0 kcalmol −1 . If we take the starting
concentration of A monosolvate (x0,AžS) to be 0.2 M, we determine from solving the cubic Eq.
(10.60) that at equilibrium x is 0.037 M, which is to say that there is about one molecule of
Bž2S for every 16 molecules of AžS. The failure of the reasonably large negative free-energy
change to lead to substantial trimerization seems paradoxical only if one forgets that that
negative number refers specifically to all species being at their standard-state concentrations
(1 M)–actual systems may be quite far from that reference point.