Essentials of Computational Chemistry

372 10 THERMODYNAMIC PROPERTIES
10.4.3 Isodesmic Equations
An alternative method for computing heats (or free energies) of formation involves
consideration of a balanced chemical equation, e.g.,
mA + nB −−−→ rC + sD (10.34)
whereA,B,C,andDaremoleculesandm, n, r, ands indicate the number of moles of
each in the balanced equation. The heat of reaction for a chemical transformation is defined
as the difference between the heats of formation of the products and those of the reactants
when these are defined relative to consistent standard states. For the reaction of Eq. (10.34),
we would have
H o
o
o
rxn,298 = [rHo f,298 (C) + sHo f,298 (D)] − [mHf,298 (A) + nHf,298 (B)] (10.35)
where we have arbitrarily selected 298 K as the temperature of interest. Note that the
standard-state symbol on the heat of reaction (as opposed to the heats of formation) does not
imply the use of elemental standard states to assign a zero of enthalpy. Because the reaction
is balanced, the standards used to define the zeroes for the heats of formation must cancel
out on the two sides of the equation. So it is equally valid to write
H o
rxn,298 = [rH298(C) + sH298(D)] − [mH298(A) + nH298(B)] (10.36)
where H298 is the quantity typically addressed theoretically, i.e., the enthalpy relative to all
nuclei and electrons infinitely separated and at rest.
Insofar as the r.h.s. of Eq. (10.35) must then be equal to the r.h.s. of Eq. (10.36), if the
experimental heats of formation for all but one of the species in Eq. (10.34) are known (say
B), we may rearrange our equality to determine this quantity as
H o
f,298 (B) =−1
n {[rH298(C) + sH298(D)] − [mH298(A) + nH298(B)]
− [rH o
f,298 (C) + sHo
f,298
o
(D)] + mHf,298 (A)} (10.37)
This technique at first seems rather cumbersome, since we must perforce compute H298 for
four different species in this example, but it has one great advantage over the apparently
simpler apriori calculation of a single heat of formation, and that is that the difficulty
in computing heats of atomization can be avoided. As noted above, computed heats of
atomization tend to be highly inaccurate unless heroic levels of theory are employed, because
the correlation energies for the electrons in the atoms and in the molecule are so enormously
different. However, assuming experimental data are available, we may select our balanced
chemical Eq. (10.34) in such a way that the various bonds on the left- and right-hand
sides are essentially identical. That being the case, we would expect bond-by-bond errors
in correlation energy to largely cancel in the computed heat of reaction (the top line on the