Essentials of Computational Chemistry

378 10 THERMODYNAMIC PROPERTIES
where {A} emphasizes computation over the population of all conformers of A (where this
set can include structures differing only by atom labeling, as detailed further in Appendix
B). Free-energy changes, then, between two species each of which exist as populations over
multiple conformers, must be computed as the difference between their averages. Note that
the formalism of Eq. (10.50) may also be applied to determine averaged transition state free
energies provided multiple transition state structures exist all of which lead to the same
product; the difference between an averaged reactant free energy and an averaged transition
state free energy defines a free energy of activation.
In fortunate instances, one conformer in a population has a free energy that is much lower
than that of any of the other possibilities. Inspection of Eq. (10.50) makes clear that in that
instance, only the low-energy term contributes significantly to the sum, in which case that
free energy may be taken as the population free energy.
10.5.4 Standard-state Conversions
Two issues associated with thermodynamic standard states bear some further attention. The
first is associated with the enthalpy of ions. Ion heats of formation may be defined based on
the heats of ionization of neutral molecules (or electron attachments thereto). For example,
one might consider a reaction like
A −−−→ A +ž + e −
(10.51)
and define the heat of formation of the radical cation A +ž as the sum of the heat of formation
of A and the enthalpy change for Eq. (10.51). In that case, however, one needs to assign a
heat of formation to the free electron. The thermal electron convention takes the free electron
as the ‘electron standard state’, i.e., its enthalpy of formation is always zero. The so-called
ion convention, on the other hand, takes the electron at rest to be the standard state (this is
the usual theoretical convention as well, recall), and predominates in the mass spectrometric
literature. The conversion between the two is straightforward, namely
H o
f,T (Xq ) = H o′
f,T (Xq ) + 5
qRT (10.52)
where superscript ‘o’ represents the thermal electron standard state, superscript ‘o ′ ’represents
the ion convention standard state, and q is the signed charge on X.
A separate issue arises in the discussion of standard-state free energies. Recall that the
entropy of translation requires a concentration specification to be included as part of the
standard-state conditions. Different tabulations of data often adopt different concentration
conventions, and it is very important that care be taken to ensure consistent comparisons.
To convert from one convention to another, we write
G o′
= G o
o Q
+ RT ln
′
(10.53)
2
Q o