Essentials of Computational Chemistry

10.5 TECHNICAL CAVEATS 379
where Q is the reaction quotient (i.e., the ratio of concentrations that appear in the equilibrium
constant) evaluated with all species at their standard-state concentrations, expressed so that
the logarithm is dimensionless. As an example, consider the gas-phase condensation reaction
A + B −−−→ C (10.54)
where we will define the ‘o’ standard state to imply all species at 1 atm pressure and the ‘o ′ ’
standard state to imply all species in the gas phase at 1 mol L−1 .IfA,B,andCareideal
1
gases, their concentration at 1 atm may be derived from the ideal gas law as 24.5 mol L−1
at 298 K. Since the reaction quotient Q is [C]/[A][B], Eq. (10.53) becomes
G o′
= G o ⎛
1
⎞
⎜
+ RT ln
1 · 1 ⎟
⎝ ⎠
24.5 · 24.5
24.5
= G o − RT ln(24.5) (10.55)
Additional standard-state issues can arise in condensed phases, and these will be dealt with
in subsequent chapters.
10.5.5 Standard-state Free Energies, Equilibrium Constants,
and Concentrations
While our focus has been primarily on thermodynamic quantities, like free energy, it should
be borne in mind that the ultimate motivation for computing free energy differences is usually
to permit calculation of chemical concentrations in actual systems. To accomplish this for a
generic equilibrium is straightforward. For example, consider the following reaction (chosen
in a completely arbitrary fashion)
A + B + C ⇀↽ 2D + E (10.56)
From the relationship between the equilibrium constant and the free energies of the reactants
and the products we may write
[D] 2 [E]
[A][B][C] = e−Go /RT
(10.57)
where the standard-state symbol on the free energy change dictates the units used for the
concentrations of the species. Thus, if we were carrying out all free energy calculations
for gas-phase species at 1 atm pressure, we would express the reactant and product
concentrations in those units. Stoichiometry then permits Eq. (10.57) to be rewritten as
(2x) 2 x
(p0,A − x)(p0,B − x)(p0,C − x) = e−Go /RT
(10.58)