Page 44 - Thermodynamics of Biochemical Reactions

38 Chapter 3 Chemical Equilibrium in Aqueous Solutions

Substituting equation 3.1-10 in 3.1-6 yields
N, N* N I
vipf = -RT viln(ci)eq = -RTln n (ci)ik (3.1-11)
i= 1 i=l i= 1
Using the nomenclature of equation 3.1-5, equation 3.1-11 can be written as

NS
A,Go = viAfGY = -RTln K (3.1 - 12)
i=O
where A,G; is the standard reaction Gibbs energy and K is the equilibrium
constant for a chemical reaction written in terms of species:

(3.1- 13)
,= 1
An equilibrium constant must always be accompanied by a chemical equation.
This equation is often used without the subscript "eq" that reminds us that the
concentrations are equilibrium values. Strictly speaking, this equation should be
written as K = lI(c,/cO)~;, but the standard concentration c0 = 1 M will be
omitted, as mentioned before equation 3.1-8. Thus the equilibrium constant will
be treated as a dimensionless quantity, as, of course, it must be if we are going to
take its logarithm.
When H,O is a reactant in a chemical reaction in dilute aqueous solutions,
its molar concentration is not included in equation 3.1-13. The reason is that in
reactions in dilute aqueous solutions the activity of water does not change
significantly. The convention is that H,O is represented in the expression for the
equilibrium constant by its activity, which is essentially unity independent of the
extent of reaction. However, Af GO(H,O) is included in the calculation of A,Go
using equation 3.1-12 and AfHO(H,O) is included in the calculation of A,Ho using
equation 3.2-13, which is given later.
To clarify the nature of the equilibrium state of a reaction system, consider
the solution reaction A = B. When one liter of ideal solution initially containing
A at 1 M is considered, the Gibbs energy of the reactants at any time is given by
0
G = nA(pu, + RTlnCA]) + n&; + RTln[B]) (3.1 - 14)
Since nA = 1 - 4 and nB = 4,

G = (1 - [)pi + (pi + RT[(1 - t)ln(l - (1 - 5)) + 4In41 (3.1-15)
At the equilibrium state of the system, the Gibbs energy is at a minimum, and the
equilibrium extent of reaction is teq. At (dG/d(), = 0,
pi - pi = -RT1n(teq/(l - teq)) = -RTlnK (3.1- 1 6)
Figure 3.1 shows a plot of the Gibbs energy G of a reaction system A = B as a
function of the extent of reaction < when pi = 20 kJ mol - and pi = 18 kJ mol

3.2 CHANGES IN THERMODYNAMIC PROPERTIES IN
CHEMICAL REACTIONS

In treating the fundamental equations of thermodynamics, chemical poten-
tials of species are always used, but in making calculations when T and P are
independent variables, chemical potentials are replaced by Gibbs energies of
formation AfGi. Therefore, we will use equation 3.1-1 0 in the form

AfGi = A,Gp + RTlnc, (3.2-1)
where Af G, is the Gibbs energy of formation of species i at concentration c, from
its elements, each in its reference state. The standard Gibbs energy of formation