A 2 The Overall Equilibrium Approach

We begin by considering an overall equilibrium of the type

If we equate chemical activities with concentrations, the equilibrium in eq. (6.3) can be described with an overall concentration equilibrium constant:

Equation (6.3) is identical to the classic representation of the overall equilibrium of a metal ion (M) with a ligand X. However, the representation is really more general. It can also be used to represent a macromolecule in equilibrium with bound anions or cations or a surfactant micelle in equilibrium with a solute used as a probe.

What eqs. (6.3) and (6.4) really represent are a series of linked or coupled equilibria. Suppose that n = 4 in eq. (6.4). Then we have the individual equilibria:

and the overall equilibrium constants is given by

If the driving force for formation of MX,, (or in general MX„) is large, the intermediate equilibria in converting M to MXA may often be neglected. In such cases, the overall equilibrium can be used to construct models for the nonlinear regression analysis. For the n = 4 case,

The overall equilibrium model is applicable for many of the systems described in this chapter. Therefore, we now derive the expressions for the relevant fractions (/¡) of components for the overall equilibrium in eq. (6.3). We begin by expressing the total concentration of M (CM) in the system as

and the total concentration of X (Cx) as

The models to be developed require the fractions of bound and free M and X. In general, the fraction of a component j will be given by fraction of j = (conc. free or bound /)/(total concentration of /). (6.13)

The fraction of free X (fx,o) is expressed as

Substituting eq. (6.15) into (6.14) and canceling \X\ from numerator and denominator gives fx'° =\+n K'\M][X\^ ■ (616)

The fraction of bound X (fx.i) can be obtained by realizing that the sum of the fractions of all the forms of X must equal 1. In this case,

Alternatively, we can begin with the expression

and use the substitution for [MX„] in eq. (6.15) to yield the required expression for fx,\ in terms of K".

Analogous procedures to those just discussed are used to obtain the fraction of free M (fM,o) and the fraction of M with X bound to it (fM.\ )• A list of these expressions for the overall equilibrium in eq. (6.3) is given in Table 6.1.

A number of other situations can be envisioned in which the linkage approach can be used to develop models. Suppose that two types of ions bind to a single macromolecule M but that X must bind before Y binds. The relevant overall equilibria are given in eqs. (6.20) and (6.21):

M + nX ^ MX,, K" = \MX„]/{\M)\X]"} (6.20)

Table 6.1 Fractions of Bound and Free M and X Derived Using the Overall Equilibrium Model in Eq. (6.3)

Fraction

Expression

fx.o

1

1 + nK"{M}{X\nl

fx.l

nK"\M\[X\"~{

1 + nK"[M}[X\nl

/«.()

1

1 + K"[Xf

flU.1

K"[X\n

1 + K"[X]"

In this case, the measured property of the macromolecular solution might depend on the fractions of all the forms M, MXn, and MXnY„. The procedures outlined in eqs. (6.11) to (6.18) again can be used to obtain the relevant fractions of each species. Examples similar to this one will be described in applications later in this chapter.

A.3. Successive or Cooperative Equilibria

The general alternative to the overall equilibrium approach is to consider all the individual equilibria involved. In the case of the system of equilibria described by eqs. (6.5) to (6.8) we would need to consider the fractions of all of the intermediate species from M to MXA, as well as each of the equilibrium constants Ku K2, K3, and K4.

In general, such individual equilibrium steps can be described by

The total concentration of X is [2]

The total concentration of M is

Procedures identical to those in Section A.2 can now be used to obtain the relevant fractions, except that the definitions of total concentrations

Table 6.2 Fractions of Bound and Free M and X Derived Using the Successive Equilibrium Model in Eq. (6.22)

Fraction Expression

Table 6.2 Fractions of Bound and Free M and X Derived Using the Successive Equilibrium Model in Eq. (6.22)

Fraction Expression

fx,o

1

[MJ^/K.W

i-0

fx. 1

[M^iK.iX]'

1=0

i-0

/m, 0

1

Ê w

i-0

Îm,\

¿=0

1-0

are those in eqs. (6.23) and (6.24). The fractions for this general case are given in Table 6.2. Examples of more complex equilibria have been discussed in detail by Wyman [2] and Kumosinski [1],

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