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THEORY OF THE ISOELECTRIC POINT
THEORY O F THE ISOELECTRIC POINT. I1 TERRELL L. HILL Department of Chemistry, University of California, Berkeley, California Received January 7, 104.8
In a previous paper (1) the’isoelectric point and some of its properties were considered. As pointed out there, however, the treatment waa exact only insofar aa it is permissible to use concentration equilibrium constants rather than thermodynamic constants. It is therefore of interest to inquire concerning the extent to which the concept of the isoelectric point retains its significance when thermodynamic constants are employed. In addition, several new topics will be discussed. The order of the first paper will be followed closely where possible, and analogous steps will consequently often be omitted. Equations referred to in the text and numbered from 1 to 49 will be found in the first reference. THE IBOILECTRIC POINT
The notation of the previous paper (1) is retained, although several new symbols will also be used. Thus we again take as our general system of interest an aqueous solution of p ampholytes, the solution being saturated with respect to g (0 q p ) . We employ here thermodynamic constants ampholytes 1, defined by the equations
<
1we make use of
aij = Kii
(j= 1,
(59)
q)
and equation 10, obtaining
or
and
(k = 1,
n i ; j = 1,
q)
(62)
The general expression for the isoelectric point then follows from equations 16, 61, 62, and 63:
m'-k
2
+ i-rtl cj
Ykj
k-ni m +"I
fi K:j i-1
IIK:j i-1
\
= 0 (64)
‘600
p
+
TERRELL L. HILL 9
(nj i-1
+ mi) unknowns (the u’s), and setting the determinant
of the
augmented matrix equal to zero, since the system is inconsistent if this determinant is not zero. A similar statement can be made about the analogous equation 17. If p = 1, equation 64 reduces to equation 52 regardless of whether p = 0 or p = 1. It was pointed out, after equation 17, that a true isoelectric point existed (using concentration equilibrium constants) only if one ampholyte were present in the solution or if the solution were saturated with respect to all ampholytes. These same restrictions evidently apply to equation 64, but in addition we have the activity coefficients of the ampholyte ions to consider. An isoelectric point has no real significance if it cannot be defined by fixed constants, hence the question arises as to the conditions under which I’ does not depend on the 7’s. Let us first investigate the simple case of but one ampholyte and assume that y k = Y - k , this being true in sufficiently dilute solutions and an excellent approximation otherwise, owing to the structural similarity of the ions of a single ampholyte. We have, then,
and we wish to know what relations must exist in order that
+
<
