Gunnar Gran suggested a way of expressing the relationship between the measured poten- tial and the analyte concentration in terms of a function that could be treated linearly.
Gran’s technique applies to a standard additions experiment as well to any titration where a reactant or product is monitored potentiometrically. Consider here the case of a standard
k k additions experiment in which an ISE for the analyte is being used to monitor cell poten-
tials. (A pH titration is considered later in this chapter.) The cell potential is shown in Eq. (4.11).
Ecell=Eo+2.303RT
zxF log(ax) +Ej=Eo+Ej+2.303RT
zxF log(𝛾x) +2.303RT
zxF log(Cx) (4.11) whereEjis the junction potential,axthe activity of the analyte, x,Cxis the concentration of the analyte, andEo is the cell potential forax=1 and no junction potential.𝛾xandzx are the activity coefficient, and the charge, respectively, for the same ion. Collecting the constant terms together, Eq. (4.11) becomes
Ecell =Ecnst+ 2.303RT
zxF log(Cx) (4.12)
For an unknown concentration of analyte,Co, in an initial volume,Vo, to which a standard solution of concentration,Cs, in a volume,V, is added the new concentration of analyte becomes
Cx= CoVo+CsV
Vo+V (4.13)
Substituting into Eq. (4.12) gives Ecell=Ecnst+2.303RT
zxF log
(CoVo+CsV Vo+V
)
=Ecnst+ 1 m⋅log
(CoVo+CsV Vo+V
)
(4.14) where, ideally, m= zxF
2.303RT. Equation (4.14) can be rearranged to give the following equation:
(Vo+V)⋅10mEcell = (CoVo+CsV)⋅10mEcnst (4.15) or
(Vo+V)⋅10mEcell = (Co
CsVo+V )
⋅10mEcnst (4.16)
A plot of the left-hand side, (Vo+V)⋅10mE, versusVyields a straight line that intersects thex-axis at a negative value ofVfor which |CsV|=CoVo. To make this more concrete, Figure 4.3 shows a Gran plot for a standard additions experiment for fluoride ions using a fluoride ISE [16]. The volume at thex-intercept is marked asVe. Hence, the original analyte concentration can be calculated from
Co=||
||CsVe Vo ||
|| (4.17)
Gran’s method has two advantages over comparing the cell potential for a sample directly to a calibration curve. First of all, the sample matrix may be too complicated to mimic accurately with the standard solutions. The difficulty of matching the matrix of the standard solutions and the sample can lead to discrepancies in the junction potential and the activity coefficients. Both of these influence the constant term,Ecnst. This term may
k k Ve
Ve 0.0 0 0.1 0.2 0.3 0.4
1 2 3
V (ml) NaF (Vo + V)ã10
4 5 6
F 23 RT–Eã
FIGURE 4.3 Gran’s method for a standard additions experiment. Fluoride standard was added in 1 ml increments to an unknown sample and the function of (Vo+V)⋅10mEwas plotted versus the added volume,V, of standard NaF, where m= ZFF
2.303RT = −F
2.303RT. The magnitude of thex-intercept, Ve, repre- sents the volume of standard that would contain the same number of moles of fluoride as in the original sample volume,Vo. Source: Adapted with permission from Liberti and Mascini [16]. Copyright 1969, American Chemical Society.
not be the same for the samples and the standard solutions. The matrices of the spiked and sample solutions are likely to be more similar andEcnstis more likely to be constant for the two. The other advantage is that several measurements go into determining the concentration of the analyte with Gran’s method. Consequently, the uncertainty in the calculated concentration is smaller for this determination.
One of the drawbacks of the approach used in Figure 4.3 is the assumption that one knows the value ofmin the coefficient of the log-term in Eq. (4.14). The coefficient com- monly differs somewhat from the ideal value of (2.303RT)/(zxF). Of course, this term can be evaluated from the slope of a calibration curve, but can one assume that the slope remains the same in the sample matrix? A more rigorous approach is to use the standard additions data to solve for the coefficient, 1/m, as well as finding the original sample concentration, Co[15]. Equation (4.16) was proposed by Gran, because it was a linear relationship that could easily be fit graphically or by using a least squares regression method. Fortunately, personal computers can handle nonlinear least squares where bothmand the original sam- ple concentration are determined as fitted parameters. A common approach is to re-express Eq. (4.14) as follows:
EN =Ecnst+ 1 m⋅log
(CoVo+CsVN Vo+VN
)
(4.18)
whereENandVNare the cell potential and added volume of standard solution after the Nth spike of the sample. Note that the cell potential for the original (unspiked) sample is
k k given by Eq. (4.19).
Eo=Ecnst+ 1
m⋅log(Co) (4.19)
Subtracting Eq. (4.19) from Eq. (4.18) yields:
(EN−Eo) = 1 mlog
(CoVo+CsVN Vo+VN
)
− 1
mlog(Co) = 1 mlog
{CoVo+CsVN (Vo+VN)Co
}
= 1 mlog
(Vo+KVN Vo+VN
)
(4.20) whereK=Cs/Co. For every sample and spiked solution all of the terms in Eq. (4.20) are known except formandK. These two terms and the uncertainties associated with them are readily obtained by fitting data using a nonlinear regression program. A full discussion of the fundamentals and implementation of nonlinear regression can be found in Refs.
[17–19].