Alexander M.Nemirovsky (


The interest to the addition method in ionometry is caused by its greater role played here than in other analytical techniques. The ionometric addition method has two major advantages. First, if fluctuation of ionic strength in the analyzed samples is unpredictable, then the common calibration curve method gives large determination errors. Using the addition method drastically changes the situation, minimizing the determination error. Secondly, there is an electrode category which, when used, might cause problems due to the potential drift. When the potential drift is intermediate, the addition method essentially reduces the determination error.

The general public is familiar with the following modifications of the addition method: the standard addition method, the double standard addition method, the GRAN method. All these methods can be sorted into two categories by an explicit criterion determining the accuracy of the results obtained. It represents the fact that some addition methods necessarily use in calculation a pre-measured value of the slope of the electrode function, and the other ones do not. According to this classification the standard addition method and the GRAN method fall into one category, while the double standard addition method falls into the other.


1. The standard addition method and the GRAN method

Before discussing individual peculiarities of this or that modification of the addition method let us describe briefly the analysis procedure. The procedure consists in adding to the analyzed sample an addition of a solution containing the same analyzed ion. For example, to determine the content of sodium ions, additions of a standard sodium solution are added. After each addition the readings of the electrodes are taken. Depending on the way the measurement results are further processed, the method is called the standard addition method of the GRAN method.

The calculation for the standard addition method looks as follows:


Cx = ΔC ( 10ΔE/S - 1)-1 ,

where Cx is the sought concentration;
ΔC is the addition size;
ΔE is the potential response to the introduction of the addition ΔC;
S is the slope of the electrode function.

The calculation by the GRAN method looks a little more complex. It consists in drawing a graph in the coordinates (W+V) 10 E/S vs. V,
where V is the volume of the introduced additions;
E is the potential value corresponding to the introduced additions V;
W is the initial sample volume.

The graph is a straight line intersecting the abscissa axis. The intersection point corresponds to the volume of the introduced sample (ΔV) which is equivalent to the sought concentration of the ion (see fig. 1). From the law of multiple proportions it follows that Cx = CstΔV / W, where Cst is the concentration of ions in the solution used to introduce the additions. There may be a number of additions, which apparently improves determination accuracy as compared to the method of standard addition.


Figure. 1



Figure. 2


It is easy to notice that in both cases the slope of the electrode function S appears. Consequently, the first stage of the addition method is the calibration of the electrodes for subsequent determination of the slope value. The absolute value of the potential is not used in the calculation because getting reliable results only requires that the slope of the calibration function be the same from sample to sample.

For additions, one can use not only the solution containing the potential determining ion, but also a solution of a compound binding the determined ion of the sample in a non-dissociating compound. This does not fundamentally change the analysis procedure. However, this case is characterized by certain features that should be taken into consideration. Those are that the graph of experimental results consists of three parts, as shown in fig. 2. The first part (A) is obtained under conditions when the concentration of the binding agent is lower than that of the potential-determining one. The next part (B) of the graph is obtained at approximately equivalent proportion of the two agents. Finally, the third part (C) of the graph corresponds to such conditions when the amount of the binding agent is greater than that of the potential-determining one. A linear extrapolation of the A part of the graph to the abscissa axis gives the value ΔV. The region B is usually not used in analytical determination.

If the titration curve is center-symmetric, then to obtain the results of the analysis one can use the C region as well. However, in that case the ordinate should be calculated as follows: (W+V)10 -E/S .

Since the GRAN method has more advantages than the standard addition method has, the following discussion will mostly concern the GRAN method.

The advantages of the GRAN method can be summarized in the following points:


1. Reduced determination error by a factor of 2 to 3 due to a greater number of measurements in one sample.

2. The addition method does not require thorough stabilization of the ionic strength in the analyzed sample because its fluctuations affect the absolute value of the potential to a greater extent than they affect the electrode function slope. Thus, the determination error is smaller in comparison with the method of the calibration curve.

3. Using a lot of electrodes is problematic because the presence of an insufficiently stable potential requires that the calibration procedure be frequently performed. Since in most cases potential drift only slightly affects the slope of the calibration function, obtaining the results by the standard addition method and the GRAN method considerably improves accuracy and simplifies the analysis procedure.

4. The standard addition method allows one to ensure correct execution of every analytical determination, which is controlled during experimental data processing. Since a number of experimental points are used in processing, drawing a straight line through them every time confirms that the mathematical form and the value of the calibration curve slope have not changed. Otherwise, the linear form of the curve is not guaranteed. Thus, the possibility to control the correctness of the analysis in each determination improves the reliability of the results obtained.

As it has been mentioned, the standard addition method allows one to make determinations in a 2 to 3 times more accurate way than does the calibration curve method. But there is a rule that has to be observed in order to obtain that determination accuracy. Excessive or too small additives reduce the determination accuracy. The optimal size of the additive should be such that it would produce a potential response of 10 to 20 mV for a monovalent ion. This rule optimizes the random inaccuracy of the analysis, but under conditions for which the addition method is often used, the systematic inaccuracy due to change of the parameters of ion-selective electrodes becomes considerable. The systematic inaccuracy in this case is completely determined by the inaccuracy due to change of the electrode function slope. If the slope has changed during experiment, then at certain conditions the relative accuracy of the determination is approximately equal to the relative accuracy due to the change of the slope.


2. The double standard addition method


The method consists in adding 2 doses of the standard solution to the analyzed sample. The sizes of the doses are equal. From the results of the measurements the following parameter is calculated:

R = ΔE2 / Δ E1, where
Δ E1 is the difference between the potential of the electrodes in the analyzed solution and that in the solution after the first addition; ΔE2 is the difference between the potential of the electrodes in the analyzed solution and that in the solution after the second addition.

Using the calculated parameter, the sought concentration value is found from a special table. Using the table is justified by the fact that finding the concentration value requires resolving the transcendent equation

R = lg(1/(1+ 2ΔV/W) + 2ΔC/Cx) / lg(1/(1+ΔV/W) + ΔC/Cx) .

It should be specified that ΔC is the concentration in the analyzed sample after the addition as if there were no more ions in the solution, i.e. ΔC = Cin ΔV / (W+ ΔV). Cin is concentration in standard solution.

Solving the transcendent equation every time it appears is difficult, therefore it is better to use the following table.

ΔV / W
ΔC/Cx 0,001 0,01 0,02 0,04 0,06 0,08 0,1 0,15 0,2
parameter R
0,1 1,9137 1,9224 1,9366 1,9861 2,0902 2,3527 3,6233 1,0112 1,2989
0,2 1,8461 1,8523 1,8608 1,8839 1,9168 1,9624 2,0255 2,3248 3,3002
0,3 1,7919 1,7968 1,8031 1,8190 1,8393 1,8648 1,8961 2,0063 2,1835
0,4 1,7473 1,7513 1,7563 1,7684 1,7833 1,8009 1,8216 1,8879 1,9786
0,5 1,7099 1,7132 1,7174 1,7271 1,7387 1,7521 1,7674 1,8142 1,8736
0,6 1,6779 1,6807 1,6842 1,6923 1,7017 1,7125 1,7246 1,7603 1,8037
0,7 1,6501 1,6526 1,6556 1,6625 1,6704 1,6793 1,6892 1,7178 1,7516
0,8 1,6258 1,6280 1,6306 1,6366 1,6433 1,6509 1,6592 1,6828 1,7103
0,9 1,6043 1,6063 1,6086 1,6138 1,6197 1,6262 1,6333 1,6533 1,6762
1 1,5851 1,5869 1,5889 1,5935 1,5987 1,6044 1,6106 1,6279 1,6474
1,1 1,5679 1,5694 1,5713 1,5754 1,5800 1,5851 1,5905 1,6056 1,6225
1,2 1,5523 1,5537 1,5553 1,5591 1,5632 1,5677 1,5725 1,5859 1,6007
1,3 1,5380 1,5393 1,5408 1,5442 1,5479 1,5520 1,5563 1,5683 1,5815
1,4 1,5250 1,5262 1,5276 1,5307 1,5341 1,5377 1,5417 1,5524 1,5642
1,5 1,5131 1,5141 1,5154 1,5182 1,5213 1,5247 1,5283 1,5380 1,5487
1,6 1,5020 1,5030 1,5042 1,5068 1,5096 1,5127 1,5160 1,5249 1,5346
1,7 1,4918 1,4927 1,4938 1,4962 1,4988 1,5017 1,5047 1,5128 1,5217
1,8 1,4822 1,4831 1,4841 1,4864 1,4888 1,4914 1,4942 1,5017 1,5099
1,9 1,4734 1,4742 1,4751 1,4772 1,4795 1,4819 1,4845 1,4915 1,4990
2 1,4651 1,4658 1,4667 1,4686 1,4708 1,4730 1,4754 1,4819 1,4889
2,1 1,4573 1,4580 1,4588 1,4606 1,4626 1,4647 1,4670 1,4730 1,4795
2,2 1,4499 1,4506 1,4514 1,4531 1,4549 1,4569 1,4591 1,4647 1,4708
2,3 1,4430 1,4436 1,4444 1,4460 1,4477 1,4496 1,4516 1,4569 1,4626
2,4 1,4365 1,4371 1,4378 1,4393 1,4410 1,4427 1,4446 1,4496 1,4549
2,5 1,4303 1,4309 1,4315 1,4330 1,4345 1,4362 1,4380 1,4427 1,4477
2,6 1,4244 1,4250 1,4256 1,4270 1,4285 1,4300 1,4317 1,4362 1,4409
2,7 1,4189 1,4194 1,4200 1,4213 1,4227 1,4242 1,4258 1,4300 1,4345
2,8 1,4136 1,4141 1,4146 1,4159 1,4172 1,4186 1,4201 1,4241 1,4284
2,9 1,4085 1,4090 1,4095 1,4107 1,4120 1,4133 1,4148 1,4186 1,4226
3 1,4037 1,4042 1,4047 1,4058 1,4070 1,4083 1,4097 1,4133 1,4171


In the presented formulas the electrode function slope is not mentioned, because that was the purpose of the development of the double standard addition method. At first glance, this is an advantage of the method, as the analysis procedure becomes simpler, but that is not true. Gaining in one thing we lose in another. First, in order to have a satisfactory analysis inaccuracy, one has to be assured of the linearity of the ion-selective electrode function. Deviations from linearity would lead to a great inaccuracy of the analysis. Thus, calibration of the electrodes is required anyway. Second, the random inaccuracy of the analysis is significantly greater than that in the GRAN method and the standard addition method. For example, an error of 1 mV in potential measurement can cause an inaccuracy in the analysis of 10-20%.

Summarizing the stated above, a conclusion arises that the double standard addition method should better be used only for electrodes with very reliable operation characteristics, such as, e.g., fluoride-selective one.