Gran plot method

Generally, in freshwaters total alkalinity determinations are carried out using additions of strong acids (either H2SO4 or HCl in the concentration ranges 0.02-0.1 mol 1 ) to an end point pH of 4.2-5.1, using an appropriate indicator or potentiometer. For accurate analysis determination a two-point or multi-point Gran plot method may be employed. 

Gran plot a linearized form of a titration curve, (p. 293) graphite furnace an electrothermal atomizer that relies on resistive heating to atomize samples, (p. 414) gravimetry any method in which the signal is a mass or change in mass. (p. 233)... 

To appreciate that an ISE must be selective if its measurements are to be useful, and how the Nernst equation can be adapted to take account of the ISE selectivity by using the concept of a selectivity coefficient (or ratio ). To leam how the problems of selectivity can be overcome by using a standard-addition method, such as drawing a Gran plot - one of the more useful of such methods. 

Experimentally, an analyst will run several standards (at constant AEj to calibrate the analysis, and will then determine the amounts of analyte in solution. A standard additions method such as a Gran plot will further enhance the accuracy of measurement (e.g. see Section 4.3.2). 

At extreme overpotentials, the current is independent of potential. This maximum current is said to be limiting, that is, current a Cbuik- It is termed the diffusion current, /j. The dependence of la on concentration, drop speed, etc., is described by the Ilkovic equation (equation (6.5)), although calibration graphs or standard addition methods (Gran plots) are preferred for more accurate analyses. 

Gran plot A commonly employed multiple-addition method, used to correct for unknown amounts of contaminant and for dilution errors (see Section 4.3.2). 

A drawback of Gran plots is the fact that all deviations from the theoretical slope value cause an error and that side reactions are not considered. The method was modified by Ingman and Still [63], who considered side reactions to a certain degree, but the equilibrium constants and the concentrations of the components involved must be known. The Gran method is, however, advantageous for determinations in the vicinity of the determination limit The extrapolation of the linear dependence yields the sum + c, where c, is the residual concentration of the test component produced by impurities, dissolution of the ISE membrane, etc. 

The fluoride ion selective electrode is the most popular means of fluoride ion determination after sample destruction by any method but it does have limitations. It can be used either directly to measure the fluoride potential6 or as an end-point detector in a potentiometric titration with a lanthanum(l II) reagent as titrant.4,7 Problems can be experienced with potential drift in direct potentiometry, especially at low fluoride ion concentrations. Titration methods often yield sluggish end points unless water miscible solvents are used to decrease solubilities and increase potentia 1 breaks and sulfate and phosphate can interfere. End-point determination can be facilitated by using a computerized Gran plotting procedure.4... 

The Gran plot (Fig. 13.2c2). This method consists of the mathematical transformation of the titration curve into straight lines via rearranged Nernst equations. Using a selective electrode that responds only to a... 

Fig. 13.2. Methods for determining the equivalence point of a potentiometric titration curve (including acid-base titrations), (a) First derivative (b) Second derivative (c) Gran plot for titration of a strong acid with a strong base Vx is the initial volume of acid and V the volume of base added.

The Gran plot is an alternative method for locating the end point in a titration. In this method, a linear plot is produced that can reveal both the acid dissociation constant and the volume of base required to reach the end point. Unlike the normal titration curve and derivative curves, which find the end point only from data located in the end point region, the Gran plot uses data far away from the end point. This can decrease the tedium of taking many measurements after dispensing very small volumes of titrant in the end point region. 

Spreadsheet Summary In Chapter 7 of Applications of Microsoft Excel in Analytical Chemistry, the first and second derivatives of an acid/base titration curve are plotted in order to better determine the titration end point. A combination plot is produced that simultaneously displays the pH versus volume curve and the second-derivative curve. Finally, an alternative plotting method, known as a Gran plot, is explored for locating the end point by a linear regression procedure. 

Dissolved organic carbon (determined by combustion at 680 °C on a Shi-rnadzu TOC-500 instrument) was about 3.5 mg/L and varied little in samples from different depths and times. Alkalinity (determined by Gran plot titrations) was 3-4 mmol/L, and pH varied with depth and time in the range 7.5-8.5 (21). Dissolved phosphate and silicate were measured in filtered samples by standard automated (Auto-Analyzer) methods (25). 

Figures 4.5-1 through 4.5-3 show thumbnail sketches such as you might embed at the top of your spreadsheet, for three different amounts of added noise. In viewing these figures you should keep in mind that the added noise is somewhat artificial, because it only affects the volume axis, and not the pH axis. (If desired you can of course add noise to the pH data as well.) Nonetheless it indicates that the Gran plot past the equivalence point (i.e., for a strong base) is the most robust. Note that noise of magnitudes such as that shown for na= 0.03 is experimentally completely unnecessary and unacceptable, and is used here only to emphasize the point that the various data analysis methods have quite different sensitivities to noise. Not everything that looks good in theory works well in practice, and the spreadsheet is a good tool to find this out.

One can of course fit experimental data to the entire, theoretical curve with a non-linear least-squares routine such as Solver. In this particular case, however, the direct, non-iterative method of using Gran plots provides a valid, simpler alternative. As illustrated below, such plots are quite linear, analogous to the Gran plots for the titration of strong acids and bases. 

H. Li, Improvement of Gran s Plot Method in Standard Addition and Subtraction Methods by a New Plot Method, Anal. Lett., 24 (1991) 473. 

Known addition, known subtraction and Gran s plot methods are other ways of using ion-selective electrodes [14,20—27]. In the standard addition to sample method, the potential, o,of the ion-selective electrode with respect to a suitable reference electrode is measured for the sample solution of volume, Fo, and total concentration, Co, of the sought species ... 

Zarate, N., M. P. Ruiz, R. Perez-Olmos, A. N. Araujo, M. Conceiqao, and B. S. M. Montenegro. 2009. Development of a sequential injection analysis system for the potentiometric determination of nitrite in meat products by using a Gran s plot method. Microchim. Acta 165 117-122. 

Gran s method is based on the idea that a linear titration curve is obtained if the concentration of the indicated ion is plotted rather than the corresponding electrode EMF. This is due to the simple relationship Titration with 10% of the equivalence volume = 10% less free ion in solution 50% equivalence volume titrated = 50% less 100% titrated = no free measured ion in solution. With titrations followed by conductance measurements a direct proportionality (not logarithmic, as with EMF measurements) is obtained between the measured ion concentration and the conductance, resulting in linear concentration curves which can be extrapolated. To get the concentration corresponding to a particular EMF one can either use a suitable calibration curve, or read the concentrations directly off of a logarithmically divided instrument scale. The empirical Nernst equation can also be mathematically manipulated into a suitable form ... 

In potentiometry with ISEs, however, the second Gran method [46] has found especially wide use, not only in titrations, but also in multiple addition methods in general. In these methods, the concentration of the test substance is plotted against the volume of the titrant or of the standard solution and thus the curve is linearized. The end-point in the titration or the determinand concentration in a multiple addition method is found as the intercept of the straight line with the volume axis. Linearization is attained by taking the antilogarithm of the Nernst equation ... 

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