Grans method

The Gran Method is particularly valuable because, like most based on linear titrations, it yields straight line segments susceptible to least squares method analysis, which leads to great accuracy in equivalence point determination. 

Let us consider a typical point in the titration of mL of a strong acid of unknown concentration with Vg mL of a known concentration, Cg M of NaOH. The pH of this solution is converted to the corresponding [H ] (= lO P ). Then the number of millimoles of H not yet neutralized, can be expressed as either [H ] (V + Vg) or (Vg - Vg), where Vg is the volume of base at the equivalence point. At every point before the equivalence point. 

A plot of [tf ] (V + Vg) vs. Vg will give a straight line intersecting the V axis at Vg = Vg. Two important advantages result (1) the values of Vg used in the titration do not have to be close to the equivalence point. Time is saved when it is not necessary to make careful, drop-wise additions in the vicinity of the equivalence point and (2) the line can be most accurately defined by use of the linear regression command /TAR /AR /DAR. The increase on precision of Vg is a direct consequence of the utilization of all of the titration data, not merely the few points taken close to the equivalence point, as is the case in the conventional titration. 

Similarly, one can use the data obtained after the equivalence by equating the two ways of expressing the excess base. 

A plot of the left-hand side of Equation 8-40 vs. Vb will also intersect the V axis at Vb = Vg. Moreover, the value of can be obtained from the slope of this linear plot. 

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Final remarks on end-point detection. In addition to our remarks above on the types of titration curves and the Henderson equation or more extended relationships, we can state that in Gran s method activity coefficients are taken into account however, these were assumed to be constant, which is incorrect, and therefore the addition of an ISA (ion strength adjuster) must be recommended (for errors of the Gran method see ref.66). 

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 ... 

Fig. 5.3. The use of the Gran method in the multiple addition method. For explanation see the text.

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. 

Figure 4.13. Emf titration curve for 154 g of seawater with v milliliters (0.1000 M HCl plus 0.4483 M NaCl) characterization of equivalence points by the Gran method. F, Fj, and F3 are Gran functions, and n, is an arbitrary scale factor. For finding the equivalence points Vx and corresponding to / = 1 and/= 0, respectively. (From Dryssen and Sillen, 1%7.)...

Because the equation describing the electrode potential in the presence of thiosulphate ions is more complicated than the simple Nernst equation the usual Gran method must be modified. The Gran function has the form ... 

This form of potential response equation implies a more complicated data processing by the modified Gran method, as indicated in the Analytical Procedure . 

Figure 8.6 Locating the Equivaience Point by Difference Titration and Gran Methods...

As we found with acid-base titrations in the last chapter, however, the most precise method for locating the endpoint is the Gran method. This method results in a linear plot which intercepts the equivalence point at the X axis. Not only can we easily find the best line through linear regression but, as mentioned earlier, the necessary points can be taken at a distance from the equivalence point making this method rapid and convenient. 

Unlike titrations previously described in this book, photometric titrations are linear since absorbance is directly proportional to concentration. To be sure, we have demonstrated how to treat pH, pM, and pE data, in the context of the Gran method, to obtain linear titrations. Remember two important benefits of linear titrations (1) it is a practical way to handle dilute solutions and (2) data can be taken more rapidly, using points on either side of the equivalence point. The best line for each of the two sets (before and after the equivalence point) is obtained by linear regression and their intercept used to locate the equivalence point. 

The common procedure of finding the endpoint, which should correspond to the equivalence volume, is based on the Gran method. This procedure is especially useful for flat and vmsymmetrical titration curves. The titration data are transformed in such a way that the titration curve consists of two linear parts - before and after the equivalence point. The intersection of those lines or their intersection with the abscissa defines the equivalence volume of the titrant (Figure 3). The linearization is based on the transformation of the Nernst equation into the form... 

Hgure 3 Titration of two-component mixture (A) titration curve (B) linearization according to the Gran method. 

Fig. 48. Calibration curve for a monovalent anion according to the Gran method...

In addition to overcoming all these problems in the actual measurement and in the evaluation method, this technique also avoids a series of chemical titration problems. These arise with other techniques when too large a solubility product or too small a complex formation constant in dilute solutions result in poorly defined end points in the usual titration curves. With the Gran method, with which an excess of titrating... 

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