Gran plot

A Gran plot (also known as Gran titration or the Gran method) is a common means of standardizing a titrate or titrant by estimating the equivalence volume or end point in a strong acid-strong base titration or in a potentiometric titration. Such plots have been also used to calibrate glass electrodes, to estimate the carbonate content of aqueous solutions, and to estimate the K a values (acid dissociation constants) of weak acids and bases from titration data. Gran plots are named after Swedish chemist Gunnar Gran, who developed the method in 1950. Gran plots use linear approximations of the a priori non-linear relationships between the measured quantity, pH or electromotive potential (emf), and the titrant volume. Other types of concentration measures, such as spectrophotometric absorbances or NMR chemical shifts, can in principle be similarly treated. These approximations are only valid near, but not at, the end point, and so the method differs from end point estimations by way of first- and second-derivative plots, which require data at the end point. Gran plots were originally devised for graphical determinations in pre-computer times, wherein an x-y plot on paper would be manually extrapolated to estimate the x-intercept.…

A Gran plot (also known as Gran titration or the Gran method) is a common means of standardizing a titrate or titrant by estimating the equivalence volume or end point in a strong acid-strong base titration or in a potentiometric titration. Such plots have been also used to calibrate glass electrodes, to estimate the carbonate content of aqueous solutions, and to estimate the K a values (acid dissociation constants) of weak acids and bases from titration data. Gran plots are named after Swedish chemist Gunnar Gran, who developed the method in 1950. Gran plots use linear approximations of the a priori non-linear relationships between the measured quantity, pH or electromotive potential (emf), and the titrant volume. Other types of concentration measures, such as spectrophotometric absorbances or NMR chemical shifts, can in principle be similarly treated. These approximations are only valid near, but not at, the end point, and so the method differs from end point estimations by way of first- and second-derivative plots, which require data at the end point. Gran plots were originally devised for graphical determinations in pre-computer times, wherein an x-y plot on paper would be manually extrapolated to estimate the x-intercept. The graphing and visual estimation of the end point have been replaced by more accurate least-squares analyses since the advent of modern computers and enabling software packages, especially spreadsheet programs with built-in least-squares functionality. E = E^0 + s \log \{H^+\} where E is a measured electrode potential, E 0 is a standard electrode potential, s is the slope, ideally equal to RT/nF, and {H + } is the activity of the hydrogen ion. The expression rearranges to [H^+]=10^\frac{E-E^0}{s} \ or \ [H^+]=10^{-pH} depending on whether the electrode is calibrated in millivolts or pH. For convenience the concentration, [H + ], is used in place of activity. In a titration of strong acid with strong alkali, the analytical concentration of the hydrogen ion is obtained from the initial concentration of acid, C i and the amount of alkali added during titration. [H^+] = \frac{C_i v_i - c_{OH} v}{v_i + v} where v i is the initial volume of solution, c OH is the concentration of alkali in the burette and v is the titre volume. Equating the two expressions for [H + ] and simplifying, the following expression is obtained C_i v_i - c_{OH} v = (v_i + v)10^\frac{E-E^0}{s} \ or \ =(v_i + v)10^{-pH} A plot of (v_i + v)10^\frac{E-E^0}{s} \ or \ (v_i + v)10^{-pH} against v will be a straight line. If E 0 and s are known from electrode calibration, where the line crosses the x-axis indicates the volume at the equivalence point, C_i v_i = c_{OH} v . Alternatively, this plot can be used for electrode calibration by finding the values of E 0 and s that give the best straight line. K_{w^{ }} = [H^+]_i[OH^-]_i where K w is the water autoprotolysis constant. If titrating an acid of initial volume v_{0^{ }} and concentration [H^+]_{0^{ }} with base of concentration [OH^-]_{0^{ }} , then at any ith point in the titration with titrant volume v_{i^{ }} , \frac{v_0 [H^+]_0-v_i[OH^-]_0}{v_0+v_i} \begin{cases} \approx [H^+]_i \text{ or } 10^{-pH_i} & \text{ when } v_{0^{ }} [H^+]_0 > v_i[OH^-]_0 \text{ (acidic region)} \\ At the equivalence point, the equivalence volume v_{e^{ }}=v_{i^{ }} . Thus, a plot of ({v_0+v_i})10^{-pH_{i}} \text{ vs. } v_{i^{ }} will have a linear region before equivalence, with slope -[OH^-]_{0^{ }} and a plot of ({v_0+v_i})10^{pH_{i}} \text{ vs. } v_{i^{ }} will have a linear region after equivalence, with slope [OH^-]_0/K_{w^{ }} both plots will have v_e=v_0 [H^+]_0/[OH^-]_{0^{ }} as intercept The equivalence volume is used to compute whichever of [H^+]_{0^{ }} or [OH^-]_{0^{ }} is unknown. The pH meter is usually calibrated with buffer solutions at known pH values before starting the titration. The ionic strength can be kept constant by judicious choice of acid and base. For instance, HCl titrated with NaOH of approximately the same concentration will replace H + with an ion (Na + ) of the same charge at the same concentration, to keep the ionic strength fairly constant. Otherwise, a relatively high concentration of background electrolyte can be used, or the activity quotient can be computed. both plots will have v_e=v_0 [OH^-]_{0^{ }}/[H^+]_0 as x-intercept Figure 1 gives sample Gran plots of a strong base-strong acid titration. K_a = \frac{[H^+]_i[A^-]_i}{[HA]_i} , we have at any ith point in the titration of a volume v_0 of acid at a concentration [HA]_0 by base of concentration [OH^-]_0 . In the linear regions away from equivalence, [HA]_i \approx \frac{v_0 [HA]_0-v_i[OH^-]_0}{v_0+v_i}   and [A^-]_i \approx \frac{v_i[OH^-]_0}{v_0+v_i}   are valid approximations, whence K_a \approx \frac{10^{-pH_i} v_i[OH^-]_0}{v_0 [HA]_0-v_i[OH^-]_0} , or K_a (v_0 \frac{[HA]_0}{[OH^-]_0}-v_i) \approx 10^{-pH_i} v_i or, because v_e = v_0 \frac{[HA]_0}{[OH^-]_0} , K_a (v_e -v_i) \approx 10^{-pH_i} v_i . A plot of 10^{-pH_i} v_i versus v_{i^{ }} will have a slope -K_{a^{ }} over the linear acidic region and an extrapolated x-intercept v_{e^{ }} , from which either [HA]_{0^{ }} or [OH^-]_{0^{ }} can be computed. The alkaline region is treated in the same manner as for a titration of strong acid. Figure 2 gives an example; in this example, the two x-intercepts differ by about 0.2 mL but this is a small discrepancy, given the large equivalence volume (0.5% error). Similar equations can be written for the titration of a weak base by strong acid (Gran, 1952; Harris, 1998). Martell and Motekaitis (1992) use the most linear regions and exploit the difference in equivalence volumes between acid-side and base-side plots during an acid-base titration to estimate the adventitious CO 2 content in the base solution. This is illustrated in the sample Gran plots of Figure 1. In that situation, the extra acid used to neutralize the carbonate, by double protonation, in volume v_{0^{ }} of titrate is (v_e-v_e^{\prime})[H^+]_{0^{ }} = 2v_0[CO_2]_0 . In the opposite case of a titration of acid by base, the carbonate content is similarly computed from (v_e^{\prime}-v_e)[OH^-]_{0^{ }} = 2v_e^{\prime}[CO_2]_0 , where v_e^{\prime} is the base-side equivalence volume (from Martell and Motekaitis). When the total CO 2 content is significant, as in natural waters and alkaline effluents, two or three inflections can be seen in the pH-volume curves owing to buffering by higher concentrations of bicarbonate and carbonate. As discussed by Stumm and Morgan (1981), the analysis of such waters can use up to six Gran plots from a single titration to estimate the multiple end points and measure the total alkalinity and the carbonate and/or bicarbonate contents. To use potentiometric (e.m.f.) measurements E_{i^{ }} in monitoring the H^{+_{ }} concentration in place of pH_{i^{ }} readings, one can trivially set -log_{10}[H^+]_i = b_0 - b_1E_{i^{ }} and apply the same equations as above, where b_{0^{ }} is the offset correction nFE_{0^{ }}/RT , and b_{1^{ }} is a slope correction nF^{ _{ }}/RT (1/59.2 pH units/mV at 25°C), such that -b_1E_{i^{ }} replaces pH_{i^{ }} . Thus, as before for a titration of strong acid by strong base, a plot of ({v_0+v_i})10^{b_1E_i} vs. v_{i^{ }} will have a linear region before equivalence, with slope -10^{b_0}[OH^-]_{0^{ }} and a plot of ({v_0+v_i})10^{-b_1E_i} vs. v_{i^{ }} will have a linear region after equivalence, with slope 10^{-b_0}[OH^-]_0/K_{w^{ }} both plots will have v_e=v_0 [H^+]_0/[OH^-]_{0^{ }} as intercept and, as before, the acid-side equivalence volume can be used to standardize whichever concentration is unknown, and the difference between acid-side and base-side equivalence volumes can be used to estimate the carbonate content Analogous plots can be drawn using data from a titration of base by acid. Based on earlier work by McBryde (1969), Gans and O'Sullivan (2000) describe an iterative approach to arrive at both b_{0^{ }} and b_{1^{ }} values in the relation -log_{10}[H^+]_i = b_0 - b_1E_{i^{ }} , from a titration of strong acid by strong base: {{ordered list |1= b_{0^{ }} is first estimated from the acidic data according to Rossotti and Rossotti (1965), and b_{1^{ }} is initially taken to have its theoretical value; |2= modified Gran function plots are drawn, using ({v_0+v_i})10^{b_1E_i - b_0} vs. v_{i^{ }} on the acidic side of equivalence and ({v_0+v_i})K_w 10^{-b_1E_i + b_0} vs. v_{i^{ }} on the alkaline side, and the equivalence volumes v_{e^{ }} and v_e^{\prime} are computed therefrom, as before; |3= as before, the difference in the equivalence volumes is used to compute the carbonate content but also to calculate an 'effective base concentration' v_e [OH^-]_{0^{ }}/v_e^{\prime} for the alkaline side of equivalence; |4= approximate [H^+]_{i^{ }} values are computed on the acid side as (v_{0^{ }} [H^+]_0-v_i[OH^-]_0)/(v_0+v_i) and on the alkaline side as (v_e [H^+]_0-v_i v_e[OH^-]_0 /v_e^{\prime})/(v_0+v_i) ; |5= the initial definition -log_{10}[H^+]_i = b_0 - b_1E_{i^{ }} is rewritten as E_{i^{ }} = b_0/b_1 + (1/b_1)log_{10}[H^+]_i , and the E_{i^{ }} data are plotted against log_{10}[H^+]_{i^{ }} , using those [H^+]_{i^{ }} values corresponding to pH values in the 2.5-4.5 and 10.7-11.5 ranges (the linear response range for a glass electrode that avoids variation of junction potentials and/or alkaline error at the pH extrema, and that additionally avoids measurement uncertainties near the equivalence point as well as computational errors from the neglect of [OH^-]_{i^{ }} on the acid side and the neglect of the carbonate/bicarbonate equilibrium on the alkaline side); a linear least-squares treatment provides 1/b_{1^{ }} as slope and b_0/b_{1^{ }} as intercept; |6= steps 2 and 3 are repeated with the new b_{0^{ }} and b_{1^{ }} values for greater precision in the equivalence volumes and the CO 2 content. }} The procedure could in principle be modified for titrations of base by acid. A computer program named GLEE (for GLass Electrode Evaluation) implements this approach on titrations of acid by base for electrode calibration. This program additionally can compute (by a separate, non-linear least-squares process) a 'correction' for the base concentration. An advantage of this method of electrode calibration is that it can be performed in the same medium of constant ionic strength which may later be used for the determination of equilibrium constants. Note that the regular Gran functions will provide the required equivalence volumes and, as b_{1^{ }} is initially set at its theoretical value, the initial estimate for b_{0^{ }} in step 1 can be had from the slope of the regular acid-side Gran function as detailed earlier. Note too that this procedure computes the CO 2 content and can indeed be combined with a complete standardization of the base, using the definition of v_{e^{ }} to compute [OH^-]_{0^{ }} . Finally, the usable pH range could be extended by solving the quadratic (v_{0{^{ }}} [H^+]_0-v_i[OH^-]_0)/(v_0+v_i) = [H^+]_i - K_w/[H^+]_i for [H^+]_{i^{ }} . Potentiometric data are also used to monitor species other than H^{+_{ }} . When monitoring any species S^{1_{ }} by potentiometry, one can apply the same formalism with -log_{10}[S^1]_i = b_0 - b_1E_{i^{ }} . Thus, a titration of a solution of another species S^{2_{ }} by species S^{1_{ }} is analogous to a pH-monitored titration of base by acid, where either ({v_0+v_i})10^{b_1E_{i}} or ({v_0+v_i})10^{-b_1E_{i}} plotted versus v_{i^{ }} will have an x-intercept v_0 [S^0]_0 /[S^1]_{0^{ }} . In the opposite titration of S^{1_{ }} by S^{2_{ }} , the equivalence volume will be v_0 [S^1]_0 /[S^0]_{0^{ }} . The significance of the slopes will depend on the interactions between the two species, whether associating in solution or precipitating together (Gran, 1952). Usually, the only result of interest is the equivalence point. However, the before-equivalence slope could in principle be used to assess the solubility product K_{sp^{ }} in the same way as K_{w^{ }} can be determined from acid-base titrations, although other ion-pair association interactions may be occurring as well. To illustrate, consider a titration of Cl − by Ag + monitored potentiometrically: \frac{v_0 [Cl^-]_0-v_i[Ag^+]_0}{v_0+v_i} \begin{cases} \approx [Cl^-]_i \text{ or } K_{sp} 10^{-b_1E_i+b_0} & \text{ when } v_{0^{ }} [Cl^-]_0 > v_i[Ag^+]_0 \text{ (before equivalence)} \\ in both plots, the x-intercept is v_e=v_0 [Cl^-]_{0^{ }}/[Ag^+]_0 Figure 3 gives sample plots of potentiometric titration data.

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