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 estimate the carbonate content of aqueous solutions, and the K_a values (acid dissociation constants) of weak acids and bases from titration data.

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 The graphing and visual estimation of the end point have been replaced by more accurate least-squares analyses since the advent of modern computers and enabling software packages, especially spreadsheet programs with built-in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.

Titrating strong acid with strong base

For a strong acid-strong base titration monitored by pH, we have at any i'th point in the titration

$K_{w^{}}=[H^{+}]_{i}[OH^{-}]_{i}$

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 i'th point in the titration with titrant volume $v_{i^{}}$,

${\frac {v_{0}[H^{+}]_{0}-v_{i}[OH^{-}]_{0}}{v_{0}+v_{i}}}{\begin{cases}\approx [H^{+}]_{i}{\text{ or }}10^{-pH_{i}}&{\text{ when }}v_{0^{}}[H^{+}]_{0}>v_{i}[OH^{-}]_{0}{\text{ (acidic region)}}\\=0&{\text{ when }}v_{0^{}}[H^{+}]_{0}=v_{i}[OH^{-}]_{0}{\text{ (equivalence point)}}\\\approx -[OH^{-}]_{i}{\text{ or }}-K_{w}10^{pH_{i}}&{\text{ when }}v_{0^{}}[H^{+}]_{0}<v_{i}[OH^{-}]_{0}{\text{ (alkaline region)}}\end{cases}}$

At the equivalence point, the equivalence volume $v_{e^{}}=v_{i^{}}$.

Thus,

${\color {Blue}\bullet }$ a plot of

$({v_{0}+v_{i}})10^{-pH_{i}}{\text{ vs. }}v_{i^{}}$

will have a linear region

before

equivalence, with slope

$-[OH^{-}]_{0^{}}$

${\color {Blue}\bullet }$ and a plot of

$({v_{0}+v_{i}})10^{pH_{i}}{\text{ vs. }}v_{i^{}}$

will have a linear region

after

equivalence, with slope

$[OH^{-}]_{0}/K_{w^{}}$

${\color {Blue}\bullet }$ 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.

Titrating strong base with strong acid

Mirror-image plots are obtained if titrating the base with the acid, and the signs of the slopes are reversed.

${\frac {v_{0}[OH^{-}]_{0}-v_{i}[H^{+}]_{0}}{v_{0}+v_{i}}}{\begin{cases}\approx [OH^{-}]_{i}{\text{ or }}K_{w}10^{pH_{i}}&{\text{ when }}v_{0^{}}[OH^{-}]_{0}>v_{i}[H^{+}]_{0}{\text{ (alkaline region)}}\\=0&{\text{ when }}v_{0^{}}[OH^{-}]_{0}=v_{i}[H^{+}]_{0}{\text{ (equivalence point)}}\\\approx -[H^{+}]_{i}{\text{ or }}-10^{-pH_{i}}&{\text{ when }}v_{0^{}}[OH^{-}]_{0}<v_{i}[H^{+}]_{0}{\text{ (acidic region)}}\end{cases}}$

Click on the image to view in full size.
Figure 1. Sample Gran plots using data from Butler (1998). Note that the filled circles indicate the data points included in the least-squares computations to give the fitted dashed lines.

Click on the image to view in full size.
Figure 2. Sample Gran plots using data from "an on-line source". Retrieved 2008-02-18. Only the region near equivalence is shown, as data far from equivalence deviate strongly from linearity. Note that the filled circles indicate the data points included in the least-squares computations to give the fitted dashed lines.

Click on the image to view in full size.
Figure 3. Sample Gran plots using data from a titration of Cl^- by Ag⁺ monitored potentiometrically. The potentials were converted to [Ag⁺] values for plotting. Note that the filled circles indicate the data points included in the least-squares computations to give the fitted dashed lines.

Hence,

${\color {Blue}\bullet }$ a plot of	$({v_{0}+v_{i}})10^{pH_{i}}{\text{ vs. }}v_{i^{}}$	will have a linear region before equivalence
with slope	$-[H^{+}]_{0}/K_{w^{}}$

${\color {Blue}\bullet }$ and a plot of	$({v_{0}+v_{i}})10^{-pH_{i}}{\text{ vs. }}v_{i^{}}$	will have a linear region after equivalence
with slope	$[H^{+}]_{0^{}}$

${\color {Blue}\bullet }$ 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.

Concentrations and Dissociation Constants of Weak Acids

The method can be used to estimate the dissociation constants of weak acids, as well as their concentrations (Gran, 1952). With an acid represented by HA, where

$K_{a}={\frac {[H^{+}]_{i}[A^{-}]_{i}}{[HA]_{i}}}$ ,

we have at any i'th 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).

Carbonate content

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 CO₂ 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 CO₂ 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.

Electrode offset

If the pH electrode is not well calibrated, an offset correction might be appropriate and can be computed from the acid-side Gran slopes. Let the offset be represented by $b\ (=pH_{0})$, then

$-log_{10}([H^{+}]_{i^{}})=b+pH_{i}$

Incorporating this into the equations above results in the following modified slopes:

• For a titration of acid by base, one obtains an acid-side slope of $-[OH^{-}]_{0^{}}10^{b}$ from which $b_{^{}}$ can be computed, and a base-side slope of $[OH^{-}]_{0^{}}10^{-b}/K_{w}$, from which $K_{w^{}}$ can be computed.
• For a titration of base by acid, as illustrated in the sample plots, the acid-side slope, $[H^{+}]_{0^{}}10^{b}$, is similarly used to compute $b_{^{}}$, and the base-side slope of $-[H^{+}]_{0^{}}10^{-b}/K_{w}$ is used to compute $K_{w^{}}$. In the sample data illustrated in Figure 1, the offset was not insignificant, at -0.054 pH units.

The x-intercepts are unaffected.

Potentiometric data

To use potentiometric measurements $E_{i^{}}$ in monitoring the $H^{+_{}}$ concentration in place of $pH_{i^{}}$ readings, one can trivially set $-log_{10}[H^{+}]_{i}=b_{0}-b_{1}E_{i^{}}$ and apply the same equations as above, where $b_{0^{}}$ is the offset correction $nFE_{0^{}}/RT$ corresponding to the $b_{^{}}$ used earlier, and $b_{1^{}}$ is a slope correction $nF^{_{}}/RT$ (1/59.2 pH units/mV at 25°C), such that $-b_{1}E_{i^{}}$ replaces $pH_{i^{}}$.

When monitoring another species $S^{1_{}}$ by potentiometry, one can apply the same formalism with $-log_{10}[S^{1}]_{i}=b_{0}-b_{1}E_{i^{}}$. 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, whence either $({v_{0}+v_{i}})10^{b_{1}E_{i}}$ or $({v_{0}+v_{i}})10^{-b_{1}E_{i}}$ 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 volumes 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: ${\frac {v_{0}[Cl^{-}]_{0}-v_{i}[Ag^{+}]_{0}}{v_{0}+v_{i}}}{\begin{cases}\approx [Cl^{-}]_{i}{\text{ or }}K_{sp}10^{-b_{1}E_{i}+b_{0}}&{\text{ when }}v_{0^{}}[Cl^{-}]_{0}>v_{i}[Ag^{+}]_{0}{\text{ (before equivalence)}}\\=0&{\text{ when }}v_{0^{}}[Cl^{-}]_{0}=v_{i}[Ag^{+}]_{0}{\text{ (equivalence point)}}\\\approx -[Ag^{+}]_{i}{\text{ or }}-10^{b_{1}E_{i}-b_{0}}&{\text{ when }}v_{0^{}}[Cl^{-}]_{0}<v_{i}[Ag^{+}]_{0}{\text{ (after equivalence)}}\end{cases}}$

Hence,

${\color {Blue}\bullet }$ a plot of

$({v_{0}+v_{i}})10^{-b_{1}E_{i}}{\text{ vs. }}v_{i^{}}$

will have a linear region before equivalence, with slope

$-[Ag^{+}]_{0}10^{-b_{0}}/K_{sp^{}}$

${\color {Blue}\bullet }$ and a plot of

$({v_{0}+v_{i}})10^{b_{1}E_{i}}{\text{ vs. }}v_{i^{}}$

will have a linear region after equivalence, with slope

$[Ag^{+}]_{0^{}}10^{b_{0}}$

${\color {Blue}\bullet }$ in both plots, the x-intercept is

$v_{e}=v_{0}[Cl^{-}]_{0^{}}/[Ag^{+}]_{0}$

Figure 3 gives sample plots of potentiometric titration data.

Non-ideal behaviour

In any titration lacking buffering components, both before-equivalence and beyond-equivalence plots should ideally cross the x axis at the same point. Non-ideal behaviour can result from measurement errors (e.g. a poorly calibrated electrode, an insufficient equilibration time before recording the electrode reading, drifts in ionic strength), sampling errors (e.g. low data densities in the linear regions) or an incomplete chemical model (e.g. the presence of titratable impurities such as carbonate in the base, or incomplete precipitation in potentiometric titrations of dilute solutions, for which Gran et al. (1981) propose alternate approaches). Buffle et al. (1972) discuss a number of error sources.

Because the $10^{pH_{i}}$ or $10^{-pH_{i}}$ terms in the Gran functions only asymptotically tend toward, and never reach, the x axis, curvature approaching the equivalence point is to be expected in all cases. However, there is disagreement among practitioners as to which data to plot, whether using only data on one side of equivalence or on both sides, and whether to select data nearest equivalence or in the most linear portions: using the data nearest the equivalence point will enable the two x-intercepts to be more coincident with each other and to better coincide with estimates from derivative plots, while using acid-side data in an acid-base titration presumably minimizes interference from titratable (buffering) impurities, such as bicarbonate/carbonate in the base (see Carbonate content), and the effect of a drifting ionic strength. In the sample plots displayed in the Figures, the most linear regions (the data represented by filled circles) were selected for the least-squares computations of slopes and intercepts. Data selection is always subjective.

Electrode calibration

Another use of the Gran plot is for electrode calibration. The advantage of this method of electrode calibration is that it can be performed in the presence of a background medium of constant ionic strength, the same medium which may later be be used for the determination of equilibrium constants. The electrode is assumed to obey a modified form of the Nernst equation.

$E=E^{0}+s\ln[H^{+}]\,$

E is a measured electrode potential, $E^{0}$ and s are calibration constants and $[H^{+}]$ is the hydrogen ion concentration. The parameter s is ideally equal to RT/F (59.1 mV at 298 K) but in practice may not be exactly equal to the ideal value. Note that the electrode response is assumed to be linear in hydrogen ion concentration (not activity). This assumption is valid when the ionic strength is effectively constant. Rearrangement of this equation gives

$[H^{+}]=10^{\frac {E^{0}-E}{s}}$

The analytical concentration of the hydrogen ion, $T_{H}$ in a titration of a strong acid, concentration $a_{H}$, with a strong base, concentration $b_{H}$ is given by

$(T_{H})_{i}={\frac {a_{H}v_{0}-b_{H}v_{i}}{v_{0}+v_{i}}}$

By convention the concentration of hydroxide ions is given as minus the corresponding concentration of hydrogen ions. Note that for this application the concentrations of acid and base must both be known. The concentrations should be directly related to primary standards.

As described above, in acid solutions with a pH of 5 or less, the concentration of hydoxide ions is negligible compared to the concentration of hydrogen ions, so $[H^{+}]=T_{H}$. Therefore

$(v_{0}+v_{i})10^{-{\frac {E^{0}-E}{s}}}=a_{H}v_{0}-b_{H}v_{i}$

and a plot of the function on the left-hand side, sometimes called the Gran function, against titre, $v_{i}$ should be a straight line cutting the x-axis at the (acid) equivalence point. The parameters $E^{0}$ and s can be obtained by the method of least squares. By a similar argument, with alkaline solutions of pH 9 or above the left-hand side of the equation

$(v_{0}+v_{i})10^{-{\frac {E-E^{0}}{s}}pK_{w}}=a_{H}v_{0}-b_{H}v_{i}$

will yield a straight line with respect to titre, cutting tha axis at the (alkaline) equivalence point. The difference between the two equivalence points can be used to calculate a carbonate content value.

A computer program, GLEE, implementing this approach to electrode calibration is available. GLass Electrode Evaluation

References

• Buffle, J., Parthasarathy, N. and Monnier, D. (1972): Errors in the Gran addition method. Part I. Theoretical Calculation of Statistical Errors; Anal. Chim. Acta 59, 427-438; Buffle, J. (1972): Anal. Chim. Acta 59, 439.
  
• Butler, J. N. (1991): Carbon Dioxide Equilibria and Their Applications; CRC Press: Boca Raton, FL.
  
• Butler, J. N. (1998): Ionic Equilibrium: Solubility and pH Calculations; Wiley-Interscience. Chap. 3.
  
• Gran, G. (1950): Determination of the equivalence point in potentiometric titrations, Acta Chemica Scandinavica, 4, 559-577.
  
• Gran, G. (1952): Determination of the equivalence point in potentiometric titrations-- Part II, Analyst, 77, 661-671.
  
• Gran, G., Johansson, A. and Johansson, S. (1981): Automatic Titration by Stepwise Addition of Equal Volumes of Titrant Part VII. Potentiometric Precipitation Titrations, Analyst, 106, 1109-1118.
  
• Harris, D. C.: Quantitative Chemical Analysis, 5th Ed.; W.H. Freeman & Co., New. York, NY, 1998.
  
• Martell, A. E. and Motekaitis, R. J.: The determination and use of stability constants, Wiley-VCH, 1992.
  
• Skoog, D. A., West, D. M., Holler, F. J. and Crouch, S. R. (2003): Fundamentals of Analytical Chemistry: An Introduction, 8th Ed., Brooks and Cole, Chap. 37.
  
• Stumm, W. and Morgan, J. J. (1981): Aquatic chemistry, 2nd Ed.; John Wiley & Sons, New York.