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 The graphing and visual estimation of the end point have been replaced by more accurate least-squares analyses since the advent of modern computers and enabling software packages, especially spreadsheet programs with built-in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

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.

Titrating strong acid with strong base

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.

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

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

The Figure at right gives sample Gran plots.

Click on the image to view in full size.
The data used to create the plots are from Butler (1998). Note that the filled circles indicate the data points included in the least-squares computations to give the fitted dashed lines.

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, the offset was not insignificant, at -0.054 pH units.

The x-intercepts are unaffected.

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 a poorly calibrated electrode, an insufficient equilibration time before recording the electrode reading, low data densities in the linear regions, the presence of titratable impurities (see next section), incomplete precipitation in potentiometric titrations of dilute solutions, and drifts in ionic strength. 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 next section), and the effect of a drifting ionic strength. In the sample plots displayed in the Figure, 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.

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 above. 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 each titration to estimate the multiple end points and measure the total alkalinity and the carbonate and/or bicarbonate contents.

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. Usually, the only result of interest is the equivalence point. However, the before-equivalence slope could be used to assess the solubility product $K_{sp^{}}$ in the same way as $K_{w^{}}$ can be determined from acid-base titrations.

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

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

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

$[HA]_{i}\approx {\frac {v_{0}[HA]_{0}-v_{i}[OH^{-}]_{0}}{v_{0}+v_{i}}}$  before equivalence and

$[A^{-}]_{i}\approx {\frac {v_{i}[OH^{-}]_{0}}{v_{0}+v_{i}}}$  after equivalence,

which are valid approximations in the linear regions away from equivalence, 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 region and an extrapolated x-intercept $v_{e^{}}$, from which either $[HA]_{0^{}}$ or $[OH^{-}]_{0^{}}$ can be computed.

General References

• 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, The Analyst, 77, 661-671.
  
• 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.