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 The graphing and visual estimation of the end point have been replaced by more accurate least-squares analyses since the advent of modern computers and enabling software packages, especially spreadsheet programs with built-in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

Basis of the calculations The Gran plot is based on the Nernst equation which can be written as

E = E 0 + s log ⁡ { H + } {\displaystyle E=E^{0}+s\log\{H^{+}\}}

where E is a measured electrode potential, E0 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 E − E 0 s o r [ H + ] = 10 − p H {\displaystyle [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, Ci and the amount of alkali added during titration.

[ H + ] = C i v i − c O H v v i + v {\displaystyle [H^{+}]={\frac {C_{i}v_{i}-c_{OH}v}{v_{i}+v}}}

where vi is the initial volume of solution, cOH 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 O H v = ( v i + v ) 10 E − E 0 s o r = ( v i + v ) 10 − p H {\displaystyle 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 E − E 0 s o r ( v i + v ) 10 − p H {\displaystyle (v_{i}+v)10^{\frac {E-E^{0}}{s}}\ or\ (v_{i}+v)10^{-pH}} against v will be a straight line. If E0 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 O H v {\displaystyle C_{i}v_{i}=c_{OH}v} . Alternatively, this plot can be used for electrode calibration by finding the values of E0 and s that give the best straight line.

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