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Micron Document

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Gran plot

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A Gran plot (also known as Gran titration or the Gran method) is a common means of standardizing a <a href="Titrate" class="mw-redirect" title="Titrate">titrate</a> or <a href="Titrant" class="mw-redirect" title="Titrant">titrant</a> by estimating the equivalence volume or <a href="Endpoint_(chemistry)" class="mw-redirect" title="Endpoint (chemistry)">end point</a> in a strong <a href="Acid" title="Acid">acid</a>-strong <a href="Base_(chemistry)" title="Base (chemistry)">base</a> <a href="Titration" title="Titration">titration</a> or in a <a href="Potentiometric_titration" title="Potentiometric titration">potentiometric titration</a>. Such plots have been also used to calibrate glass electrodes, to estimate the carbonate content of aqueous solutions, and to estimate the Ka values (<a href="Acid_dissociation_constant" title="Acid dissociation constant">acid dissociation constants</a>) of weak acids and bases from titration data. Gran plots are named after Swedish chemist Gunnar Gran, who developed the method in 1950.<a href="#cite_note-1">[1]</a>

Gran plots use linear approximations of the a priori non-linear relationships between the measured quantity, <a href="PH" title="PH">pH</a> or <a href="Electromotive_force" title="Electromotive force">electromotive potential</a> (emf), and the titrant volume. Other types of concentration measures, such as spectrophotometric <a href="Absorbance" title="Absorbance">absorbances</a> or NMR <a href="Chemical_shift" title="Chemical shift">chemical shifts</a>, 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-<a href="Derivative" title="Derivative">derivative</a> 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 The graphing and visual estimation of the end point have been replaced by more accurate <a href="Least-squares" class="mw-redirect" title="Least-squares">least-squares</a> analyses since the advent of modern computers and enabling software packages, especially spreadsheet programs with built-in least-squares functionality.

<div class="mw-heading mw-heading2"><h2 id="Basis_of_the_calculations">Basis of the calculations</h2></div>

The Gran plot is based on the <a href="Nernst_equation" title="Nernst equation">Nernst equation</a> which can be written as

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<annotation encoding="application/x-tex">{\displaystyle E=E^{0}+s\log\{H^{+}\}}</annotation>

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</math><img src="./aaed674cf588fa685d73b11b4561076b9dd70e08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.951ex; height:3.176ex;" alt="{\displaystyle E=E^{0}+s\log\{H^{+}\}}" loading="lazy"></dd></dl>

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

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<annotation encoding="application/x-tex">{\displaystyle [H^{+}]=10^{\frac {E-E^{0}}{s}}\ or\ [H^{+}]=10^{-pH}}</annotation>

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</math><img src="./54cc2b0dad52a2a38d4eae5d61668859a598aff0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.753ex; height:4.343ex;" alt="{\displaystyle [H^{+}]=10^{\frac {E-E^{0}}{s}}\ or\ [H^{+}]=10^{-pH}}" loading="lazy"></dd></dl>

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.

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<annotation encoding="application/x-tex">{\displaystyle [H^{+}]={\frac {C_{i}v_{i}-c_{OH}v}{v_{i}+v}}}</annotation>

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</math><img src="./4fa60de2bc0b02925c56225af1ad7af24904981e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:21.152ex; height:5.676ex;" alt="{\displaystyle [H^{+}]={\frac {C_{i}v_{i}-c_{OH}v}{v_{i}+v}}}" loading="lazy"></dd></dl>

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

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<annotation encoding="application/x-tex">{\displaystyle C_{i}v_{i}-c_{OH}v=(v_{i}+v)10^{\frac {E-E^{0}}{s}}\ or\ =(v_{i}+v)10^{-pH}}</annotation>

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</math><img src="./37d866c7141268570feb48b3965d273e2451fce6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.656ex; height:4.343ex;" alt="{\displaystyle C_{i}v_{i}-c_{OH}v=(v_{i}+v)10^{\frac {E-E^{0}}{s}}\ or\ =(v_{i}+v)10^{-pH}}" loading="lazy"></dd></dl>

A plot of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (v_{i}+v)10^{\frac {E-E^{0}}{s}}\ or\ (v_{i}+v)10^{-pH}}">

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<annotation encoding="application/x-tex">{\displaystyle (v_{i}+v)10^{\frac {E-E^{0}}{s}}\ or\ (v_{i}+v)10^{-pH}}</annotation>

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</math><img src="./ae129159fdbb46916e26051432083cb2854a95f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.15ex; height:4.343ex;" alt="{\displaystyle (v_{i}+v)10^{\frac {E-E^{0}}{s}}\ or\ (v_{i}+v)10^{-pH}}" loading="lazy"> 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, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C_{i}v_{i}=c_{OH}v}">

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<annotation encoding="application/x-tex">{\displaystyle C_{i}v_{i}=c_{OH}v}</annotation>

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</math><img src="./5b46b01356e37161daaf40c8b011539df533837b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.567ex; height:2.509ex;" alt="{\displaystyle C_{i}v_{i}=c_{OH}v}" loading="lazy">. Alternatively, this plot can be used for electrode calibration by finding the values of E0 and s that give the best straight line.

<div class="mw-heading mw-heading2"><h2 id="Titrating_strong_acid_with_strong_base">Titrating strong acid with strong base</h2></div>

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

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<annotation encoding="application/x-tex">{\displaystyle K_{w^{}}=[H^{+}]_{i}[OH^{-}]_{i}}</annotation>

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</math><img src="./1db0b94c20344944dea92ee4878afb135017aaef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:19.669ex; height:3.009ex;" alt="{\displaystyle K_{w^{}}=[H^{+}]_{i}[OH^{-}]_{i}}" loading="lazy"></dd></dl>

where Kw is the water <a href="Self-ionization_of_water" title="Self-ionization of water">autoprotolysis</a> constant.

If titrating an acid of initial volume <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{0^{}}}">

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<annotation encoding="application/x-tex">{\displaystyle v_{0^{}}}</annotation>

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</math><img src="./831145d1e93ea5ed1ade316301b4529b3b843ba2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.182ex; height:2.009ex;" alt="{\displaystyle v_{0^{}}}" loading="lazy"> and concentration <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [H^{+}]_{0^{}}}">

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<annotation encoding="application/x-tex">{\displaystyle [H^{+}]_{0^{}}}</annotation>

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</math><img src="./1ad86c50bb5c87cbc6ce8e6140089390b1797c64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.962ex; height:3.009ex;" alt="{\displaystyle [H^{+}]_{0^{}}}" loading="lazy"> with base of concentration <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [OH^{-}]_{0^{}}}">

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<annotation encoding="application/x-tex">{\displaystyle [OH^{-}]_{0^{}}}</annotation>

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</math><img src="./349ebe8d364545dc81a278f8396263ae21b7c12b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.735ex; height:3.009ex;" alt="{\displaystyle [OH^{-}]_{0^{}}}" loading="lazy">, then at any i'th point in the titration with titrant volume <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{i^{}}}">

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<annotation encoding="application/x-tex">{\displaystyle v_{i^{}}}</annotation>

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<dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\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}}}">

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<mtext> (acidic region)</mtext>

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<mtext> (equivalence point)</mtext>

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<mtext> (alkaline region)</mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\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}}}</annotation>

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</math><img src="./dc9534301f6128c37f6a45e275d7beadb97a34bb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:100.455ex; height:9.176ex;" alt="{\displaystyle {\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}}}" loading="lazy"></dd></dl>

At the <a href="Equivalence_point" title="Equivalence point">equivalence point</a>, the equivalence volume <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{e^{}}=v_{i^{}}}">

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<annotation encoding="application/x-tex">{\displaystyle v_{e^{}}=v_{i^{}}}</annotation>

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</math><img src="./69a989bd50388a4d8b2edba95d8344a9db0b43de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.152ex; height:2.009ex;" alt="{\displaystyle v_{e^{}}=v_{i^{}}}" loading="lazy">.

Thus,

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<annotation encoding="application/x-tex">{\displaystyle ({v_{0}+v_{i}})10^{-pH_{i}}{\text{ vs. }}v_{i^{}}}</annotation>

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</math><img src="./931991f6f9bc6ebd618ea235b70d10bf6c966cb9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.291ex; height:3.176ex;" alt="{\displaystyle ({v_{0}+v_{i}})10^{-pH_{i}}{\text{ vs. }}v_{i^{}}}" loading="lazy"> will have a linear region before equivalence, with slope <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -[OH^{-}]_{0^{}}}">

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<annotation encoding="application/x-tex">{\displaystyle -[OH^{-}]_{0^{}}}</annotation>

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</math><img src="./30990190f192ab893969327d787e617348fe2441.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.544ex; height:3.009ex;" alt="{\displaystyle -[OH^{-}]_{0^{}}}" loading="lazy"></li>

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<annotation encoding="application/x-tex">{\displaystyle ({v_{0}+v_{i}})10^{pH_{i}}{\text{ vs. }}v_{i^{}}}</annotation>

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</math><img src="./754ab0af824b04294ade6663ff6de8eb16ae54d7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.013ex; height:3.176ex;" alt="{\displaystyle ({v_{0}+v_{i}})10^{pH_{i}}{\text{ vs. }}v_{i^{}}}" loading="lazy"> will have a linear region after equivalence, with slope <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [OH^{-}]_{0}/K_{w^{}}}">

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<annotation encoding="application/x-tex">{\displaystyle [OH^{-}]_{0}/K_{w^{}}}</annotation>

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</math><img src="./af930d58632586406096e06ab513b848df232ae7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.28ex; height:3.009ex;" alt="{\displaystyle [OH^{-}]_{0}/K_{w^{}}}" loading="lazy"></li>

<li>both plots will have <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{e}=v_{0}[H^{+}]_{0}/[OH^{-}]_{0^{}}}">

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<annotation encoding="application/x-tex">{\displaystyle v_{e}=v_{0}[H^{+}]_{0}/[OH^{-}]_{0^{}}}</annotation>

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The equivalence volume is used to compute whichever of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [H^{+}]_{0^{}}}">

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<annotation encoding="application/x-tex">{\displaystyle [H^{+}]_{0^{}}}</annotation>

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</math><img src="./1ad86c50bb5c87cbc6ce8e6140089390b1797c64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.962ex; height:3.009ex;" alt="{\displaystyle [H^{+}]_{0^{}}}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [OH^{-}]_{0^{}}}">

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<annotation encoding="application/x-tex">{\displaystyle [OH^{-}]_{0^{}}}</annotation>

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The pH meter is usually calibrated with buffer solutions at known pH values before starting the titration. The <a href="Ionic_strength" title="Ionic strength">ionic strength</a> 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 <a href="Activity_coefficient" title="Activity coefficient">activity quotient</a> can be computed.<a href="#cite_note-mudrift-2">[2]</a>

<div class="mw-heading mw-heading2"><h2 id="Titrating_strong_base_with_strong_acid">Titrating strong base with strong acid</h2></div>

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

<dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\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}}}">

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<mtext> (alkaline region)</mtext>

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<mtext> (equivalence point)</mtext>

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<mtext> (acidic region)</mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\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}}}</annotation>

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</math><img src="./34d2024e286f5db906831a2e8c22a87571c2eb0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:96.579ex; height:9.176ex;" alt="{\displaystyle {\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}}}" loading="lazy"></dd></dl>

Hence,

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<annotation encoding="application/x-tex">{\displaystyle ({v_{0}+v_{i}})10^{pH_{i}}{\text{ vs. }}v_{i^{}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle -[H^{+}]_{0}/K_{w^{}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle ({v_{0}+v_{i}})10^{-pH_{i}}{\text{ vs. }}v_{i^{}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle [H^{+}]_{0^{}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle v_{e}=v_{0}[OH^{-}]_{0^{}}/[H^{+}]_{0}}</annotation>

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Figure 1 gives sample Gran plots of a strong base-strong acid titration.

<div class="mw-heading mw-heading2"><h2 id="Concentrations_and_dissociation_constants_of_weak_acids">Concentrations and dissociation constants of weak acids</h2></div>

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

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<annotation encoding="application/x-tex">{\displaystyle K_{a}={\frac {[H^{+}]_{i}[A^{-}]_{i}}{[HA]_{i}}}}</annotation>

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we have at any i'th point in the titration of a volume <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{0}}">

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</math><img src="./528d40dd3e1f5de41850626c0b99d0a7f538f8ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.735ex; height:3.009ex;" alt="{\displaystyle [OH^{-}]_{0}}" loading="lazy">. In the linear regions away from equivalence,

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<annotation encoding="application/x-tex">{\displaystyle [HA]_{i}\approx {\frac {v_{0}[HA]_{0}-v_{i}[OH^{-}]_{0}}{v_{0}+v_{i}}}}</annotation>

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</math><img src="./19224a237e91bfe16ce79a2973ddb89dbba5f4c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:30.674ex; height:6.009ex;" alt="{\displaystyle [HA]_{i}\approx {\frac {v_{0}[HA]_{0}-v_{i}[OH^{-}]_{0}}{v_{0}+v_{i}}}}" loading="lazy">  and</dd>

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<annotation encoding="application/x-tex">{\displaystyle [A^{-}]_{i}\approx {\frac {v_{i}[OH^{-}]_{0}}{v_{0}+v_{i}}}}</annotation>

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are valid approximations, whence

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<annotation encoding="application/x-tex">{\displaystyle K_{a}\approx {\frac {10^{-pH_{i}}v_{i}[OH^{-}]_{0}}{v_{0}[HA]_{0}-v_{i}[OH^{-}]_{0}}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle K_{a}(v_{0}{\frac {[HA]_{0}}{[OH^{-}]_{0}}}-v_{i})\approx 10^{-pH_{i}}v_{i}}</annotation>

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</math><img src="./77dc2909198804c04da9d24a259197c9d439d359.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:32.084ex; height:6.509ex;" alt="{\displaystyle K_{a}(v_{0}{\frac {[HA]_{0}}{[OH^{-}]_{0}}}-v_{i})\approx 10^{-pH_{i}}v_{i}}" loading="lazy"> or, because <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{e}=v_{0}{\frac {[HA]_{0}}{[OH^{-}]_{0}}}}">

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<annotation encoding="application/x-tex">{\displaystyle v_{e}=v_{0}{\frac {[HA]_{0}}{[OH^{-}]_{0}}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle K_{a}(v_{e}-v_{i})\approx 10^{-pH_{i}}v_{i}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle 10^{-pH_{i}}v_{i}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle [OH^{-}]_{0^{}}}</annotation>

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</math><img src="./349ebe8d364545dc81a278f8396263ae21b7c12b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.735ex; height:3.009ex;" alt="{\displaystyle [OH^{-}]_{0^{}}}" loading="lazy"> can be computed.<a href="#cite_note-mudrift-2">[2]</a> The alkaline region is treated in the same manner as for a <a href="#Titrating_strong_acid_with_strong_base">titration of strong acid</a>. 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).

<div class="mw-heading mw-heading2"><h2 id="Carbonate_content">Carbonate content</h2></div>

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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{0^{}}}">

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<annotation encoding="application/x-tex">{\displaystyle v_{0^{}}}</annotation>

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</math><img src="./831145d1e93ea5ed1ade316301b4529b3b843ba2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.182ex; height:2.009ex;" alt="{\displaystyle v_{0^{}}}" loading="lazy"> of titrate is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (v_{e}-v_{e}^{\prime })[H^{+}]_{0^{}}=2v_{0}[CO_{2}]_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle (v_{e}-v_{e}^{\prime })[H^{+}]_{0^{}}=2v_{0}[CO_{2}]_{0}}</annotation>

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</math><img src="./cd551caef6ac3c109713851f5b6345c2fcaffeb2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:28.248ex; height:3.009ex;" alt="{\displaystyle (v_{e}-v_{e}^{\prime })[H^{+}]_{0^{}}=2v_{0}[CO_{2}]_{0}}" loading="lazy">. In the opposite case of a titration of acid by base, the carbonate content is similarly computed from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (v_{e}^{\prime }-v_{e})[OH^{-}]_{0^{}}=2v_{e}^{\prime }[CO_{2}]_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle (v_{e}^{\prime }-v_{e})[OH^{-}]_{0^{}}=2v_{e}^{\prime }[CO_{2}]_{0}}</annotation>

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</math><img src="./cb1956588a4c55722d610e836490ea8ea159e414.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:29.966ex; height:3.009ex;" alt="{\displaystyle (v_{e}^{\prime }-v_{e})[OH^{-}]_{0^{}}=2v_{e}^{\prime }[CO_{2}]_{0}}" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{e}^{\prime }}">

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<annotation encoding="application/x-tex">{\displaystyle v_{e}^{\prime }}</annotation>

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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.

<div class="mw-heading mw-heading2"><h2 id="Potentiometric_monitoring_of_H+">Potentiometric monitoring of H+</h2></div>

To use potentiometric (e.m.f.) measurements <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{i^{}}}">

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<annotation encoding="application/x-tex">{\displaystyle E_{i^{}}}</annotation>

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</math><img src="./a094c9ae0353e63c8fb31b50bb78f156d4243609.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.515ex; height:2.509ex;" alt="{\displaystyle E_{i^{}}}" loading="lazy"> in monitoring the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{+_{}}}">

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<annotation encoding="application/x-tex">{\displaystyle H^{+_{}}}</annotation>

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</math><img src="./f2fea756e3b80c895d39da4420375bcc89e5602c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.614ex; height:2.509ex;" alt="{\displaystyle H^{+_{}}}" loading="lazy"> concentration in place of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle pH_{i^{}}}">

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<annotation encoding="application/x-tex">{\displaystyle pH_{i^{}}}</annotation>

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</math><img src="./acd33cb50acd060b21137107a4f92e729e8df4ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:3.99ex; height:2.509ex;" alt="{\displaystyle pH_{i^{}}}" loading="lazy"> readings, one can trivially set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -log_{10}[H^{+}]_{i}=b_{0}-b_{1}E_{i^{}}}">

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<annotation encoding="application/x-tex">{\displaystyle -log_{10}[H^{+}]_{i}=b_{0}-b_{1}E_{i^{}}}</annotation>

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</math><img src="./70741e6ba1b626302187955e8b1d32ce102eba20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.879ex; height:3.009ex;" alt="{\displaystyle -log_{10}[H^{+}]_{i}=b_{0}-b_{1}E_{i^{}}}" loading="lazy"> and apply the same equations as above, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{0^{}}}">

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<annotation encoding="application/x-tex">{\displaystyle b_{0^{}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle nFE_{0^{}}/RT}</annotation>

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</math><img src="./0cb8e652b27a3fdbe996e625f9527e8c388f9216.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.468ex; height:2.843ex;" alt="{\displaystyle nFE_{0^{}}/RT}" loading="lazy">, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{1^{}}}">

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<annotation encoding="application/x-tex">{\displaystyle b_{1^{}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle nF^{_{}}/RT}</annotation>

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</math><img src="./db63716c393548dbd26072c6ce9256f7343c8dbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.004ex; height:2.843ex;" alt="{\displaystyle nF^{_{}}/RT}" loading="lazy"> (1/59.2 pH units/mV at 25°C), such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -b_{1}E_{i^{}}}">

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<annotation encoding="application/x-tex">{\displaystyle -b_{1}E_{i^{}}}</annotation>

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</math><img src="./787e6d340e47fd032ce0b6740703199bdeaea9ca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.375ex; height:2.509ex;" alt="{\displaystyle -b_{1}E_{i^{}}}" loading="lazy"> replaces <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle pH_{i^{}}}">

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<annotation encoding="application/x-tex">{\displaystyle pH_{i^{}}}</annotation>

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Thus, as before for a titration of strong acid by strong base,

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<annotation encoding="application/x-tex">{\displaystyle ({v_{0}+v_{i}})10^{b_{1}E_{i}}}</annotation>

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</math><img src="./732762c101255d919eadc6a0c1b3a8beb7bc52eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:14.691ex; height:3.176ex;" alt="{\displaystyle ({v_{0}+v_{i}})10^{b_{1}E_{i}}}" loading="lazy"> vs. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{i^{}}}">

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<annotation encoding="application/x-tex">{\displaystyle v_{i^{}}}</annotation>

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</math><img src="./9a000888a46ec71b926d34217aff45ff0a0c7d9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.927ex; height:2.009ex;" alt="{\displaystyle v_{i^{}}}" loading="lazy"> will have a linear region before equivalence, with slope <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -10^{b_{0}}[OH^{-}]_{0^{}}}">

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<annotation encoding="application/x-tex">{\displaystyle -10^{b_{0}}[OH^{-}]_{0^{}}}</annotation>

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</math><img src="./d8a85a80049e3f6c5297bbf42f8deccbce01821c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.638ex; height:3.176ex;" alt="{\displaystyle -10^{b_{0}}[OH^{-}]_{0^{}}}" loading="lazy"></li>

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<annotation encoding="application/x-tex">{\displaystyle ({v_{0}+v_{i}})10^{-b_{1}E_{i}}}</annotation>

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</math><img src="./2ab7267c3784b988d5fccd8ca58daa23dd4161ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.969ex; height:3.176ex;" alt="{\displaystyle ({v_{0}+v_{i}})10^{-b_{1}E_{i}}}" loading="lazy"> vs. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{i^{}}}">

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<annotation encoding="application/x-tex">{\displaystyle v_{i^{}}}</annotation>

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</math><img src="./9a000888a46ec71b926d34217aff45ff0a0c7d9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.927ex; height:2.009ex;" alt="{\displaystyle v_{i^{}}}" loading="lazy"> will have a linear region after equivalence, with slope <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10^{-b_{0}}[OH^{-}]_{0}/K_{w^{}}}">

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<annotation encoding="application/x-tex">{\displaystyle 10^{-b_{0}}[OH^{-}]_{0}/K_{w^{}}}</annotation>

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</math><img src="./86b814278e54ca0355316067e831e769e5519e08.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.653ex; height:3.176ex;" alt="{\displaystyle 10^{-b_{0}}[OH^{-}]_{0}/K_{w^{}}}" loading="lazy"></li>

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<annotation encoding="application/x-tex">{\displaystyle v_{e}=v_{0}[H^{+}]_{0}/[OH^{-}]_{0^{}}}</annotation>

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</math><img src="./f5699e3929db80c1d0387b0fa7ec845a08cc88c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.266ex; height:3.009ex;" alt="{\displaystyle v_{e}=v_{0}[H^{+}]_{0}/[OH^{-}]_{0^{}}}" loading="lazy"> 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 <a href="#Carbonate_content">carbonate content</a></li></ul>

Analogous plots can be drawn using data from a titration of base by acid.

<div class="mw-heading mw-heading2"><h2 id="Electrode_calibration">Electrode calibration</h2></div>

Note that the above analysis requires prior knowledge of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{0^{}}}">

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<annotation encoding="application/x-tex">{\displaystyle b_{1^{}}}</annotation>

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If a pH electrode is not well calibrated, an offset correction can be computed in situ from the acid-side Gran slope:

<ul><li>For a titration of acid by base, the acid-side slope (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -[OH^{-}]_{0^{}}10^{b_{0}}}">

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<annotation encoding="application/x-tex">{\displaystyle -[OH^{-}]_{0^{}}10^{b_{0}}}</annotation>

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</math><img src="./c089bb758310a6f737e29853a0b075e980378b32.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.638ex; height:3.176ex;" alt="{\displaystyle -[OH^{-}]_{0^{}}10^{b_{0}}}" loading="lazy">) can serve to compute <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{0^{}}}">

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<annotation encoding="application/x-tex">{\displaystyle b_{0^{}}}</annotation>

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<annotation encoding="application/x-tex">{\displaystyle [OH^{-}]_{0^{}}}</annotation>

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</math><img src="./349ebe8d364545dc81a278f8396263ae21b7c12b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.735ex; height:3.009ex;" alt="{\displaystyle [OH^{-}]_{0^{}}}" loading="lazy"> or using the value given by the equivalence volume. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{w^{}}}">

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<annotation encoding="application/x-tex">{\displaystyle K_{w^{}}}</annotation>

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<li>For a titration of base by acid, as illustrated in the sample plots, the acid-side slope (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [H^{+}]_{0^{}}10^{b_{0}}}">

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<annotation encoding="application/x-tex">{\displaystyle [H^{+}]_{0^{}}10^{b_{0}}}</annotation>

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</math><img src="./fd321617413b9314c1a63743dfc2fdcf7b11eea1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.056ex; height:3.176ex;" alt="{\displaystyle [H^{+}]_{0^{}}10^{b_{0}}}" loading="lazy">) is similarly used to compute <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{0^{}}}">

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<annotation encoding="application/x-tex">{\displaystyle b_{0^{}}}</annotation>

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</math><img src="./75a16f0ecef25df7dea0168c329effc3ebfc1312.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.052ex; height:2.509ex;" alt="{\displaystyle b_{0^{}}}" loading="lazy"> and the base-side slope (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -[H^{+}]_{0^{}}10^{-b_{0}}/K_{w}}">

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<annotation encoding="application/x-tex">{\displaystyle -[H^{+}]_{0^{}}10^{-b_{0}}/K_{w}}</annotation>

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</math><img src="./0ead3ce9876a94a8bd46d7cfe73e707bc2e7707a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.687ex; height:3.176ex;" alt="{\displaystyle -[H^{+}]_{0^{}}10^{-b_{0}}/K_{w}}" loading="lazy">) is used to compute <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{w^{}}}">

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<annotation encoding="application/x-tex">{\displaystyle K_{w^{}}}</annotation>

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</math><img src="./a52e4c8cdaa2753bbeed0cc10ebcf87578b438e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.382ex; height:2.509ex;" alt="{\displaystyle K_{w^{}}}" loading="lazy"> using a known value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [H^{+}]_{0^{}}}">

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<annotation encoding="application/x-tex">{\displaystyle [H^{+}]_{0^{}}}</annotation>

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In the sample data illustrated in Figure 1, this offset correction was not insignificant, at -0.054 pH units.

The value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{1^{}}}">

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<annotation encoding="application/x-tex">{\displaystyle b_{1^{}}}</annotation>

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</math><img src="./a1136d1fc732ce8f632023e6084b9538b5db970b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.052ex; height:2.509ex;" alt="{\displaystyle b_{1^{}}}" loading="lazy">, however, may deviate from its theoretical value and can only be assessed by a proper calibration of the electrode. Calibration of an electrode is often performed using buffers of known pH, or by performing a titration of strong acid with strong base. In that case, a constant ionic strength can be maintained, and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [H^{+_{}}]_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle [H^{+_{}}]_{i}}</annotation>

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</math><img src="./becbf5d2e21654200c46fa31d0f2c3145403f992.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.708ex; height:3.009ex;" alt="{\displaystyle [H^{+_{}}]_{i}}" loading="lazy"> is known at all titration points if both <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [H_{}^{+}]_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle [H_{}^{+}]_{0}}</annotation>

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</math><img src="./650afe4f973213f6c9c51c2c92009f87a08a4372.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.735ex; height:3.009ex;" alt="{\displaystyle [OH_{}^{-}]_{0}}" loading="lazy"> are known (and should be directly related to <a href="Primary_standard" title="Primary standard">primary standards</a>). For instance, Martell and Motekaitis (1992) calculated the pH value expected at the start of the titration, having earlier titrated the acid and base solutions against primary standards, then adjusted the pH electrode reading accordingly, but this does not afford a slope correction if one is needed.

Based on earlier work by McBryde (1969), Gans and O'Sullivan (2000) describe an iterative approach to arrive at both <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{0^{}}}">

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<annotation encoding="application/x-tex">{\displaystyle -log_{10}[H^{+}]_{i}=b_{0}-b_{1}E_{i^{}}}</annotation>

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</math><img src="./75a16f0ecef25df7dea0168c329effc3ebfc1312.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.052ex; height:2.509ex;" alt="{\displaystyle b_{0^{}}}" loading="lazy"> is first estimated from the acidic data according to Rossotti and Rossotti (1965), and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{1^{}}}">

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</math><img src="./a1136d1fc732ce8f632023e6084b9538b5db970b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.052ex; height:2.509ex;" alt="{\displaystyle b_{1^{}}}" loading="lazy"> is initially taken to have its theoretical value;</li><li>modified Gran function plots are drawn, using <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({v_{0}+v_{i}})10^{b_{1}E_{i}-b_{0}}}">

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</math><img src="./de031ccff3c5cf32c6362777a1570bbc071a05c3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.506ex; height:3.176ex;" alt="{\displaystyle ({v_{0}+v_{i}})10^{b_{1}E_{i}-b_{0}}}" loading="lazy"> vs. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{i^{}}}">

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</math><img src="./9a000888a46ec71b926d34217aff45ff0a0c7d9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.927ex; height:2.009ex;" alt="{\displaystyle v_{i^{}}}" loading="lazy"> on the acidic side of equivalence and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({v_{0}+v_{i}})K_{w}10^{-b_{1}E_{i}+b_{0}}}">

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</math><img src="./3dd480b99dd172801e479ae23f096d32af75a265.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.167ex; height:3.176ex;" alt="{\displaystyle ({v_{0}+v_{i}})K_{w}10^{-b_{1}E_{i}+b_{0}}}" loading="lazy"> vs. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{i^{}}}">

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</math><img src="./9a000888a46ec71b926d34217aff45ff0a0c7d9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.927ex; height:2.009ex;" alt="{\displaystyle v_{i^{}}}" loading="lazy"> on the alkaline side, and the equivalence volumes <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{e^{}}}">

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</math><img src="./f43a1a147f239d8a49e49f5df248cda706d8374b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.126ex; height:2.009ex;" alt="{\displaystyle v_{e^{}}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{e}^{\prime }}">

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</math><img src="./884b017ad977eb14220a52a104b918d76760829d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.126ex; height:2.509ex;" alt="{\displaystyle v_{e}^{\prime }}" loading="lazy"> are computed therefrom, as before;</li><li>as before, the difference in the equivalence volumes is used to compute the <a href="#Carbonate_content">carbonate content</a> but also to calculate an 'effective base concentration' <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{e}[OH^{-}]_{0^{}}/v_{e}^{\prime }}">

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</math><img src="./b5a35c98ffe984a3ac4995bf249effd05b1fed3e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.15ex; height:3.009ex;" alt="{\displaystyle v_{e}[OH^{-}]_{0^{}}/v_{e}^{\prime }}" loading="lazy"> for the alkaline side of equivalence;</li><li>approximate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [H^{+}]_{i^{}}}">

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<annotation encoding="application/x-tex">{\displaystyle (v_{0^{}}[H^{+}]_{0}-v_{i}[OH^{-}]_{0})/(v_{0}+v_{i})}</annotation>

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</math><img src="./4dc107a833fc669533394df6901c516255bd6e46.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.378ex; height:3.009ex;" alt="{\displaystyle (v_{0^{}}[H^{+}]_{0}-v_{i}[OH^{-}]_{0})/(v_{0}+v_{i})}" loading="lazy"> and on the alkaline side as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (v_{e}[H^{+}]_{0}-v_{i}v_{e}[OH^{-}]_{0}/v_{e}^{\prime })/(v_{0}+v_{i})}">

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</math><img src="./721aaf6302494a767750d0c7c4f176e14345f69e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:37.736ex; height:3.009ex;" alt="{\displaystyle (v_{e}[H^{+}]_{0}-v_{i}v_{e}[OH^{-}]_{0}/v_{e}^{\prime })/(v_{0}+v_{i})}" loading="lazy">;</li><li>the initial definition <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -log_{10}[H^{+}]_{i}=b_{0}-b_{1}E_{i^{}}}">

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</math><img src="./70741e6ba1b626302187955e8b1d32ce102eba20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:24.879ex; height:3.009ex;" alt="{\displaystyle -log_{10}[H^{+}]_{i}=b_{0}-b_{1}E_{i^{}}}" loading="lazy"> is rewritten as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{i^{}}=b_{0}/b_{1}+(1/b_{1})log_{10}[H^{+}]_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle E_{i^{}}=b_{0}/b_{1}+(1/b_{1})log_{10}[H^{+}]_{i}}</annotation>

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</math><img src="./81d60b752e3a97207ac1564c74b77021bb2b8748.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.42ex; height:3.009ex;" alt="{\displaystyle E_{i^{}}=b_{0}/b_{1}+(1/b_{1})log_{10}[H^{+}]_{i}}" loading="lazy">, and the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle E_{i^{}}}">

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</math><img src="./a094c9ae0353e63c8fb31b50bb78f156d4243609.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.515ex; height:2.509ex;" alt="{\displaystyle E_{i^{}}}" loading="lazy"> data are plotted against <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle log_{10}[H^{+}]_{i^{}}}">

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</math><img src="./aa262e87a4d416162b2e3ce1d9c18980afb6f1f8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.514ex; height:3.009ex;" alt="{\displaystyle log_{10}[H^{+}]_{i^{}}}" loading="lazy">, using those <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [H^{+}]_{i^{}}}">

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</math><img src="./bcac7a06f0f699d71340713329cd33e33c623abb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.708ex; height:3.009ex;" alt="{\displaystyle [H^{+}]_{i^{}}}" loading="lazy"> 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [OH^{-}]_{i^{}}}">

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</math><img src="./b550c62051cba4fe4c8b63f03fe0412520b5a78c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.481ex; height:3.009ex;" alt="{\displaystyle [OH^{-}]_{i^{}}}" loading="lazy"> on the acid side and the neglect of the carbonate/bicarbonate equilibrium on the alkaline side); a linear least-squares treatment provides <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 1/b_{1^{}}}">

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</math><img src="./9e08087f7d27262aaa661cb0660a26bd1dc789ec.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.377ex; height:2.843ex;" alt="{\displaystyle 1/b_{1^{}}}" loading="lazy"> as slope and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{0}/b_{1^{}}}">

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</math><img src="./9ae17943b0bbd5a74bb76e13c02a44850cf8e2dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.266ex; height:2.843ex;" alt="{\displaystyle b_{0}/b_{1^{}}}" loading="lazy"> as intercept;</li><li>steps 2 and 3 are repeated with the new <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{0^{}}}">

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The procedure could in principle be modified for titrations of base by acid. A computer program named <a rel="nofollow" class="external text" href="http://www.hyperquad.co.uk/glee.htm">GLEE</a> (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 <a href="Ionic_strength" title="Ionic strength">ionic strength</a> which may later be used for the <a href="Determination_of_equilibrium_constants" title="Determination of equilibrium constants">determination of equilibrium constants</a>.

Note that the regular Gran functions will provide the required equivalence volumes and, as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{1^{}}}">

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</math><img src="./75a16f0ecef25df7dea0168c329effc3ebfc1312.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.052ex; height:2.509ex;" alt="{\displaystyle b_{0^{}}}" loading="lazy"> 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 CO2 content and can indeed be combined with a complete standardization of the base, using the definition of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{e^{}}}">

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</math><img src="./f43a1a147f239d8a49e49f5df248cda706d8374b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.126ex; height:2.009ex;" alt="{\displaystyle v_{e^{}}}" loading="lazy"> to compute <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [OH^{-}]_{0^{}}}">

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</math><img src="./349ebe8d364545dc81a278f8396263ae21b7c12b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.735ex; height:3.009ex;" alt="{\displaystyle [OH^{-}]_{0^{}}}" loading="lazy">. Finally, the usable pH range could be extended by solving the quadratic <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (v_{0{^{}}}[H^{+}]_{0}-v_{i}[OH^{-}]_{0})/(v_{0}+v_{i})=[H^{+}]_{i}-K_{w}/[H^{+}]_{i}}">

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<annotation encoding="application/x-tex">{\displaystyle (v_{0{^{}}}[H^{+}]_{0}-v_{i}[OH^{-}]_{0})/(v_{0}+v_{i})=[H^{+}]_{i}-K_{w}/[H^{+}]_{i}}</annotation>

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<div class="mw-heading mw-heading2"><h2 id="Potentiometric_monitoring_of_other_species">Potentiometric monitoring of other species</h2></div>

Potentiometric data are also used to monitor species other than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{+_{}}}">

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</math><img src="./129a2e9e888bf4d56c6fe90c70f4dfaaf2ec65fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.576ex; height:2.676ex;" alt="{\displaystyle S^{1_{}}}" loading="lazy"> by potentiometry, one can apply the same formalism with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -log_{10}[S^{1}]_{i}=b_{0}-b_{1}E_{i^{}}}">

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</math><img src="./bdcdb2dbf4bb1e3adae9a8d420ee793d276f39b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.841ex; height:3.176ex;" alt="{\displaystyle -log_{10}[S^{1}]_{i}=b_{0}-b_{1}E_{i^{}}}" loading="lazy">. Thus, a titration of a solution of another species <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{2_{}}}">

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</math><img src="./2b825e77bf32ed8dc13993f2b8fa3bbee1614461.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.576ex; height:2.676ex;" alt="{\displaystyle S^{2_{}}}" loading="lazy"> by species <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{1_{}}}">

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</math><img src="./129a2e9e888bf4d56c6fe90c70f4dfaaf2ec65fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.576ex; height:2.676ex;" alt="{\displaystyle S^{1_{}}}" loading="lazy"> is analogous to a pH-monitored titration of base by acid, where either <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle ({v_{0}+v_{i}})10^{b_{1}E_{i}}}">

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</math><img src="./c7cb445111bc6469dc5c85492d9997f48d3ca1c9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.192ex; height:3.176ex;" alt="{\displaystyle v_{0}[S^{0}]_{0}/[S^{1}]_{0^{}}}" loading="lazy">. In the opposite titration of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{1_{}}}">

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</math><img src="./129a2e9e888bf4d56c6fe90c70f4dfaaf2ec65fd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.576ex; height:2.676ex;" alt="{\displaystyle S^{1_{}}}" loading="lazy"> by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S^{2_{}}}">

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</math><img src="./8795d1a16a8e7bad358482bacee448b6cfd1f5c0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.192ex; height:3.176ex;" alt="{\displaystyle v_{0}[S^{1}]_{0}/[S^{0}]_{0^{}}}" loading="lazy">. 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 <a href="Solubility_product" class="mw-redirect" title="Solubility product">solubility product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{sp^{}}}">

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</math><img src="./cef2092d5d36ce2ef2e75bee23c8250a2e6b0ef8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.803ex; height:2.843ex;" alt="{\displaystyle K_{sp^{}}}" loading="lazy"> in the same way as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{w^{}}}">

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<annotation encoding="application/x-tex">{\displaystyle K_{w^{}}}</annotation>

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</math><img src="./a52e4c8cdaa2753bbeed0cc10ebcf87578b438e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.382ex; height:2.509ex;" alt="{\displaystyle K_{w^{}}}" loading="lazy"> can be determined from acid-base titrations, although other ion-pair association interactions may be occurring as well.<a href="#cite_note-GranVII-3">[3]</a>

To illustrate, consider a titration of Cl− by Ag+ monitored potentiometrically:

<dl><dd><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\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}}}">

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<mtext> (before equivalence)</mtext>

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<mtext> (equivalence point)</mtext>

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<mtext> (after equivalence)</mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\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}}}</annotation>

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</math><img src="./911677d9d43bd0d8faee7d987a44dde0c690a06e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.787ex; margin-bottom: -0.218ex; width:98.988ex; height:9.176ex;" alt="{\displaystyle {\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}}}" loading="lazy"></dd></dl>

Hence,

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</math><img src="./e148646841e0ae5f8455b20864e31b9f8d6c84ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.848ex; height:3.176ex;" alt="{\displaystyle ({v_{0}+v_{i}})10^{-b_{1}E_{i}}{\text{ vs. }}v_{i^{}}}" loading="lazy"> will have a linear region before equivalence, with slope <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -[Ag^{+}]_{0}10^{-b_{0}}/K_{sp^{}}}">

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</math><img src="./9b9c5abe2ea483461e18a92ca2d2da8f1f748219.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.867ex; height:3.343ex;" alt="{\displaystyle -[Ag^{+}]_{0}10^{-b_{0}}/K_{sp^{}}}" loading="lazy"></li>

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</math><img src="./4d40ca066c554d1bc1ef31bfb6d0c53a8a2e0d05.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.57ex; height:3.176ex;" alt="{\displaystyle ({v_{0}+v_{i}})10^{b_{1}E_{i}}{\text{ vs. }}v_{i^{}}}" loading="lazy"> will have a linear region after equivalence, with slope <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [Ag^{+}]_{0^{}}10^{b_{0}}}">

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<annotation encoding="application/x-tex">{\displaystyle [Ag^{+}]_{0^{}}10^{b_{0}}}</annotation>

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<li>in both plots, the x-intercept is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{e}=v_{0}[Cl^{-}]_{0^{}}/[Ag^{+}]_{0}}">

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<annotation encoding="application/x-tex">{\displaystyle v_{e}=v_{0}[Cl^{-}]_{0^{}}/[Ag^{+}]_{0}}</annotation>

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Figure 3 gives sample plots of potentiometric titration data.

<div class="mw-heading mw-heading2"><h2 id="Non-ideal_behaviour">Non-ideal behaviour</h2></div>

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 <a href="#Carbonate_content">carbonate</a> 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 <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10^{pH_{i}}}">

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<annotation encoding="application/x-tex">{\displaystyle 10^{pH_{i}}}</annotation>

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</math><img src="./3c640403541f63eb985189a6734b41ccf3670dd9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.375ex; height:2.676ex;" alt="{\displaystyle 10^{pH_{i}}}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10^{-pH_{i}}}">

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<annotation encoding="application/x-tex">{\displaystyle 10^{-pH_{i}}}</annotation>

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</math><img src="./ed437c8e746cc0fe18e57a5641e61b7ff5942a09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.653ex; height:2.676ex;" alt="{\displaystyle 10^{-pH_{i}}}" loading="lazy"> 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:<a href="#cite_note-4">[4]</a><a href="#cite_note-5">[5]</a> 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 <a href="#Carbonate_content">Carbonate content</a>), 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.

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

<ul><li>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.</li>

<li>Butler, J. N. (1991): Carbon Dioxide Equilibria and Their Applications; CRC Press: Boca Raton, FL.</li>

<li>Butler, J. N. (1998): Ionic Equilibrium: Solubility and pH Calculations; Wiley-Interscience. Chap. 3.</li>

<li>Gans, P. and O'Sullivan, B. (2000): GLEE, a new computer program for glass electrode calibration; Talanta, 51, 33–37.</li>

<li>Gran, G. (1950): Determination of the equivalence point in potentiometric titrations, Acta Chemica Scandinavica, 4, 559-577.</li>

<li>Gran, G. (1952): Determination of the equivalence point in potentiometric titrations—Part II, Analyst, 77, 661-671.</li>

<li>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.</li>

<li>Harris, D. C.: Quantitative Chemical Analysis, 5th Ed.; W.H. Freeman & Co., New. York, NY, 1998.</li>

<li>Martell, A. E. and Motekaitis, R. J.: The determination and use of stability constants, Wiley-VCH, 1992.</li>

<li>McBryde, W. A. E. (1969): Analyst, 94, 337.</li>

<li>Rossotti, F. J. C. and Rossotti, H. (1965): J. Chem. Ed., 42, 375</li>

<li>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.</li>

<li>Stumm, W. and Morgan, J. J. (1981): Aquatic chemistry, 2nd Ed.; John Wiley & Sons, New York.</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

<li id="cite_note-1"><a href="#cite_ref-1">^</a> <cite id="CITEREFGranDahlenborgLaurellRottenberg1950" class="citation journal cs1">Gran, Gunnar; Dahlenborg, Hans; Laurell, S.; Rottenberg, Max (1950). <a rel="nofollow" class="external text" href="http://actachemscand.org/doi/10.3891/acta.chem.scand.04-0559">"Determination of the Equivalent Point in Potentiometric Titrations"</a>. Acta Chemica Scandinavica. 4: 559–577. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.3891%2Facta.chem.scand.04-0559">10.3891/acta.chem.scand.04-0559</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0904-213X">0904-213X</a>.</cite>

</li>

<li id="cite_note-mudrift-2">^ <a href="#cite_ref-mudrift_2-0">a</a> <a href="#cite_ref-mudrift_2-1">b</a> If the ionic strength is not constant, one can correct this expression for the changing activity quotient. See Harris (1998) or this <cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://chemistry.depaul.edu/wwolbach/205_207/9.pdf">"on-line document"</a> (PDF). Retrieved 2008-02-17.</cite>

</li>

<li id="cite_note-GranVII-3"><a href="#cite_ref-GranVII_3-0">^</a> Gran et al. (1981) give a more detailed treatment that takes into account other complex species in a titration of Cl− by Ag+ (Ag2Cl+ and AgCl2−, notably) and in other precipitation titrations, in order to compute equivalence volumes of dilute solutions, when precipitation is incomplete.

</li>

<li id="cite_note-4"><a href="#cite_ref-4">^</a> <cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20081217001306/http://academic.georgefox.edu/~cchamber/analytical/NeutTit.pdf">"On-line document"</a> (PDF). Archived from <a rel="nofollow" class="external text" href="http://academic.georgefox.edu/~cchamber/analytical/NeutTit.pdf">the original</a> (PDF) on December 17, 2008. Retrieved 2008-02-17.</cite>) and W. Wolbach of Depaul University (<cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://chemistry.depaul.edu/wwolbach/205_207/9.pdf">"On-line document"</a> (PDF). Retrieved 2008-02-17.</cite>) recommend using last 10-20% volume data before the equivalence point, and the latter recognizes that base-side Gran plots from titrations of acid by base (i.e after the equivalence point) can be used to assess the CO2 content in the base. Similarly, W. E. Brewer and J. L. Ferry of the University of South Carolina recommend using those data within 10% before equivalence (<cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.chem.sc.edu/analytical/chem321L/labs/Expt5.pdf">"On-line document"</a> (PDF). Retrieved 2008-02-17.</cite>). K. Kuwata of Macalester College recommends that students choose whichever data region gives the straightest line before equivalence (<cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://www.macalester.edu/~kuwata/classes/2005-06/Chem%20222/Gran%20Plot%20Lab%202006.pdf">"On-line document"</a> (PDF). Retrieved 2008-02-17.</cite>). D. L. Zellmer of the California State University at Fresno asks students to plot data from both sides of equivalence, using data furthest from equivalence, and to assess the errors in order to determine whether or not the two estimates of the equivalence volumes are significantly different (pH data: <cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://zimmer.csufresno.edu/~davidz/Chem102/pHGrans/pHGrans.html">"On-line document"</a>. Retrieved 2008-02-17.</cite>; potentiometric titration of chloride ion with silver ion: <cite class="citation web cs1"><a rel="nofollow" class="external text" href="http://zimmer.csufresno.edu/~davidz/Chem102/GransPlot/GransPlot.html">"On-line document"</a>. Retrieved 2008-02-17.</cite>).

</li>

<li id="cite_note-5"><a href="#cite_ref-5">^</a> Butler (1991) discusses the issue of data selection, and also examines interferences from titratable impurities such as borate and phosphate.

</li>

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