National Field Manual for the Collection of Water-Quality Data

6.6.4 MEASUREMENT

TWRI Book 9
(REVISED 9/2001)

Alkalinity, ANC, and concentrations of bicarbonate, carbonate, and hydroxide species are most commonly determined by analyzing acidimetric-titration data with either the inflection point titration method (section 6.6.4.B) or the Gran function plot method (section 6.6.4.C).

	The Inflection Point Titration (IPT) method is adequate for most waters and study needs. Difficulty in identifying the inflection points using the IPT method increases as the ratio of organic acids to carbonate species increases, or as the alkalinity decreases.
	The Gran function plot (Gran) method is recommended for water in which the alkalinity or ANC is expected to be less than about 0.4 meq/L (20 mg/L as CaCO3), or in which conductivity is less than 100 µS/cm, or if there are appreciable noncarbonate contributors or measurable concentrations of organic acids

The Fixed Endpoint method (titration to pH 4.5) rarely is used and is no longer recommended by the USGS for determining alkalinity values because it is less accurate than the IPT and Gran methods. This decrease in accuracy is evident particularly for low concentrations of total carbonate species and for water with significant organic and other noncarbonate contributors to alkalinity or ANC.

6.6.4.A TITRATION SYSTEMS AND PROCEDURES

Titration procedures are identical for surface-water and ground-water determinations on filtered or unfiltered aliquots of fresh to saline water samples. Become familiar with the information and detailed instructions for the buret and digital titration systems and the IPT and Gran methods before proceeding with the titration.

Titration System

Select the titration system to be used.

	The digital titration system is convenient but tends to be less precise and less accurate than the buret system because of mechanical inadequacies. Good technique is necessary to produce acceptable results.
	The buret system can be cumbersome and fragile in the field, and requires experience to execute with precision and accuracy.
	A micrometer buret can achieve accuracy to 0.1 mg/L (routine determinations are reported to whole numbers).

Sample volume and acid normality

The methods as presented in this manual require electrometric titration of a sample with incremental additions of a strong acid (commonly H₂SO₄) of known normality. Suggested combinations of titrant normality and sample volume for various ranges of alkalinity or ANC values are given in table 6.6–2. These ranges can overlap at the thresholds indicated and should not be interpreted as absolute. Generally, 1.600N acid is too strong for most samples and only is used when alkalinity or ANC is greater than 4.0 meq/L (200 mg/L as CaCO₃). A more dilute acid (0.01639N) commonly is used with the buret system.

Select the size of the delivery and measurement vessels according to the volume of sample needed. Use a volumetric pipet for an alkalinity sample and a graduated cylinder or digital balance for an ANC sample. When selecting the measurement vessel:

	50 mL of sample in a 100-mL beaker is typical for most routine work.
	Use 100 mL (or more) of sample in a 150-mL (or larger) beaker for samples with low alkalinity or ANC.
	Use 25 mL or less of a sample in a 50-mL beaker for samples with high alkalinity or ANC. Larger volumes of sample may be used in combination with higher normality titrant.

To pipet the sample for alkalinity determination:

A small volume of sample will remain in the tip of class A “TD” volumetric pipets—do not blow it out.

1. Suspend the pipet tip vertically in a beaker, touching neither the walls nor the contents of the receiving vessel.
2. Allow the sample to drain freely until the liquid it contains reaches the bottom of the pipet.
3. Touch the pipet tip to the beaker wall until the flow from the pipet stops—leave the tip in contact with the beaker wall for an additional 10 seconds after the flow stops.

Stirring method

When titrating, stirring helps to establish a uniform mixture of sample and titrant, and an equilibrium between sensors and sample. Select a stirring method and use a consistent technique.

	If using a magnetic stirrer, stir the sample slowly and continuously, using a small stir bar; avoid creating a vortex and large streaming potentials. If using a digital titrator, keep the delivery tube immersed throughout the procedure but keep the aperture of the tube away from the stir bar to avoid bleeding acid from the tube to the sample between titrant additions.
	If swirling the sample to mix, make the pH measurement as the sample becomes quiescent, after each addition of titrant.
	Avoid splashing the sample out of the beaker or onto the beaker walls. Droplets on the beaker walls can be rinsed down with deionized water. If you splash the sample out of the beaker, you must start over.
	Avoid splashing the sample out of the beaker or onto the beaker walls. Droplets on the beaker walls can be rinsed down with deionized water. If you splash the sample out of the beaker, you must start over.
	Allow sufficient time between titrant additions for the pH value displayed on the instrument to equilibrate. Emphasis should be placed on maintaining a consistent technique (titrant additions every 15 - 30 seconds) rather than waiting for the instrument to “lock on” to a particular pH value.

Titration methods

Select and plan your method of titration.

	IPT method. Titrate cautiously on both sides of the expected equivalence points (fig. 6.6-1).
	-	If concentrations of contributing carbonate species will be determined and the initial pH is greater than 8.1: titrate to a pH of about 8.0, carefully using small increments of acid. These data are important in determining the carbonate equivalence point.
	-	If concentrations of contributing carbonate species will not be determined: titrate rapidly at first, adding relatively large acid increments to bring the pH to about 5.5. Do not skip the pH range above 5.5 completely, or you may pass the equivalence point.
	-	If a pH is below 5.5: titrate slowly, using small increments of acid. This region is important in determining the bicarbonate equivalence point. Titrate to a pH of 4.0 or lower (3.5 if the sample alkalinity or ANC range is unknown or if the sample contains high concentrations of noncarbonate contributors, such as organic acids).
	Gran method. Collect data between and beyond the expected equivalence points (fig. 6.6-1).
	-	If concentrations of contributing carbonate species will be determined: record titration points throughout the entire pH range of the titration. A good rule of thumb is to collect data along the titration curve roughly every 0.2 to 0.3 pH units. The points on the titration curve that are somewhat removed from the carbonate and bicarbonate equivalence points are used by the Gran method (Pankow, 1991).
	-	If concentrations of contributing carbonate species will not be determined: it is not necessary to develop incremental titration points above a pH of about 5.5.
	-	Titrate to a pH of 3.5 or lower (3.0 or less if the sample alkalinity or ANC range is unknown). A sufficient number of titration points beyond the equivalence point are needed to ensure accuracy.

Quality-control (QC) procedures

	Verify your accuracy and ability to reproduce the alkalinity or ANC determinations by using reference samples and repeating the titration periodically on duplicate or triplicate samples. The frequency and distribution of QC determinations are established by study requirements.
	Rule of thumb—QC samples should be collected and titrated no less than every tenth sample. The determination on a filtered sample should be reproducible within ±5 percent when titrating a duplicate aliquot from the same batch of sample filtrate.
	–	For filtered samples with alkalinity less than 0.4 meq/L (20 mg/L as CaCO3), reproducibility should be between 5 and 10 percent.
	–	If the alkalinity is about 0.02 meq/L or less, differences between duplicate samples are likely to exceed 10 percent because of rounding errors alone. Such rounding errors may be reduced by using a larger sample volume or a lower normality of titrant.

	Compare ANC with alkalinity values—When interferences are absent, titration on an unfiltered sample often results in a determination identical to or within 5 percent of the filtered sample and can be used as the QC check.
	–	If filtered and unfiltered values fail the ±5-percent criterion, perform the QC check on the filtered sample.
	–	Reproducibility of the ANC determination to within 5 percent on duplicate unfiltered samples can be problematical when the sample has large amounts of particulate matter—extend the quality-assurance criterion to ±10 percent.

Digital titrator

Be thoroughly familiar with the operation of the digital titrator before field use. The procedures in this section describe use of the Hach^® digital titrator, for which the Hach Company has provided accuracy and precision data. A plunger in the digital titrator forces acid in the titrant cartridge into the delivery tube. The plunger is controlled by a main-drive screw, which in turn is controlled by rotation of the delivery knob. The delivery knob controls the volume of titrant delivered through the delivery tube, as indicated by a digital counter. To minimize errors caused by uncertainty in the volume of titrant dispensed to the sample, titration procedures must account for the accuracy and precision of the titrant-delivery system.

To use the digital titrator:

1.	Equilibrate titrant temperature to sample temperature.
2.	Assemble the digital titrator.
	•	Depress the plunger-release button and retract the plunger.
	•	Insert the titrant cartridge into the titrator and twist the cartridge one-quarter turn to lock it into position.
	•	Carefully depress the plunger-release button and push the plunger forward until it makes contact with the Teflon® seal inside the cartridge.
3.	Remove the vinyl cap from the cartridge (save the cap) and insert the straight end of the delivery tube into the cartridge.
	•	Do not push the delivery tube beyond the cartridge tip. Do not alter the delivery tube.
	•	Use of a new delivery tube is recommended for each assembly of the titrator. Discard a tube that shows wear.
	•	If tubes are reused, store them in separate, clean plastic bags after rinsing with DIW. Do not reuse a tube for a different titrant normality.
4.	To ensure that no air bubbles or water are in the delivery tube, hold the titrator with the cartridge tip up and turn the delivery knob to force a few drops of titrant through the end of the delivery tube. Rinse tube exterior with DIW and blot acid or water droplets from the tube before inserting it into the sample.
5.	Set the digital counter to zero using the counter-reset knob (taking care not to turn the delivery knob).
6.	Transfer the selected volume of the sample (pipet for alkalinity, graduated cylinder or digital balance for ANC) to a clean beaker. If a magnetic stirrer is used, place a clean, dry, small stir bar into the beaker before transferring the sample to the beaker. Do not use a magnetic stirrer for sample conductivity <100 µS/cm. Place beaker on stirrer.
7.	Rinse the pH and temperature sensors with DIW. Gently blot with lint-free paper any water droplets adhering to the sensors.
8.	Insert the sensors into the beaker.
	•	Do not let sensors touch the bottom or wall of the beaker.
	•	The amount of sample in the beaker must be sufficient to cover the junction of the reference electrode, the electrode bulb, and the temperature sensor.
9.	Measure the initial pH and temperature while gently stirring or after gently swirling the sample.
	•	Do not splash sample onto the beaker wall or out of the beaker.
	•	Minimize the vortex caused by magnetic stirring, and ensure that the stir bar does not hit the pH electrodes.
	•	Record on the field form the pH and temperature values, the initial counter reading (it should read “0000”), the titrant normality, the time, and the sample volume.
10.	Immerse the end of the titrant delivery tube in the sample. To prevent bleeding of the titrant from the delivery tube, keep the aperture of the delivery tube away from the stir bar.
11.	Begin titration. If using a magnetic stirrer, stir the sample slowly and continuously. Measure pH after each addition of titrant, and after the acid and sample are mixed homogeneously. If a magnetic stirrer is not used, swirl to mix the sample and acid after each addition of titrant. Allow 15 to 30 seconds after each addition for equilibration, then record the pH and counter readings.
	•	>pH 8.1—To determine the carbonate equivalence point using the IPT method, slowly add the titrant in small (but no less than three digital-count) increments until the pH of the sample is less than 8.0. Larger increments can be used for samples containing high carbonate concentrations.
	•	pH < 8.1 and 5.0—If using the IPT method, titrate with larger increments to pH ~5.0 (5.5 for sample alkalinity or ANC <0.4 meq/L (<20 mg/L as CaCO3) or sample conductivity <100 µS/cm). Do not skip this pH region entirely; the equivalence point might be passed if too much acid is added. If using the Gran method, collect data points every 0.2 to 0.3 pH units in this region.
	•	pH < 5.0—To determine the bicarbonate equivalence point with the IPT method, cautiously add the titrant in small (but no less than three digital-count) increments from pH 5.0 to4.0. (The most sensitive part of the titration curve is between pH 4.8 and 4.3 for many natural waters.) If using the Gran method, extend the titration to pH 3.5 ( 3.0 for samples high in organic acids or other noncarbonate contributors, or when the alkalinity or ANC range is unknown). The Gran method relies on these low pH points beyond the equivalence point.

Buret titrator

When using a buret, exercise caution to ensure that the acid does not evaporate or become contaminated with extrinsic matter or moisture. The titrant temperature should be equilibrated to the sample temperature before use. Always empty the buret after each use. Never reuse the titrant solution; dispose of the solution properly.

To titrate with a buret:

1.	Fill a clean, dry buret with sulfuric acid titrant (0.01639N or other known concentration).
	•	Use a 10-mL semimicroburet with 0.05-mL graduations and a Teflon® stopcock for samples with alkalinity or ANC less than 4 meq/L (200 mg/L as CaCO3).
	•	Use a 25-mL buret with 0.1-mL graduations and a Teflon® stopcock for samples with alkalinity or ANC of 4 meq/L (200 mg/L as CaCO3) or greater and when the sample pH exceeds 8.1.
	•	If greater accuracy is desired, use a Gilmont®-type micrometer buret.
2.	Make sure no air bubbles are trapped in the buret or the buret stopcock. Record on field forms the sulfuric acid normality and initial buret reading.
3.	Transfer the selected volume of sample (pipet for alkalinity, graduated cylinder or digital balance for ANC) to a clean beaker. Do not pipet by mouth.
	•	If a magnetic stirrer is used, place a clean, dry, small stir bar into the beaker before transferring the sample to the beaker. Place the beaker on a magnetic stirrer.
	•	Do not use a magnetic stirrer if sample conductivity is <100 µS/cm.
4.	Rinse the pH and temperature sensors with DIW. Gently blot water droplets adhering to the sensors with lint-free paper (residual DIW will not affect the determination).
5.	Insert the sensors into the beaker.
	•	Do not let sensors touch the bottom or wall of the beaker.
	•	Sample depth in the beaker must be sufficient to cover the junction of the reference electrode, the electrode bulb, and the temperature sensor.
6.	Measure the initial pH and temperature while gently stirring or after gently swirling the sample.
	•	Do not splash the sample onto the beaker wall or out of the beaker.
	•	Minimize the vortex caused by magnetic stirring, and ensure that the stir bar does not hit the pH electrodes.
	•	Record on the field form the pH and temperature values, the initial buret reading, the titrant normality, the time, and the sample volume.
7.	Begin titration. If using a magnetic stirrer, stir the sample slowly and continuously. Measure pH after each addition of titrant, and after the acid and sample are mixed homogeneously. If a magnetic stirrer is not used, swirl to mix the sample and acid after each addition of titrant. Allow 15 to 30 seconds after each addition for equilibration, then record the pH and buret readings.
	•	>pH 8.1—To determine the carbonate equivalence point using the IPT method, add the titrant drop by drop in 0.01-mL increments until the pH is less than 8.0. Larger increments can be used for samples containing high carbonate concentrations.
	•	>pH < 8.1 and 5.0—If using the IPT method, titrate with larger increments to pH ~5.0 (5.5 for sample with alkalinity or ANC <0.4 meq/L (<20 mg/L as CaCO3) or with sample conductivity <100 µS/cm). Do not skip this pH region entirely; the equivalence point might be passed if too much acid is added. If using the Gran method, collect data points every 0.2 to 0.3 pH units in this region.
	•	>pH < 5.0—To determine the bicarbonate equivalence point with the IPT method, cautiously add the titrant drop by drop in 0.0 1-mL increments from pH 5.0 to 4.0 or less (the most sensitive part of the titration curve is between pH 4.8 and 4.3 for many natural waters). If using the Gran method, extend the titration to pH 3.5 or less (3.0 or less for samples high in organic acids and other noncarbonate contributors, or when the alkalinity or ANC range is unknown). The Gran method relies on these low pH points beyond the equivalence point.
		TECHNICAL NOTE: 0.01 mL of a standard 0.05-mL drop of titrant tends to remain on the buret tip. To dispense a 0.01-mL titrant drop, quickly rotate the stopcock through 180 degrees (one-half turn) and then rinse the titrant from the buret tip into the beaker with a small quantity of DIW.
8.	Analyze the titration data to determine the carbonate and bicarbonate equivalence points using the IPT method (section 6.6.4.B) and/or the Gran method (section 6.6.4.C).
9.	Calculate the sample alkalinity or ANC and the concentrations of the carbonate species from the equivalence points, as described in section 6.6.5.

6.6.4.B INFLECTION POINT TITRATION METHOD

The inflection point titration (IPT) method uses the inflection points of the titration curve to determine equivalence points. Whereas for many natural water samples these points are near pH 8.3 and 4.5, it is more accurate to calculate their values from the titration data.

Inflection points are the points of maximum rate of change in pH per volume of titrant added. Near equivalence points, rapid pH changes occur with small additions of titrant. For this reason, the titration must be performed slowly and cautiously near the expected equivalence points, using small incremental additions of titrant. The relative error of the determinations can be within ±4 percent if the equivalence point is recognizable within ±0.3 pH unit of the true equivalence point.

To determine the inflection point(s), you can either construct a titration curve by plotting the change in pH divided by the change in titrant volume against the volume of titrant added to the sample, or calculate such values in a table or spreadsheet.

	Graphing the titration curve always is advisable. Such plots are helpful in uncovering typographical errors and spurious maxima that might confuse the detection of the inflection point(s).
	More than one inflection point in close proximity indicates that the true inflection point has been missed. If this occurs, titrate a duplicate sample using smaller acid increments near the inflection point or use a Gran plot. Note that if the acid increments are too small, the location of the inflection points may become masked by noise in the data.
	If the maximum rate of change in pH per volume of titrant occurs at two or more points near an equivalence point (two or more points are “tied” for the maximum value), then determine the location of the equivalence point as the middle of the range where ties were produced. For example, if the maximum rate occurs at digital counts 120, 122, and 126, then the calculated location of the equivalence point is at digital count 123.
	If no clear inflection point(s) can be determined easily, interferences from weak organic acids are likely—use the Gran method.

Example IPT–1A shows the results of an IPT analysis of a typical titration using a digital titrator. Example IPT–1B shows similar results from a buret titration. Both of these titrations have inflection points at both the carbonate and bicarbonate equivalence points. Example IPT–2 uses the IPT method on a sample with low alkalinity; that sample has only one inflection point.

Example IPT–1A: IPT method using the digital titrator

A titration was performed on a natural water sample from the South Diamond Canal inflow to the Donner und Blitzen River, Oregon. The data are plotted in figure 6.6–2 and listed in table 6.6–3. Using the IPT method, the maximum rates of change of pH per volume of titrant added occur at pH 8.27 and 4.50 (at counts 186 and 1014). Because these slopes represent changes between two points, the actual inflection points are located between counts 183 and 186 for the carbonate equivalence point and between 1011 and 1014 for the bicarbonate equivalence point. Thus, the calculated digital-counter values for the inflection points are 184.5 (185) and 1012.5 (1013). The error in using 1014 rather than 1013 typically is insignificant but the larger the increments used, the greater the error. Calculation of the correct inflection point (Section 6.6.5) results in reduced errors; alkalinity for this ample is calculated at 101.3 mg/L as CaCo₃.

Figure 6.6-2. Plot of data for an inflection point titration using a digital titrator (Example 1PT-1A).

Example IPT–1B: IPT method using the buret system

If the sample in Example IPT–1A had been titrated with the buret system, the titration data would be similar to that plotted in figure 6.6–3 and given in table 6.6–4. Using the IPT method, the maximum rates of change of pH per volume of titrant added occur at pH 8.27 and 4.50 (at 1.86 and 10.14 mL of acid added). Because these slopes are changes between two points, the actual inflection points are located between 1.83 and 1.86 mL for the carbonate equivalence point, and between 10.11 and 10.14 mL for the bicarbonate equivalence point. The calculated titrant volumes for the inflection points, therefore, are 1.85 and 10.13 mL. The calculated aklalinity (Section 6.6.5) that results is 101.3 mg/L as CaCo₃.

Figure 6.6-3. Plot of data for an inflection point titration using a buret (Example IPT-1B).

Example IPT–2: IPT method for a low alkalinity sample

A water sample collected from Little Abiqua Creek near Scotts Mills, Oregon, provides a good example for a low-alkalinity titration. In this titration, 200 mL of filtered sample were titrated with 0.16N titrant and analyzed by the IPT method. The results are shown in figure 6.6–4 and table 6.6–5. The bicarbonate equivalence point was found near a pH of 5.25, between digital counts 134 and 137. The correct digital-counter value at the inflection point, therefore, is 136 (135.5) counts. This results in a calculated alkalinity of 6.8 mg/L as CaCO₃.

Figure 6.6–4. Plot of data for an inflection point titration of a low-alkalinity sample (Example IPT–2).

6.6.4.C GRAN FUNCTION PLOT METHOD

Gran function plots commonly are used to determine alkalinity and ANC in sea water, low ionic-strength water, water with low carbonate concentrations, and water with measurable concentrations of organic compounds. Gran’s method does not rely on the presence of inflection points in the titration curve; therefore, it particularly is useful for waters with low alkalinity.

Using the known chemistry of carbonic acid and some simplifying assumptions, Gran’s method linearizes a set of functions that describe parts of the titration curve (Gran, 1952). The linearizing assumptions used by Gran’s method are valid only for data that are some distance away from the equivalence points (Pankow, 1991).

	Collect titration points throughout the entire pH range of the titration. A good rule of thumb is to collect data along the titration curve roughly every 0.2 to 0.3 pH unit.
	Titrate to a pH of 3.5 or lower (3.0 or less if the alkalinity or ANC range is unknown for the waters sampled). A sufficient number of titration points beyond the bicarbonate equivalence point are needed to ensure the accuracy of the calculation.

Gran Functions

During an alkalinity titration (carbonate system), the hydrogen ions added convert carbonate to bicarbonate and then bicarbonate to carbonic acid. The titration continues until no more species are reacting. When this process is complete, additional hydrogen ions will be in excess in the solution. The F₁ Gran function plot identifies the point at which all alkalinity has been titrated and hydrogen ions begin to be in excess. Beyond the bicarbonate equivalence point, the shape of the curve is determined by hydrogen ions in excess of all hydrogen ion acceptors in the sample. Similar relations are used with the Gran functions in other parts of the titration curve.

Two Gran functions can be calculated for each equivalence point in the titration. Including the equivalence points for hydroxide, carbonate, and bicarbonate, six Gran functions (F₁ through F₆) are useful for analyzing titration data from natural water samples. The functions for the hydroxide equivalence point, however, commonly are not used. Derivations of these Gran functions are available (Stumm and Morgan, 1981).

For an acidimetric titration, the six Gran functions are where

F₁ = (V_o + V_t ) (10^–pH ) / = (V_t – V_s ) C_a

F₂ = (V_s – V_t ) 10^–pH = (V_t – V_w )K₁’

F₃ = (V_t – V_w ) 10^pH = (V_s – V_t ) / K₁’

F₄ = (V_s – 2V_w + V_t 10^pH = (V_w – V_t ) / K₂

F₅ = (V_w – V_t ) 10^–pH = (V_t – V_x ) K₂’

F₆ = (V_o + V_t ) 10^pH = (V_x – V_t ) C_a / K_w’ ,

where

Vo	is	initial volume of the sample;
Vt	is	volume of acid titrant added;
Vs	is	titrant volume at the bicarbonate equivalence point;
Vw	is	titrant volume at the carbonate equivalence point;
Vx	is	titrant volume at the hydroxide equivalence point;
Ca	is	normality of the acid titrant;
	is	activity coefficient for H+;
K1’	is	first acid dissociation constant for H2CO3, corrected for the activity of carbonate species;
K2’	is	second acid dissociation constant for H2CO3, corrected for theactivity of carbonate species; and
Kw’	is	acid dissociation constant for water, corrected for the activity of hydroxide.

Note that if these functions are calculated in the correct sequence, the function value on the left side of each equation will consist of known values. By extrapolating these function values to zero, the right side of each equation can be set to zero and can be used to solve for a previously unknown equivalence point.

	The F1 function, followed by F2, is the most commonly used Gran function.
	The Gran functions F1 and F3 are useful in determining the bicarbonate equivalence point because F1 and F3 are zero when Vt = Vs. Similarly, F2 and F4 are used to determine the carbonate equivalence point (Vt = Vw), whereas F5 and F6 can be used to determine the hydroxide equivalence point (Vt =Vx).

Gran Function Plots

Gran function plots are made by plotting each of the Gran functions against titrant volume and fitting a line through the data points for each function in a particular pH region.

	The F1 function is valid for the pH range just below the bicarbonate equivalence point. The F1 data only become linear somewhat beyond the bicarbonate equivalence point; therefore, it is often necessary to titrate the sample down to a pH between 3.5 and 3.0. For systems with measurable concentrations of organic acids, titrate to pH 2.5 (Baedecker and Cozzarelli, 1992).
	Functions F2 and F3 are valid in the pH range between the carbonate and bicarbonate endpoints.
	Functions F4 and F5 are valid for pH values between the carbonate and hydroxide endpoints.
	Function F6 is valid for pH values higher than the hydroxide endpoint.

Equivalence points are found by extrapolating each function to where it crosses the x-axis. An idealized Gran analysis would result in a plot such as that shown in figure 6.6–5. The F₁ and F₂ functions are the most commonly used Gran functions.

Figure 6.6–5. An idealized Gran function plot, showing six Gran functions.

The F₁ function must be calculated first. In order to calculate F₂, one needs to know the location of the bicarbonate equivalence point, V_s, which is obtained from the solution of F₁.

Similarly, the value of V_w from an analysis of F₂ is necessary to plot and analyze F₃. Function F₄ requires both V_s and V_w from the results of F₂ and either F₁ or F₃. Function F₅ requires a value for V_w from the results of either F₂ or F₄.

TECHNICAL NOTE: The Gran functions also can be used to extract useful information from their fitted slopes, such as the values of Ca and the various acid dissociation constants. Further explanation can be found in Stumm and Morgan (1981) and Pankow (1991).

To prepare a Gran function plot:

1.	Bicarbonate equivalence point (Gran function F1). Plot (Vo + Vt) 10–pH against the titrant volume, Vt. (This formulation ignores activity corrections, setting = 1.0.)
	•	When developing this function, Vo and Vt must be in the same units (probably mL).
	•	The value of Vt on the x-axis can be in either milliliter or digital counts.
2.	Extrapolate a straight line through the data in the region beyond the bicarbonate equivalence point to where it meets the x-axis at F1 = (Vo + Vt) 10–pH = 0 or Vt = Vs. That point is the bicarbonate equivalence point. See Example Gran–1 for an illustration of steps 1 and 2.
3.	Carbonate equivalence point (Gran function F2). Using the value of Vs from step 2, plot (Vs – Vt) 10–pH against the titrant volume, Vt, in the region between the carbonate and bicarbonate equivalence points. Make sure to use the same units for Vt and Vs in developing this function.
4.	Extrapolate a straight line through the data in this region to where it meets the x-axis at F2 = (Vs – Vt) 10–pH = 0 or Vt = Vw. That point is the carbonate equivalence point.
Skip steps 5 and 6 (F5 and F6 Gran functions) if the initial sample pH is less than approximately 10.3.
5.	Hydroxide equivalence point (Gran function F5). Using the value of Vw from step 4, plot (Vw – Vt) 10–pH against the titrant volume, Vt, in the region above the carbonate equivalence point. Use the same units for Vt and Vw in developing this function.
6.	Extrapolate a straight line through the data in the region between the carbonate and hydroxide equivalence points to where it meets the x-axis at F5 = (Vw – Vt) 10–pH = 0 or Vt = Vx. That point is the hydroxide equivalence point.
At this point, you can either stop, or try to verify your values of Vs, Vw and Vx by plotting additional Gran functions. To continue plotting, follow these optional steps:
7.	In the region between the carbonate and bicarbonate equivalence points, plot (Vt – Vw) 10pH against the titrant volume, Vt. This is a plot of the Gran function F3.
8.	Extrapolate a straight line in this region to where it meets the x-axis at F3 = (Vt – Vw) 10pH = 0 or Vt = Vs to get another estimate of the bicarbonate equivalence point.
9.	In the region between the carbonate and hydroxide equivalence points, plot (Vs – 2Vw + Vt) 10pH against the titrant volume, Vt. This is a plot of the Gran function F4.
10.	Extrapolate a straight line in this region to where it meets the x-axis at F4 = (Vs – 2Vw + Vt) 10pH = 0 or Vt = Vw to obtain another estimate of the carbonate equivalence point.
11.	In the region above the hydroxide equivalence point, plot (Vo + Vt) 10pH against the titrant volume, Vt. This is a plot of the Gran function F6.
12.	Extrapolate a straight line in this region to where it meets the x-axis at F6 = (Vo + Vt) 10pH = 0 or Vt = Vx to obtain another estimate of the hydroxide equivalence point.

Example Gran–1: Gran function plot of F₁ only

Gran function plots are useful for samples with low alkalinity. Using the titration data obtained from a sample of Little Abiqua Creek near Scotts Mills, Oregon (from Example IPT–2), a Gran function plot is easily prepared. The necessary calculations are shown in table 6.6-6. The results are plotted in figure 6.6-6. In the region beyond the equivalence point in figure 6.6-6, a straight line results. Extrapolation of this straight line to (V_o+V_t)10^–pH = 0 locates the equivalence point. The extrapolated straight line intercept at (V_o+V_t) 10^–pH = 0 on figure 6.6-6 is at 139.5 digital counts of titrant added, corresponding to a bicarbonate equivalence point at a pH of approximately 5.18. The calculated alkalinity by this method is 7.0 mg/L as CaCO₃, in excellent agreement with the value of 6.8 mg/L as CaCO₃ calculated by the IPT method in Example IPT–2.

Figure 6.6–6. Example of a Gran function plot using F1 to determine the bicarbonate equivalence point.

Example Gran–2: Gran function plot using F₁ through F₄

In this example, the same titration data that were used in Example IPT–1A in section 6.6.4.B are analyzed using the Gran method. The sample pH (9.35) was not high enough to justify the use of Gran functions F₅ and F₆, so only functions F₁ through F₄ are used.

The results are shown in figure 6.6–7, and indicate a bicarbonate equivalence point at 1005 counts (from F₃, only 2 data points could be used for F₁), and a carbonate equivalence point at about 209 counts (F₂ estimated the equivalence point at 214, F₄ at 204). Although this is not an ideal data set for Gran function analysis (the titration did not extend to pH values lower than about 4.0), the Gran method was able to provide reasonable estimates of the equivalence points.

Using a bicarbonate equivalence point of 1005 counts as determined by the Gran method, an acid titrant normality of 0.16N, and a sample volume of 100 mL, the calculated alkalinity of this sample is 100.5 mg/L as CaCO₃ (see section 6.6.5). Using an equivalence point of 1013 counts for this sample from the analysis in Example IPT–1A (inflection point method), the calculated alkalinity would be 101.3 mg/L as CaCO₃. The agreement between these two methods is very good, producing a discrepancy of less than 1 percent. Results from either method would be reported as 101 mg/L as CaCO₃.

Figure 6.6–7. Example of a Gran function plot using F1, F2, F3, and F4 to determine carbonate and bicarbonate equivalence points.