Appendix - Example A3: An Acid/Base Titration
Summary
Goal
A solution of hydrochloric acid (HCl) is standardised against a solution of sodium hydroxide (NaOH) with known content.
Measurement Procedure
A solution of hydrochloric acid (HCl) is titrated against a solution of sodium hydroxide (NaOH), which has been standardised against the titrimetric standard potassium hydrogen phthalate (KHP), to determine its concentration. The stages of the procedure are shown in figure a3.1.
Measurand:
where the symbols are as given in Table A3.1 and the value of 1000 is a conversion factor from ml to litres.
Table A3.1: Acid-base Titration values and uncertainties
| Description | Value x | Standard uncertainty u(x) | Relative standard uncertainty u(x)/x | |
| rep | Repeatability | 1 | 0.001 | 0.001 |
| mKHP | Weight of KHP | 0.3888 g | 0.00013 g | 0.00033 |
| PKHP | Purity of KHP | 1.0 | 0.00029 | 0.00029 |
| VT2 | Volume of NaOH for HCl titration | 14.89 ml | 0.015 ml | 0.0010 |
| VT1 | Volume of NaOH for KHP titration | 18.64 ml | 0.016 ml | 0.00086 |
| MKHP | Molar mass of KHP | 204.2212 g mol-1 | 0.0038 g mol-1 | 0.000019 |
| VHCl | HCl aliquot for NaOH titration | 15 ml | 0.011 ml | 0.00073 |
| cHCl | HCl solution concentration | 0.10139 mol l-1 | 0.00016 mol l-1 | 0.0016 |
Identification of the Uncertainty Sources:
The relevant uncertainty sources are shown in figure a3.2.
Quantification of the Uncertainty Components
The final uncertainty is estimated as 0.00016 mol l-1. Table a3.1 summarises the values and their uncertainties; Figure a3.3 shows the values diagrammatically.
Figure A3.3: Uncertainty contributions in acid-base titration
The values of u(y,xi)=(
y/xi).u(xi) are taken from table a3.3
Axample A3: An Acid/Base Titration: Detailed Discussion
A3.1 Introduction
This example discusses a sequence of experiments to determine the concentration of a solution of hydrochloric acid (HCl). In addition, a number of special aspects of the titration technique are highlighted. The HCl is titrated against solution of sodium hydroxide (NaOH), which was freshly standardised with potassium hydrogen phthalate (KHP). As in the previous example (A2) it is assumed that the HCl concentration is known to be of the order of 0.1 mol 1-1 and that the end-point of the titration is determined by an automatic titration system using the shape of the pH-curve. This evaluation gives the measurement uncertainty in terms of the SI units of measurement.
A3.2 Step 1: Specification
A detailed description of the measurement procedure is given in the first step. It compromises a listing of the measurement steps and a mathematical statement of the measurand.
Procedure
The determination of the concentration of the HCl solution consists of the following stages (See also figure a3.4):
- The titrimetric standard potassium hydrogen phthalate (KHP) is dried to ensure the purity quoted in the supplier's certificate. Approximately 0.388 g of the dried standard is then weighed to achieve a titration volume of 19 ml NaOH.
- The KHP titrimetric standard is dissolved with
50 ml of ion free water and then titrated using the NaOH solution. A titration system controls automatically the addition of NaOH and samples the pH-curve. The end-point is evaluated from the shape of the recorded curve. - 15 ml of the HCl solution is transferred by means of a volumetric pipette. The HCl solution is diluted with de-ionised water to give 50 ml solution in the titration vessel.
- The same automatic titrator performs the measurement of HCl solution.
calculation:
The measurand is the concentration of the HCl solution, cHCl. It depends on the mass of KHP, its purity, its molecular weight, the volumes of NaOH at the end-point of the two titrations and the aliquot of HCl.:
where
cHCl :concentration of the HCl solution [mol 1-1]
1000 :conversion factor [ml] to [l]
mKHP :mass of KHP taken [g]
PKHP :purity of KHP given as mass fraction
VT2 :volume of NaOH solution to titrate HCl [ml]
VT1 :volume of NaOH solution to titrate KHP [ml]
MKHP: molar mass of KHP [g mol-1]
VHCl :volume of HCl titrated with NaOH solution [ml]
A3.3 Step 2: Identifying and Analysing Uncertainty Sources
The different uncertainty sources and their influence on the measurand are best analysed by visualising them first in a cause and effect diagram (figure a3.5).
Because a repeatability estimate is available from validation studies for the procedure as a whole, there is no need to consider all the repeatability contributions individually. They are therefore grouped into one contribution (shown in the revised cause and effect diagram in figure a3.5).
The influences on the parameters VT2, VT1, mKHP, PKHP and MKHP have been discussed extensively in the previous example, therefore only the new influence quantities of VHCl will be dealt with in more detail in this section.
Volume VHCl:
15 ml of the investigated HCl solution is to be transferred by means of a volumetric pipette. The delivered volume of the HCl from the pipette is subject to the same three sources of uncertainty as all the volumetric measuring devices.
- The variability or repeatability of the delivered volume
- The uncertainty in the stated volume of the pipette
- The solution temperature differing from the calibration temperature of the pipette.
A3.4 Step 3: Quantifying Uncertainty Components
The goal of this step is to quantify each uncertainty source analysed in step 2. The quantification of the branches or rather of the different components was described in detail in the previous two examples. Therefore only a summary for each of the different contributions will be given.
Repeatability:
The method validation shows a repeatability for the determination of 0.1% (as %rsd). This value can be used directly for the calculation of the combined standard uncertainty associated with the different repeatability terms.
Mass mKHP:
Calibration/linearity: The balance manufacturer quotes ±0.15 mg for the linearity contribution. This value represents the maximum difference between the actual mass on the pan and the reading of the scale. The linearity contribution is assumed to show a rectangular distribution and is converted to a standard uncertainty:
The contribution for the linearity has to be accounted for twice, once for the tare and once for the gross mass, leading to an uncertainty u(mKHP) of
- NOTE 1: The contribution is applied twice because no assumptions are made about the form of the non-linearity. The non-linearity is accordingly treated as a systematic effect on each weighing, which varies randomly in magnitude across the measurement range.
- NOTE 2: Buoyancy correction is not considered because all weighing results are quoted on the conventional basis for weighing in air [h.18]. The remaining uncertainties are too small to consider. Note 1 in appendix g refers.
Purity PKHP:
PKHP is given in the supplier's certificate as 100% ±0.05%. The quoted uncertainty is taken as a rectangular distribution, so the standard uncertainty u(PKHP) is
Volume VT2:
- Calibration: Figure given by the manufacturer (±0.03 ml) and approximated to a triangular distribution
- Temperature: The possible temperature variation is within the limits of ±4°C and approximated to a rectangular distribution
- Bias of the end-point detection: A bias between the determined end-point and the equivalence-point due to atmospheric CO2 can be prevented by performing the titration under Argon. No uncertainty allowance is made.
VT2 is found to be 14.89 ml and combining the two contributions to the uncertainty u(VT2) of the volume VT2 gives a value of
.
.
Volume VT1:
All contributions except the one for the temperature are the same as for VT2
- Calibration:
- Temperature: The approximate volume for the titration of 0.3888 g KHP is 19 ml NaOH, therefore its uncertainty contribution is
. - Bias: Negligible
VT1 is found to be 18.64 ml with a standard uncertainty u(VT1) of
Molar Mass MKHP:
Atomic weights and listed uncertainties (from current IUPAC tables) for the constituent elements of KHP (C8H5O4K) are:
| Element | Atomic weight | Quoted uncertainty | Standard uncertainty |
| C | 12.0107 | ±0.0008 | 0.00046 |
| H | 1.00794 | ±0.00007 | 0.000040 |
| O | 15.9994 | ±0.0003 | 0.00017 |
| K | 39.0983 | ±0.0001 | 0.000058 |
For each element, the standard uncertainty is found by treating the IUPAC quoted uncertainty as forming the bounds of a rectangular distribution. The corresponding standard uncertainty is therefore obtained by dividing those values by 
.
The molar mass MKHP for KHP and its uncertainty u(MKHP)are, respectively:
NOTE: The single atom contributions are not independent. The uncertainty for the atom contribution is therefore calculated by multiplying the standard uncertainty of the atomic weight by the number of atoms.
Volume VHCl:
- Calibration: Uncertainty stated by the manufacturer for a 15 ml pipette as ±0.02 ml and approximated with a triangular distribution:
. - Temperature: The temperature of the laboratory is within the limits of ±4°C. Using a rectangular temperature distribution gives a standard uncertainty of
= 0.007 ml.
Combining these contributions gives
A3.5 Step 4: Calculating the Combined Standard Uncertainty
cHClis given by
NOTE: The repeatability estimate is, in this example, treated as a relative effect; the complete model equation is therefore
All the intermediate values of the two step experiment and their standard uncertainties are collected in table A3.2.
Table a3.2: Acid-base titration values and uncertainties (2-step procedure)
| Description | Value x | Standard uncertainty u(x) | Relative standard uncertainty u(x)/x | |
| rep | Repeatability | 1 | 0.001 | 0.001 |
| mKHP | Mass of KHP | 0.3888 g | 0.00012 g | 0.00031 |
| PKHP | Purity of KHP | 1.0 | 0.00029 | 0.00029 |
| VT2 | Volume of NaOH for HCl titration | 14.89 ml | 0.014 ml | 0.0094 |
| VT1 | Volume of NaOH for KHP titration | 18.64 ml | 0.015 ml | 0.00080 |
| MKHP | Molar mass of KHP | 204.2212 g mol-1 | 0.0038 g mol-1 | 0.000019 |
| VHCl | HCl aliquot for NaOH titration | 15 ml | 0.011 ml | 0.00073 |
Using these values:
The uncertainties associated with each component are combined accordingly:
A spreadsheet method (see Appendix E) can be used to simplify the above calculation of the combined standard uncertainty. The spreadsheet filled in with the appropriate values is shown in table a3.3, with an explanation.
Table A3.3: Acid-base titration - spreadsheet calculation of uncertainty
| " | A | B | C | D | E | F | G | H | I |
| 1 | Rep | m(KHP) | P(KHP) | V(T2) | V(T1) | M(KHP) | V(HCl) | ||
| 2 | Value | 1.0 | 0.3888 | 1.0 | 14.89 | 18.64 | 204.2212 | 15 | |
| 3 | Uncertainty | 0.001 | 0.00012 | 0.00029 | 0.014 | 0.015 | 0.0038 | 0.011 | |
| 4 | |||||||||
| 5 | rep | 1.0 | 1.001 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 | 1.0 |
| 6 | m(KHP) | 0.3888 | 0.3888 | 0.38892 | 0.3888 | 0.3888 | 0.3888 | 0.3888 | 0.3888 |
| 7 | P(KHP) | 1.0 | 1.0 | 1.0 | 1.00029 | 1.0 | 1.0 | 1.0 | 1.0 |
| 8 | V(T2) | 14.89 | 14.89 | 14.89 | 14.89 | 14.904 | 14.89 | 14.89 | 14.89 |
| 9 | V(T1) | 18.64 | 18.64 | 18.64 | 18.64 | 18.64 | 18.655 | 18.64 | 18.64 |
| 10 | M(KHP) | 204.2212 | 204.2212 | 204.2212 | 204.2212 | 204.2212 | 204.2212 | 204.2250 | 204.2212 |
| 11 | V(HCl) | 15 | 15 | 15 | 15 | 15 | 15 | 15 | 15.011 |
| 12 | |||||||||
| 13 | c(HCl) | 0.101387 | 0.101489 | 0.101418 | 0.101417 | 0.101482 | 0.101306 | 0.101385 | 0.101313 |
| 14 | u(y,xi) | 0.000101 | 0.000031 | 0.000029 | 0.000095 | -0.000082 | -0.0000019 | -0.000074 | |
| 15 | u(y)2, u(y,xi)2 |
3.34E-8 | 1.03E-8 | 9.79E-10 | 8.64E-10 | 9.09E-9 | 6.65E-9 | 3.56E-12 | 5.52E-9 |
| 16 | |||||||||
| 17 | u(c(HCl)) | 0.00018 |
The values of the parameters are given in the second row from C2 to I2. Their standard uncertainties are entered in the row below (C3-I3). The spreadsheet copies the values from C2-I2 into the second column from B5 to B11. The result (c(HCl)) using these values is given in B13. The C5 shows the value of the repeatability from C2 plus its uncertainty given in C3. The result of the calculation using the values C5-C11 is given in C13. The columns D to I follow a similar procedure. The values shown in the row 14 (C14-I14) are the differences of the row (C13-H13) minus the value given in B13. In row 15 (C15-I15) the values of row 14 (C14-I14) are squared and summed to give the value shown in B15. B17 gives the combined standard uncertainty, which is the square root of B15.
The sizes of the different contributions can be compared using a histogram. Figure a3.6 shows the values of the contributions |u(y,xi)| from table a3.3.
Figure A3.6: Uncertainties in Acid-Base Titration
The expanded uncertainty U(cHCl) is calculated by multiplying the combined standard uncertainty by a coverage factor of 2:
The concentration of the HCl solution is
(0.1014 ±0.0004) mol 1-1
A3.6 Special Aspects of the Titration Example
Three special aspects of the titration experiment will be dealt with in this second part of the example. It is interesting to see what effect changes in the experimental set up or in the implementation of the titration would have on the final result and its combined standard uncertainty.
Influence of a Mean Room Temperature of 25°C
For routine analysis, analytical chemists rarely correct for the systematic effect of the temperature in the laboratory on the volume. This question considers the uncertainty introduced by the corrections required.
The volumetric measuring devices are calibrated at a temperature of 20°C. But rarely does any analytical laboratory have a temperature controller to keep the room temperature that level. For illustration, consider correction for a mean room temperature of 25°C.
The final analytical result is calculated using the corrected volumes and not the calibrated volumes at 20°C. A volume is corrected for the temperature effect according to
where
V' :actual volume at the mean temperature T
V :volume calibrated at 20°C
:expansion coefficient of an aqueous solution [°C-1]
T :observed temperature in the laboratory [°C]
The equation of the measurand has to be rewritten:
Including the temperature correction terms gives:
This expression can be simplified by assuming that the mean temperature T and the expansion coefficient of an aqueous solution are the same for all three volumes
This gives a slightly different result for the HCl concentration at 20°C:
The figure is still within the range given by the combined standard uncertainty of the result at a mean temperature of 20°C, so the result is not significantly affected. Nor does the change affect the evaluation of the combined standard uncertainty, because a temperature variation of ±4°C at the mean room temperature of 25°C is still assumed.
Visual End-Point Detection
A bias is introduced if the indicator phenolphthalein is used for visual end-point detection, instead of an automatic titration system extracting the equivalence-point from the pH curve. The change of colour from transparent to red/purple occurs between pH 8.2 and 9.8 leading to an excess volume, introducing a bias compared to the end-point detection employing a pH meter. Investigations have shown that the excess volume is around 0.05 ml with a standard uncertainty for the visual detection of the end-point of approximately 0.03 ml. The bias arising from the excess volume has to be considered in the calculation of the final result. The actual volume for the visual end-point detection is given by
where
VT1;Ind :volume from a visual end-point detection
VT1 :volume at the equivalence-point
VExcess:excess volume needed to change the colour of phenolphthalein
The volume correction quoted above leads to the following changes in the equation of the measurand
The standard uncertainties u(VT2) and u(VT1) have to be recalculated using the standard uncertainty of the visual end-point detection as the uncertainty component of the repeatability of the end-point detection.
The combined standard uncertainty
is considerable larger than before.
Triple Determination to obtain the Final Result
The two step experiment is performed three times to obtain the final result. The triple determination is expected to reduce the contribution from repeatability, and hence reduce the overall uncertainty.
As shown in the first part of this example, all the run to run variations are combined to one single component, which represents the overall experimental repeatability as shown in the in the cause and effect diagram (figure a3.5).
The uncertainty components are quantified in the following way:
Mass mKHP:
Linearity:
Purity PKHP:
Purity:
Volume VT2:
Calibration:
Temperature:
Repeatability:
The quality log of the triple determination shows a mean long term standard deviation of the experiment of 0.001 (as RSD). It is not recommended to use the actual standard deviation obtained from the three determinations because this value has itself an uncertainty of 52%. The standard deviation of 0.001 is divided by the square root of to obtain the standard uncertainty of the triple determination (three independent measurements)
(as RSD)
Volume VHCl:
Calibration:
Temperature:
Molar Mass MKHP:
Volume VT1:
Calibration:
Temperature:
All the values of the uncertainty components are summarised in table a3.4. The combined standard uncertainty is 0.00016 mol 1-1, which is a very modest reduction due to the triple determination. The comparison of the uncertainty contributions in the histogram, shown in figure a3.7, highlights some of the reasons for that result. Though the repeatability contribution is much reduced, the volumetric uncertainty contributions remain, limiting the improvement.
Table A3.4: Replicated acid-base titration values and uncertainties
| Description | Value x | Standard uncertainty u(x) | Relative standard uncertainty u(x)/x | |
| rep | Repeatability of the determination | 1.0 | 0.00058 | 0.00058 |
| mKHP | Mass of KHP | 0.3888 g | 0.00013 g | 0.00033 |
| PKHP | Purity of KHP | 1.0 | 0.00029 | 0.00029 |
| VT2 | Volume of NaOH for HCl titration | 14.90 ml | 0.014 ml | 0.00094 |
| VT1 | Volume of NaOH for KHP titration | 18.65 ml | 0.015 ml | 0.0008 |
| MKHP | Molar mass of KHP | 204.2212 g mol-1 | 0.0038 g mol-1 | 0.000019 |
| VHCl | HCl aliquot for NaOH titration | 15 ml | 0.01 ml | 0.00067 |
Figure A3.7: Replicated Acid-Base Titration Values and Uncertainties
