Appendix - Example A2: Standardising a Sodium Hydroxide Solution

Summary

Goal
A solution of sodium hydroxide (NaOH) is standardised against the titrimetric standard potassium hydrogen phthalate (KHP).

Measurement Procedure
The titrimetric standard (KHP) is dried and weighed. After the preparation of the NaOH solution the sample of the titrimetric standard (KHP) is dissolved and then titrated using the NaOH solution. The stages in the procedure are shown in the flow chart figure a2.1.

Measurand:

    where

    cNaOH :concentration of the NaOH solution [mol l-1]

    1000 :conversion factor [ml] to [l]

    mKHP :weight of the titrimetric standard KHP [g]

    PKHP :purity of the titrimetric standard given as mass fraction

    MKHP :molar mass of KHP [g mol-1]

    VT :titration volume of NaOH solution [ml]

Identification of the Uncertainty Sources:
The relevant uncertainty sources are shown as a cause and effect diagram in figure a2.2.

Quantification of the Uncertainty Components
The different uncertainty contributions are given in table a2.1, and shown diagrammatically in figure a2.3. The combined standard uncertainty for the 0.10214 mol l-1

NaOH solution is 0.00010 mol l-1

Table A2.1: Values and Uncertainties in NaOH Standardisation

   Description Value
x		 Standard uncertainty
u		 Relative standard uncertainty u(x)/x
rep Repeatability 1.0 0.0005 0.0005
mKHP Mass of KHP 0.3888 g 0.00013 g 0.00033
PKHP Purity of KHP 1.0 0.00029 0.00029
MKHP Molar mass of KHP 204.2212 g mol-1 0.0038 g mol-1 0.000019
VT Volume of NaOH for KHP titration 18.64 ml 0.013 ml 0.0007
cNaOH NaOH solution 0.10214 g mol-1 0.00010 g mol-1 0.00097

Figure A2.3: Contributions to titration uncertainty

The values of u(y,xi)=(y/xi).u(xi) are taken from table a2.3

Example A2: Standardising a Sodium Hydroxide Solution: Detailed Discussion

A2.1 Introduction

This second introductory example discusses an experiment to determine the concentration of a solution of sodium hydroxide (NaOH). The NaOH is titrated against the titrimetric standard potassium hydrogen phthalate (KHP). It is assumed that the NaOH concentration is known to be of the order of 0.1 mol 1-1. The end-point of the titration is determined by an automatic titration system using a combined pH-electrode to measure the shape of the pH-curve. The functional composition of the titrimetric standard potassium hydrogen phthalate (KHP), which is the number of free protons in relation to the overall number of molecules, provides traceability of the concentration of the NaOH solution to the SI system.

A2.2 Step 1: Specification

The aim of the first step is to describe the measurement procedure. This description consists of a listing of the measurement steps and a mathematical statement of the measurand and the parameters upon which it depends.

Procedure:
The measurement sequence to standardise the NaOH solution has the following stages.

The separate stages are:

  1. The primary standard potassium hydrogen phthalate (KHP) is dried according to the supplier's instructions. The instructions are given in the supplier's catalogue, which also states the purity of the titrimetric standard and its uncertainty. A titration volume of approximately 19 ml of 0.1 mol 1-1 solution of NaOH entails weighing out an amount as close as possible to


    The weighing is carried out on a balance with a last digit of 0.1 mg.

  2. A 0.1 mol 1-1 solution of sodium hydroxide is prepared. In order to prepare 1 l of solution, it is necessary to weigh out 4 g NaOH. However, since the concentration of the NaOH solution is to be determined by assay against the primary standard KHP and not by direct calculation, no information on the uncertainty sources connected with the molecular weight or the mass of NaOH taken is required.
  3. The weighed quantity of the titrimetric standard KHP is dissolved with 50 ml of ion-free water and then titrated using the NaOH solution. An automatic titration system controls the addition of NaOH and records the pH-curve. It also determines the end-point of the titration from the shape of the recorded curve.

Calculation:
The measurand is the concentration of the NaOH solution, which depends on the mass of KHP, its purity, its molecular weight and the volume of NaOH at the end-point of the titration

where

cNaOH :concentration of the NaOH solution [mol 1-1]
1000 :conversion factor [ml] to [l]
mKHP:mass of the titrimetric standard KHP [g]
PKHP :purity of the titrimetric standard given as mass fraction
MKHP :molar mass of KHP [g mol-1]
VT :titration volume of NaOH solution [ml]


A2.3 Step 2: Identifying and Analysing Uncertainty Sources

The aim of this step is to identify all major uncertainty sources and to understand their effect on the measurand and its uncertainty. This has been shown to be one of the most difficult step in evaluating the uncertainty of analytical measurements, because there is a risk of

neglecting uncertainty sources on the one hand and an the other of double-counting them. The use of a cause and effect diagram (Appendix D) is one possible way to help prevent this happening. The first step in preparing the diagram is to draw the four parameters of the equation of the measurand as the main branches.

Afterwards, each step of the method is considered and any further influence quantity is added as a factor to the diagram working outwards from the main effect. This is carried out for each branch until effects become sufficiently remote, that is, until effects on the result are negligible.

Mass mKHP:
Approximately 388 mg of KHP are weighed to standardise the NaOH solution. The weighing procedure is a weight by difference. This means that a branch for the determination of the tare (mtare) and another branch for the gross weight (mgross) have to be drawn in the cause and effect diagram. Each of the two weighings is subject to run to run variability and the uncertainty of the calibration of the balance. The calibration itself has two possible uncertainty sources: the sensitivity and the linearity of the calibration function. If the weighing is done on the same scale and over a small range of weight then the sensitivity contribution can be neglected.

All these uncertainty sources are added into the cause and effect diagram (see Figure A2.6).

Purity PKHP:
The purity of KHP is quoted in the supplier's catalogue to be within the limits of 99.95% and 100.05%. PKHP is therefore 1.0000 ±0.0005. There is no other uncertainty source if the drying procedure was performed according to the suppliers specification.

Molar Mass MKHP:
Potassium hydrogen phthalate (KHP) has the empirical formula C8H5O4K The uncertainty in the molar mass of the compound can be determined by combining the uncertainty in the atomic weights of its constituent elements. A table of atomic weights including uncertainty estimates is published biennially by IUPAC in the Journal of Pure and Applied Chemistry. The molar mass can be calculated directly from these; the cause and effect diagram (figure a2.7) omits the individual atomic masses for clarity

Volume VT:
The titration is accomplished using a 20 ml piston burette. The delivered volume of NaOH from the piston burette is subject to the same three uncertainty sources as the filling of the volumetric flask in the previous example. These uncertainty sources are the repeatability of the delivered volume, the uncertainty of the calibration of that volume and the uncertainty resulting from the difference between the temperature in the laboratory and that of the calibration of the piston burette. In addition there is the contribution of the end-point detection, which has two uncertainty sources.

  1. The repeatability of the end-point detection, which is independent of the repeatability of the volume delivery.
  2. The possibility of a systematic difference between the determined end-point and the equivalence point (bias), due to carbonate absorption during the titration and inaccuracy in the mathematical evaluation of the end-point from the titration curve.

These items are included in the cause and effect diagram shown in figure a2.7.

A2.4 Step 3: Quantifying Uncertainty Components

In step 3, the uncertainty from each source identified in step 2 has to be quantified and then converted to a standard uncertainty. All experiments always include at least the repeatability of the volume delivery of the piston burette and the repeatability of the weighing operation. Therefore it is reasonable to combine all the repeatability contributions into one contribution for the overall experiment and to use the values from the method validation to quantify its size, leading to the revised cause and effect diagram in figure a2.8.

The method validation shows a repeatability for the titration experiment of 0.05%. This value can be directly used for the calculation of the combined standard uncertainty.

Mass mKHP:
The relevant weighings are:

  • container and KHP: 60.5450 g (observed)
  • container less KHP: 60.1562 g (observed)
  • KHP: 0.3888 g (calculated)
Because of the combined repeatability term identified above, there is no need to take into account the weighing repeatability. Any systematic offset across the scale will also cancel. The uncertainty therefore arises solely from the balance linearity uncertainty.

Linearity: The calibration certificate of the balance quotes ±0.15 mg for the linearity. This value is the maximum difference between the actual mass on the pan and the reading of the scale. The balance manufacture's own uncertainty evaluation recommends the use of a rectangular distribution to convert the linearity contribution to a standard uncertainty.

The balance linearity contribution is accordingly

This contribution has to be counted twice, once for the tare and once for the gross weight, because each is an independent observation and the linearity effects are not correlated.

This gives for the standard uncertainty u(mKHP) of the mass mKHP, a value of

  • NOTE 1: 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.
  • NOTE 2: There are other difficulties when weighing a titrimetric standard. A temperature difference of only 1°C between the standard and the balance causes a drift in the same order of magnitude as the repeatability contribution. The titrimetric standard has been completely dried, but the weighing procedure is carried out at a humidity of around 50 % relative humidity, so adsorption of some moisture is expected.

Purity PKHP:
PKHP is 1.0000 ±0.0005. The supplier gives no further information concerning the uncertainty in the catalogue. Therefore this uncertainty is taken as having a rectangular distribution, so the standard uncertainty

u(PKHP) is .

Molar Mass MKHP:
From the latest IUPAC table, the atomic weights and listed uncertainties 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 separate element contributions to the molar mass, together with the uncertainty contribution for each, are:

  Calculation Result Standard uncertainty
C8 8x12.0107 96.0856 0.0037
H5 5x1.00794 5.0397 0.00020
O4 4x15.9994 63.9976 0.00068
K 1x39.0983 39.0983 0.000058

The uncertainty in each of these values is calculated by multiplying the standard uncertainty in the previous table by the number of atoms.

This gives a molar mass for KHP of

As this expression is a sum of independent values, the standard uncertainty u(MKHP) is a simple square root of the sum of the squares of the contributions:

NOTE: Since the element contributions to MKHP are simply the sum of the single atom contributions, it might be expected from the general rule for combing uncertainty contributions that the uncertainty for each element contribution would be calculated from the sum of squares of the single atom contributions, that is, for carbon, . Recall, however, that this rule applies only to independent contributions, that is, contributions from separate determinations of the value. In this case, the total is obtained by multiplying the a single value by 8. Notice that the contributions from different elements are independent, and will therefore combine in the usual way.

Volume VT:

  1. Repeatability of the volume delivery: As before, the repeatability has already been taken into account via the combined repeatability term for the experiment.
  2. Calibration: The limits of accuracy of the delivered volume are indicated by the manufacturer as a ± figure. For a 20 ml piston burette this number is typically ±0.03 ml. Assuming a triangular distribution gives a

    .

    Note: The ISO Guide (F.2.3.3) recommends adoption of a triangular distribution if there are reasons to expect values in the centre of the range being more likely than those near the bounds. For the glassware in examples a1 and a2, a triangular distribution has been assumed (see the discussion under Volume uncertainties in example a1).

  3. Temperature: The uncertainty due to the lack of temperature control is calculated in the same way as in the previous example, but this time taking a possible temperature variation of ±3°C (with a 95% confidence). Again using the coefficient of volume expansion for water as 2.1 x 10-4 °C-1 gives a value of

    Thus the standard uncertainty due to incomplete temperature control is 0.006 ml.

    NOTE: When dealing with uncertainties arising from incomplete control of environmental factors such as temperature, it is essential to take account of any correlation in the effects on different intermediate values. In this example, the dominant effect on the solution temperature is taken as the differential heating effects of different solutes, that is, the solutions are not equilibrated to ambient temperature. Temperature effects on each solution concentration at STP are therefore uncorrelated in this example, and are consequently treated as independent uncertainty contributions.

  4. Bias of the end-point detection: The titration is performed under a layer of Argon to exclude any bias due to the absorption of CO2 in the titration solution. This approach follows the principle that it is better to prevent any bias than to correct for it. There are no other indications that the end-point determined from the shape of the pH-curve does not correspond to the equivalence-point, because a strong acid is titrated with a strong base. Therefore it is assumed that the bias of the end-point detection and its uncertainty are negligible.

    VT is found to be 18.64 ml and combining the remaining contributions to the uncertainty u(VT) of the volume VT gives a value of

A2.5 Step 4: Calculating the Combined Standard Uncertainty

cNaOH is given by

The values of the parameters in this equation, their standard uncertainties and their relative standard uncertainties are collected in Table A2.2

Table A2.2: Values and uncertainties for titration

   Description Value x Standard uncertainty u(x) Relative standard uncertainty u(x)/x
rep Repeatability 1.0 0.0005 0.0005
mKHP Mass of KHP 0.3888 g 0.00013 g 0.00033
PKHP Purity of KHP 1.0 0.00029 0.00029
MKHP Molar mass of KHP 204.2212 g mol-1 0.0038 g mol-1 0.000019
VT Volume of NaOH for KHP titration 18.64 ml 0.013 ml 0.0007

Using the values given above:

For a multiplicative expression (as above) the standard uncertainties are used as follows:

Spreadsheet software is used to simplify the above calculation of the combined standard uncertainty (see appendix e.2). The spreadsheet filled in with the appropriate values is shown as table a2.3, which appears with additional explanation.

It is instructive to examine the relative contributions of the different parameters. The contributions can easily be visualised using a histogram. Figure a2.9 shows the calculated values |u(y,xi)| from table a2.3.

Figure A2.9: Uncertainty Contributions in NaOH Standardisation

Table A2.3: Spreadsheet Calculation of Titration Uncertainty

"  A B C D E F G
1     Rep m(KHP) P(KHP) M(KHP) V(T)
2   Value 1.0 0.388 1.0 204.2212 18.64
3   Uncertainty 0.0005 0.00013 0.00029 0.0038 0.013
4              
5 rep 1.0 1.0005 1.0 1.0 1.0 1.0
6 m(KHP) 0.3888 0.3888 0.38893 0.3888 0.3888 0.3888
7 P(KHP) 1.0 1.0 1.0 1.00029 1.0 1.0
8 M(KHP) 204.2212 204.2212 204.2212 204.2212 204.2250 204.2212
9 v(T) 18.64 18.64 18.64 18.64 18.64 18.653
10              
11 c(NaOH) 0.102136 0.102187 0.102170 0.102166 0.102134 0.102065
12  u(y,xi)   0.000051 0.000034 0.000030 -0.00002 -0.000071
13 u(y)2,u(y,xi)2 9.72E-9 2.62E-9 1.16E-9 9E-10 4E-12 5.041E-9
14              
15 u(c(NaOH)) 0.000099          

The values of the parameters are given in the second row from C2 to G2. Their standard uncertainties are entered in the row below (C3-G3). The spreadsheet copies the values from C2-G2 into the second column from B5 to B9. The result (c(NaOH)) using these values is given in B11. C5 shows the value of the repeatability from C2 plus its uncertainty given in C3. The result of the calculation using the values C5-C9 is given in C11. The columns D and G follow a similar procedure. The values shown in the row 12 (C12-G12) are the differences of the row (C11-G11) minus the value given in B11. In row 13 (C13-G13) the values of row 12 (C12-G12) are squared and summed to give the value shown in B13. B15 gives the combined standard uncertainty, which is the square root of B13.

The contribution of the uncertainty of the titration volume VT is by far the largest followed by the repeatability. The weighing procedure and the purity of the titrimetric standard show the same order of magnitude, whereas the uncertainty in the molar mass is again nearly an order of magnitude smaller.

A2.6 Step 5: Re-evaluate the Significant Components

The contribution of V(T) is the largest one. The volume of NaOH for titration of KHP (V(T)) itself is affected by four influence quantities: the repeatability of the volume delivery, the calibration of the piston burette, the difference between the operation and calibration temperature of the burette and the repeatability of the end-point detection. Checking the size of each contribution, the calibration is by far the largest. Therefore this contribution needs to be investigated more thoroughly.

The standard uncertainty of the calibration of V(T) was calculated from the data given by the manufacturer assuming a triangular distribution. The influence of the choice of the shape of the distribution is shown in Table A2.4.

According to the ISO Guide 4.3.9 Note 1:

For a normal distribution with expectation and standard deviation , the interval ±3 encompasses approximately 99.73 percent of the distribution. Thus, if the upper and lower bounds + and - define 99.73 percent limits rather than 100 percent limits, Xi can be assumed to be approximately normally distributed rather than there being no specific knowledge about Xi [between the bounds], then u2(xi) = 2/9. By comparison, the variance of a symmetric rectangular distribution of the half-width is 2/3 ... and that of a symmetric triangular distribution of the half-width is 2/6 ... The magnitudes of the variances of the three distributions are surprisingly similar in view of the differences in the assumptions upon which they are based.

Thus the choice of the distribution function of this influence quantity has little effect on the value of the combined standard uncertainty (uc(cNaOH)) and it is adequate to assume that it is triangular.

The expanded uncertainty U(cNaOH) is obtained by multiplying the combined standard uncertainty by a coverage factor of 2.

Thus the concentration of the NaOH solution is (0.1021 ±0.0002) mol 1-1.

Table A2.4: Effect of Different Distribution Assumptions

Distribution factor u(V(T;cal)) (ml) u(V(T)) (ml) uc(cNaOH)
Rectangular 0.017 0.019 0.00011 mol l-1
Triangular 0.012 0.015 0.00009 mol l-1
NormalNOTE1 0.010 0.013 0.000085 mol l-1

Note 1: The factor of arises from the factor of 3 in Note 1 of ISO Guide 4.3.9.