An acid/base titration

This is the example A3 of the EURACHEM / CITAC Guide "Quantifying Uncertainty in Analytical Measurement", Second Edition.

A solution of hydrochloric acid (HCl) is standardised against a solution of sodium hydroxide (NaOH) with known content. The standardisation of the NaOH solution is similar to example A2.

Model Equation:

{calculation of the uncertainty of VT2}

VT2 = VT2 nominal * fVT2-calibration * fVT2-temperature;

{calculation of the uncertainty of VT1}

VT1 = VT1 nominal * fVT1-calibration * fVT1-temperature;

{calculation of the uncertainty of VHCl}

VHCl = VHCl nominal * fVHCl-calibration * fVHCl-temperature;

{molar mass of KHP}

MKHP = 8 * MC + 5 * MH + 4 * MO + MK;

{calculation of the the HCl concentration}

cHCl = ( kmL * mKHP * PKHP * VT2) / (VT1 * MKHP * VHCl ) * frepeatability;

List of Quantities:

Quantity Unit Definition
VT2 mL Volume of NaOH for HCl titration
VT2 nominal mL Nominal volume of NaOH for HCl titration
fVT2-calibration   Uncertainty contribution to VT2 due to instrument calibration
fVT2-temperature   Uncertainty contribution to VT2 due to temperature variation
VT1 mL Volume of NaOH for KHP titration
VT1 nominal mL Nominal volume of NaOH for KHP titration
fVT1-calibration   Uncertainty contribution to VT1 due to instrument calibration
fVT1-temperature   Uncertainty contribution to VT1 due to temperature variation
VHCl mL HCl aliquot for NaOH titration
VHCl nominal mL Nominal volume of HCl for NaOH titration
fVHCl-calibration   Uncertainty contribution to VHCl due to pipette calibration
fVHCl-temperature   Uncertainty contribution to VHCl due to temperature variation
MKHP g/mol Molar mass of KHP
MC g/mol Atomic weight of carbon
MH g/mol Atomic weight of hydrogen
MO g/mol Atomic weight of oxygen
MK g/mol Atomic weight of potassium
cHCl mol/L HCl solution concentration
kmL mL/L Conversion factor 1000 mL = 1 L
mKHP g Weight of KHP
PKHP   Purity of KHP
frepeatability   Uncertainty contribution attributed to repeatability

VT2 nominal: Constant
Value: 14.89 mL
The nominal volume is not associated with any uncertainties. The uncertainty of the real volume of the burette has three components, repeatability, calibration and temperature. The latter two are included in the uncertainty budget as separate factors. Repeatability of the volume delivery is taken into account via the combined repeatability term for the experiment, frepeatability. Another factor influencing the result of the titration, which can also be attributed to the automatic titration system, of which the burette is one part, is the bias of the end-point detection. The titration is performed under a protective atmosphere (Ar) to prevent absorption of CO2, which would bias the titration. No further uncertainty contributions are introduced to cover the bias of the end-point detection.

fVT2-calibration: Type B triangular distribution
Value: 1
Halfwidth of Limits: =0.03/14.89
The limits of accuracy for a 20 mL piston burette are indicated by the manufacturer as typically ±0.03 ml. No further statement is made about the level of confidence or the underlying distribution. An assumption is necessary to work with this uncertainty statement. In this case a triangular distribution is assumed. Since fVT2-calibration is a multiplicative factor to the nominal volume, which is only used to introduce the calibration uncertainty, it has the value 1. The halfwidth of limits corresponds to the relative uncertainty as stated by the manufacturer (i.e. 0.03 mL / 14.89 mL).

fVT2-temperature: Type B rectangular distribution
Value: 1
Halfwidth of Limits: =2.1e-4*4
The laboratory temperature can vary by ±4°C. The uncertainty of the volume due to temperature variations can be calculated from the estimate of the possible temperature range and the coefficient of the volume expansion. The volume expansion of the liquid is considerably larger than that of the burette, so only the volume expansion of the liquid is considered. The coefficient of volume expansion for water is 2.1·10-4°C-1. This leads to a possible volume variation of ±(15 · 4 · 2.1·10-4) mL. A rectangular distribution is assumed for the temperature variation Since fVT2-temperature is a multiplicative factor to the nominal volume, which is only used to introduce the temperature uncertainty, it has the value 1. Its uncertainty is calculated as the possible volume variation divided by the volume dispensed.

VT1 nominal: Constant
Value: 18.64 mL
The nominal volume is not associated with any uncertainties. The uncertainty of the real volume of the burette has three components, repeatability, calibration and temperature. The latter two are included in the uncertainty budget as separate factors. Repeatability of the volume delivery is taken into account via the combined repeatability term for the experiment, frepeatability. Another factor influencing the result of the titration, which can also be attributed to the automatic titration system, of which the burette is one part, is the bias of the end-point detection. The titration is performed under a protective atmosphere (Ar) to prevent absorption of CO2, which would bias the titration. No further uncertainty contributions are introduced to cover the bias of the end-point detection.

fVT1-calibration: Type B triangular distribution
Value: 1
Halfwidth of Limits: =0.03/18.64
The limits of accuracy for a 20 mL piston burette are indicated by the manufacturer as typically ±0.03 ml. No further statement is made about the level of confidence or the underlying distribution. An assumption is necessary to work with this uncertainty statement. In this case a triangular distribution is assumed. Since fVT1-calibration is a multiplicative factor to the nominal volume, which is only used to introduce the calibration uncertainty, it has the value 1. The halfwidth of limits corresponds to the relative uncertainty as stated by the manufacturer (i.e. 0.03 mL / 18.64 mL).

fVT1-temperature: Type B rectangular distribution
Value: 1
Halfwidth of Limits: =2.1e-4*4
The laboratory temperature can vary by ±4°C. The uncertainty of the volume due to temperature variations can be calculated from the estimate of the possible temperature range and the coefficient of the volume expansion. The volume expansion of the liquid is considerably larger than that of the burette, so only the volume expansion of the liquid is considered. The coefficient of volume expansion for water is 2.1·10-4°C-1. This leads to a possible volume variation of ±(19 · 4 · 2.1·10-4) mL. A rectangular distribution is assumed for the temperature variation Since fVT1-temperature is a multiplicative factor to the nominal volume, which is only used to introduce the temperature uncertainty, it has the value 1. Its uncertainty is calculated as the possible volume variation divided by the volume dispensed.

VHCl nominal: Constant
Value: 15 mL
The nominal volume is not associated with any uncertainties. The uncertainty of the real volume of the pipette has three components, repeatability, calibration and temperature. The latter two are included in the uncertainty budget as separate factors. Repeatability of the volume delivery is taken into account via the combined repeatability term for the experiment, frepeatability.

fVHCl-calibration: Type B triangular distribution
Value: 1
Halfwidth of Limits: =0.02/15
The uncertainty stated by the manufacturer for a 15 mL pipette is ±0.02 mL. No further statement is made about the level of confidence or the underlying distribution. An assumption is necessary to work with this uncertainty statement. In this case a triangular distribution is assumed. Since fVHCl-calibration is a multiplicative factor to the nominal volume, which is only used to introduce the calibration uncertainty, it has the value 1. The halfwidth of limits corresponds to the relative uncertainty as stated by the manufacturer (i.e. 0.02 mL / 15 mL).

fVHCl-temperature: Type B rectangular distribution
Value: 1
Halfwidth of Limits: =2.1e-4*4
The laboratory temperature can vary by ±4°C. The uncertainty of the volume due to temperature variations can be calculated from the estimate of the possible temperature range and the coefficient of the volume expansion. The volume expansion of the liquid is considerably larger than that of the pipette, so only the volume expansion of the liquid is considered. The coefficient of volume expansion for water is 2.1·10-4°C-1. This leads to a possible volume variation of ±(15 · 4 · 2.1·10-4) mL. A rectangular distribution is assumed for the temperature variation Since fVHCl-temperature is a multiplicative factor to the nominal volume, which is only used to introduce the temperature uncertainty, it has the value 1. Its uncertainty is calculated as the possible volume variation divided by the volume dispensed.

MC: Type B rectangular distribution
Value: 12.0107 g/mol
Halfwidth of Limits: 0.0008 g/mol
The atomic weight of carbon and its uncertainty are taken from data listed in the latest IUPAC table of atomic weights. The IUPAC quoted data is considered to be of rectangular distribution.

MH: Type B rectangular distribution
Value: 1.00794 g/mol
Halfwidth of Limits: 0.00007 g/mol
The atomic weight of hydrogen and its uncertainty are taken from data listed in the latest IUPAC table of atomic weights. The IUPAC quoted data is considered to be of rectangular distribution.

MO: Type B rectangular distribution
Value: 15.9994 g/mol
Halfwidth of Limits: 0.0003 g/mol
The atomic weight of oxigen and its uncertainty are taken from data listed in the latest IUPAC table of atomic weights. The IUPAC quoted data is considered to be of rectangular distribution.

MK: Type B rectangular distribution
Value: 39.0983 g/mol
Halfwidth of Limits: 0.0001 g/mol
The atomic weight of potassium and its uncertainty are taken from data listed in the latest IUPAC table of atomic weights. The IUPAC quoted data is considered to be of rectangular distribution.

kmL: Constant
Value: 1000 mL/L
mKHP: Type B normal distribution
Value: 0.3888 g
Expanded Uncertainty: =sqrt(2*sqr(0.00015/sqrt(3)))
Coverage Factor: 1
Repeatability of the wheighing is taken into account via the combined repeatability term, frepeatability. Any systematic offset across the scale will also cancel due to the wheighing by difference. The only contributing source of uncertainty is the linearity of the balance. The calibration certficate of the balance quotes ±0.15 mg for the linearity. The manufacturer recommends using a rectangular distribution to convert this linearity contribution into a standard uncertatiny. This uncertainty is accounted for twice, once for the tare and once for the gross mass.

PKHP: Type B rectangular distribution
Value: 1
Halfwidth of Limits: 0.05 %
In the supplier's catalogue, the purity of the KHP is given as 100%±0.05%. No further information concerning the uncertainty is given. Therefore this value is assumed to be of rectangular distribution.

frepeatability: Type B normal distribution
Value: 1
Expanded Uncertainty: 0.1 %
Coverage Factor: 1
All uncertainty contributions due to repeatability of one of the operations are combined in this factor. It includes at least the repeatability of the wheighings and of the volumes delivered by the burette and the pipette. The magnitude of this uncertainty contribution is assessed during the method validation stage. The data shows that the overall repeatability of the titration experiment is 0.1%. Since frepeatability is a multiplicative factor to the result, which is only used to introduce the repeatability uncertainty, it has the value 1 with an uncertainty of 0.1%.

Interim Results:

Quantity Value Standard
Uncertainty
VT2 14.8900 mL 0.0142 mL
VT1 18.6400 mL 0.0152 mL
VHCl 15.0000 mL 0.0109 mL
MKHP 204.22120 g/mol 3.77·10-3 g/mol

Uncertainty Budgets:

cHCl: HCl solution concentration
Quantity Value Standard
Uncertainty
Distribution Sensitivity
Coefficient
Uncertainty
Contribution
Index
VT2 nominal 14.89 mL          
fVT2-calibration 1.000000 823·10-6 triangular 0.10 83·10-6 mol/L 20.5 %
fVT2-temperature 1.000000 485·10-6 rectangular 0.10 49·10-6 mol/L 7.1 %
VT1 nominal 18.64 mL          
fVT1-calibration 1.000000 657·10-6 triangular -0.10 -67·10-6 mol/L 13.1 %
fVT1-temperature 1.000000 485·10-6 rectangular -0.10 -49·10-6 mol/L 7.1 %
VHCl nominal 15.0 mL          
fVHCl-calibration 1.000000 544·10-6 triangular -0.10 -55·10-6 mol/L 9.0 %
fVHCl-temperature 1.000000 485·10-6 rectangular -0.10 -49·10-6 mol/L 7.1 %
MC 12.010700 g/mol 462·10-6 g/mol rectangular -4.0·10-3 -1.8·10-6 mol/L 0.0 %
MH 1.0079400 g/mol 40.4·10-6 g/mol rectangular -2.5·10-3 -100·10-9 mol/L 0.0 %
MO 15.999400 g/mol 173·10-6 g/mol rectangular -2.0·10-3 -340·10-9 mol/L 0.0 %
MK 39.0983000 g/mol 57.7·10-6 g/mol rectangular -500·10-6 -29·10-9 mol/L 0.0 %
kmL 1000.0 mL/L          
mKHP 0.388800 g 122·10-6 g normal 0.26 32·10-6 mol/L 3.0 %
PKHP 1.000000 289·10-6 rectangular 0.10 29·10-6 mol/L 2.5 %
frepeatability 1.00000 1.00·10-3 normal 0.10 100·10-6 mol/L 30.4 %
cHCl 0.101387 mol/L 184·10-6 mol/L

Results:

Quantity Value Expanded
Uncertainty
Coverage
factor
Coverage
cHCl 0.10139 mol/L 370·10-6 mol/L 2.00 95% (t-table 95.45%)

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