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 V_{T2}}
V_{T2} = V_{T2}_{ }_{nominal} * f_{VT2}_{}_{calibration} * f_{VT2}_{}_{temperature};
{calculation of the uncertainty of V_{T1}}
V_{T1} = V_{T1}_{ }_{nominal} * f_{VT1}_{}_{calibration} * f_{VT1}_{}_{temperature};
{calculation of the uncertainty of V_{HCl}}
V_{HCl} = V_{HCl}_{ }_{nominal} * f_{VHCl}_{}_{calibration} * f_{VHCl}_{}_{temperature};
{molar mass of KHP}
M_{KHP} = 8 * M_{C} + 5 * M_{H} + 4 * M_{O} + M_{K};
{calculation of the the HCl concentration}
c_{HCl} = ( k_{mL} * m_{KHP} * P_{KHP} * V_{T2}) / (V_{T1} * M_{KHP} * V_{HCl} ) * f_{repeatability}; 
List of Quantities:
Quantity

Unit

Definition


V_{T2}

mL

Volume of NaOH for HCl titration

V_{T2}_{ }_{nominal}

mL

Nominal volume of NaOH for HCl titration

f_{VT2}_{}_{calibration}


Uncertainty contribution to V_{T2} due to instrument calibration

f_{VT2}_{}_{temperature}


Uncertainty contribution to V_{T2} due to temperature variation

V_{T1}

mL

Volume of NaOH for KHP titration

V_{T1}_{ }_{nominal}

mL

Nominal volume of NaOH for KHP titration

f_{VT1}_{}_{calibration}


Uncertainty contribution to V_{T1} due to instrument calibration

f_{VT1}_{}_{temperature}


Uncertainty contribution to V_{T1} due to temperature variation

V_{HCl}

mL

HCl aliquot for NaOH titration

V_{HCl}_{ }_{nominal}

mL

Nominal volume of HCl for NaOH titration

f_{VHCl}_{}_{calibration}


Uncertainty contribution to V_{HCl} due to pipette calibration

f_{VHCl}_{}_{temperature}


Uncertainty contribution to V_{HCl} due to temperature variation

M_{KHP}

g/mol

Molar mass of KHP

M_{C}

g/mol

Atomic weight of carbon

M_{H}

g/mol

Atomic weight of hydrogen

M_{O}

g/mol

Atomic weight of oxygen

M_{K}

g/mol

Atomic weight of potassium

c_{HCl}

mol/L

HCl solution concentration

k_{mL}

mL/L

Conversion factor 1000 mL = 1 L

m_{KHP}

g

Weight of KHP

P_{KHP}


Purity of KHP

f_{repeatability}


Uncertainty contribution attributed to repeatability

V_{T2}_{ }_{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, f
_{repeatability}. 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 endpoint detection. The titration is performed under a protective atmosphere (Ar) to prevent absorption of CO
_{2}, which would bias the titration. No further uncertainty contributions are introduced to cover the bias of the endpoint detection.
f_{VT2}_{}_{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 f
_{VT2}_{}_{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).
f_{VT2}_{}_{temperature}: 
Type B rectangular distribution
Value: 1
Halfwidth of Limits: =2.1e4*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 f
_{VT2}_{}_{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.
V_{T1}_{ }_{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, f
_{repeatability}. 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 endpoint detection. The titration is performed under a protective atmosphere (Ar) to prevent absorption of CO
_{2}, which would bias the titration. No further uncertainty contributions are introduced to cover the bias of the endpoint detection.
f_{VT1}_{}_{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 f
_{VT1}_{}_{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).
f_{VT1}_{}_{temperature}: 
Type B rectangular distribution
Value: 1
Halfwidth of Limits: =2.1e4*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 f
_{VT1}_{}_{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.
V_{HCl}_{ }_{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, f
_{repeatability}.
f_{VHCl}_{}_{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 f
_{VHCl}_{}_{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).
f_{VHCl}_{}_{temperature}: 
Type B rectangular distribution
Value: 1
Halfwidth of Limits: =2.1e4*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 f
_{VHCl}_{}_{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.
M_{C}: 
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.
M_{H}: 
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.
M_{O}: 
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.
M_{K}: 
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.
k_{mL}: 
Constant
Value: 1000 mL/L

m_{KHP}: 
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, f
_{repeatability}. 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.
P_{KHP}: 
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.
f_{repeatability}: 
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 f
_{repeatability} 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


V_{T2}

14.8900 mL

0.0142 mL

V_{T1}

18.6400 mL

0.0152 mL

V_{HCl}

15.0000 mL

0.0109 mL

M_{KHP}

204.22120 g/mol

3.77·10^{3} g/mol

Uncertainty Budgets:
c
_{HCl}: HCl solution concentration
Quantity

Value

Standard Uncertainty

Distribution

Sensitivity Coefficient

Uncertainty Contribution

Index


V_{T2}_{ }_{nominal}

14.89 mL






f_{VT2}_{}_{calibration}

1.000000

823·10^{6}

triangular

0.10

83·10^{6} mol/L

20.5 %

f_{VT2}_{}_{temperature}

1.000000

485·10^{6}

rectangular

0.10

49·10^{6} mol/L

7.1 %

V_{T1}_{ }_{nominal}

18.64 mL






f_{VT1}_{}_{calibration}

1.000000

657·10^{6}

triangular

0.10

67·10^{6} mol/L

13.1 %

f_{VT1}_{}_{temperature}

1.000000

485·10^{6}

rectangular

0.10

49·10^{6} mol/L

7.1 %

V_{HCl}_{ }_{nominal}

15.0 mL






f_{VHCl}_{}_{calibration}

1.000000

544·10^{6}

triangular

0.10

55·10^{6} mol/L

9.0 %

f_{VHCl}_{}_{temperature}

1.000000

485·10^{6}

rectangular

0.10

49·10^{6} mol/L

7.1 %

M_{C}

12.010700 g/mol

462·10^{6} g/mol

rectangular

4.0·10^{3}

1.8·10^{6} mol/L

0.0 %

M_{H}

1.0079400 g/mol

40.4·10^{6} g/mol

rectangular

2.5·10^{3}

100·10^{9} mol/L

0.0 %

M_{O}

15.999400 g/mol

173·10^{6} g/mol

rectangular

2.0·10^{3}

340·10^{9} mol/L

0.0 %

M_{K}

39.0983000 g/mol

57.7·10^{6} g/mol

rectangular

500·10^{6}

29·10^{9} mol/L

0.0 %

k_{mL}

1000.0 mL/L






m_{KHP}

0.388800 g

122·10^{6} g

normal

0.26

32·10^{6} mol/L

3.0 %

P_{KHP}

1.000000

289·10^{6}

rectangular

0.10

29·10^{6} mol/L

2.5 %

f_{repeatability}

1.00000

1.00·10^{3}

normal

0.10

100·10^{6} mol/L

30.4 %

c_{HCl}

0.101387 mol/L

184·10^{6} mol/L

Results:
Quantity

Value

Expanded Uncertainty

Coverage factor

Coverage


c_{HCl}

0.10139 mol/L

370·10^{6} mol/L

2.00

95% (ttable 95.45%)
