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 end-point 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 end-point 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: =2.1e-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 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 end-point 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 end-point 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: =2.1e-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 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: =2.1e-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 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% (t-table 95.45%) |