Standardising a sodium hydroxide solution
This is the example A2 of the EURACHEM / CITAC Guide "Quantifying Uncertainty in Analytical Measurement", Second Edition.
A solution of sodium hydroxide (NaOH) is standardised against the titrimetric standard potassium hydrogen phtalate (KHP).
The titrimetric standard KHP is dried and weighed. After the preparation of the NaOH solution a sample of the KHP is dissolved and then titrated using the NaOH solution.
Model Equation:
{Molar mass of KHP}
M_{KHP} = 8 * M_{C} + 5 * M_{H} + 4 * M_{O} + M_{K};
{Volume delivered by the piston burette, excluding repeatability}
V_{T} = V_{nominal} * f_{V}_{}_{calibration} * f_{V}_{}_{temperature};
{mass of KHP weighed, excluding repeatability}
m_{KHP} = m_{container}_{ }_{and}_{ }_{KHP}  m_{container}_{ }_{less}_{ }_{KHP};
{calculation of the concentration of the NaOH solution}
c_{NaOH} = (k_{mL} * m_{KHP} * P_{KHP}) / (M_{KHP} * V_{T}) * f_{repeatability};

List of Quantities:
Quantity  Unit  Definition 

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 
V_{T}  mL  Volume delivered by piston burette 
V_{nominal}  mL  Volume indicated by burette 
f_{V}_{}_{calibration}  Uncertainty contribution to the volume due to uncertainty in calibration of the burette  
f_{V}_{}_{temperature}  Uncertainty contribution to the volume due to temperature variation  
m_{KHP}  g  Mass of KHP weighed 
m_{container}_{ }_{and}_{ }_{KHP}  g  Mass of container and KHP 
m_{container}_{ }_{less}_{ }_{KHP}  g  Mass of container less KHP 
c_{NaOH}  mol/L  Concentration of the sodium hydroxide solution 
k_{mL}  mL/L  Conversion factor 1000 ml = 1L 
P_{KHP}  Purity of the KHP  
f_{repeatability}  Repeatability of the titration 
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.
V_{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. There are no indications that the endpoint determined from the shape of the pHcurve does not correspond to the equivalencepoint, because a strong acid is titrated with a strong base. No further uncertainty contributions are introduced to cover the bias of the endpoint detection.
f_{V}_{}_{calibration}: 
Type B triangular distribution Value: 1 Halfwidth of Limits: 0.0015 
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_{V}_{}_{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 / 20 mL).
f_{V}_{}_{temperature}: 
Type B normal distribution Value: 1 Expanded Uncertainty: 0.0003 Coverage Factor: 1 
The laboratory temperature can vary by ±3°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}, which is used here also for the NaOH solution. This leads to a possible volume variation of ±(19 · 3 · 2.1·10^{4}/1.96) mL = ±0.006 mL. A rectangular distribution is assumed for the temperature variation Since f_{V}_{}_{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 0.006 mL / 19 mL.
m_{container}_{ }_{and}_{ }_{KHP}: 
Type B rectangular distribution Value: 60.5450 g Halfwidth of Limits: 0.00015 g 
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 not correlated with the uncertainty of m_{container}_{ }_{less}_{ }_{KHP,} since they are independent observations and the linearity effects are not correlated.
m_{container}_{ }_{less}_{ }_{KHP}: 
Type B rectangular distribution Value: 60.1562 g Halfwidth of Limits: 0.00015 g 
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 not correlated with the uncertainty of m_{container}_{ }_{and}_{ }_{KHP,} since they are independent observations and the linearity effects are not correlated.
k_{mL}: 
Constant Value: 1000 mL/L 
P_{KHP}: 
Type B rectangular distribution Value: 1 Halfwidth of Limits: 0.0005 
In the supplier's catalogue, the purity of the KHP is given as 1.0000±0.0005. 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: 0.0005 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 volume delivered by the burette. 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 0.05%. 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 0.0005.
Interim Results:
Quantity  Value 
Standard Uncertainty 

M_{KHP}  204.22120 g/mol  3.77·10^{3} g/mol 
V_{T}  18.6400 mL  0.0127 mL 
m_{KHP}  0.388800 g  122·10^{6} g 
Uncertainty Budgets:
c_{NaOH}: Concentration of the sodium hydroxide solution
Quantity  Value 
Standard Uncertainty 
Distribution 
Sensitivity Coefficient 
Uncertainty Contribution 
Index 

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}  350·10^{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 % 
V_{nominal}  18.64 mL  
f_{V}_{}_{calibration}  1.000000  612·10^{6}  triangular  0.10  63·10^{6} mol/L  41.8 % 
f_{V}_{}_{temperature}  1.000000  300·10^{6}  normal  0.10  31·10^{6} mol/L  10.0 % 
m_{container}_{ }_{and}_{ }_{KHP}  60.5450000 g  86.6·10^{6} g  rectangular  0.26  23·10^{6} mol/L  5.5 % 
m_{container}_{ }_{less}_{ }_{KHP}  60.1562000 g  86.6·10^{6} g  rectangular  0.26  23·10^{6} mol/L  5.5 % 
k_{mL}  1000.0 mL/L  
P_{KHP}  1.000000  289·10^{6}  rectangular  0.10  29·10^{6} mol/L  9.3 % 
f_{repeatability}  1.000000  500·10^{6}  normal  0.10  51·10^{6} mol/L  27.8 % 
c_{NaOH}  0.1021362 mol/L  96.8·10^{6} mol/L 
Results:
Quantity  Value 
Expanded Uncertainty 
Coverage factor 
Coverage 

c_{NaOH}  0.10214 mol/L  190·10^{6} mol/L  2.00  95% (ttable 95.45%) 