Standardising a Sodium Hydroxide Solution (A2)
This section is based on a measurement described in Appendix 2 of the 3rd edition of the EURACHEM / CITAC Guide [1].
The CITAC Guide gives a careful discussion of the uncertainty analysis leading to particular numerical values. The following shows only how the subsequent calculation can be preformed using GTC.
The measurement
The concentration of a solution of NaOH is to be determined. The NaOH is titrated against the titrimetric standard potassium hydrogen phthalate (KHP).
The measurand can be expressed as
- where
\(c_\mathrm{NaOH}\) is the concentration expressed in mol/L,
\(1000\) is a volume conversion factor from mL to L,
\(m_\mathrm{KHP}\) is the mass of the titrimetric standard in g,
\(P_\mathrm{KHP}\) is the purity of the titrimetric standard as a mass fraction,
\(M_\mathrm{KHP}\) is the molar mass of KHP in g/mol,
\(V_\mathrm{T}\) is the titration volume of NaOH solution in mL.
The uncertainty contributions
Section A2.4 of the CITAC Guide provides numerical estimates of influence quantities, which can be used to define uncertain numbers for the calculation.
The mass \(m_\mathrm{KHP}\) is determined from the difference of two weighings with balance linearity as the only source of measurement error considered. However, a linearity error occurs twice: once in the tare weighing and once in the gross weighing. So in the calculations we introduce the nett weight as a number (0.3888) and the uncertainty contribution is found by taking the difference of uncertain numbers representing the errors that occur during the weighings (if the raw observations were available, they might have been used to define u_lin_tare and u_lin_gross) [2].
>>> u_lin_tare = ureal(0,type_b.uniform(0.15E-3),label='u_lin_tare')
>>> u_lin_gross = ureal(0,type_b.uniform(0.15E-3),label='u_lin_gross')
>>> u_m_KHP = u_lin_gross - u_lin_tare
>>> m_KHP = result( 0.3888 + u_m_KHP, label='m_KHP' )
The purity \(P_\mathrm{KHP}\) is [3]
>>> P_KHP = ureal(1.0,type_b.uniform(0.0005),label='P_KHP')
The molar mass \(m_\mathrm{KHP}\) is calculated from IUPAC data and the number of each constituent element in the KHP molecule \(\mathrm{C}_8\mathrm{H}_5\mathrm{O}_4\mathrm{K}\).
>>> M_C = ureal(12.0107,type_b.uniform(0.0008),label='M_C')
>>> M_H = ureal(1.00794,type_b.uniform(0.00007),label='M_H')
>>> M_O = ureal(15.9994,type_b.uniform(0.0003),label='M_O')
>>> M_K = ureal(39.0983,type_b.uniform(0.0001),label='M_K')
>>> M_KHP = result( 8*M_C + 5*M_H + 4*M_O + M_K, label='M_KHP' )
The volume term \(V_\mathrm{T2}\) is affected by contributions from calibration error and temperature.
>>> uV_T_cal = ureal(0,type_b.triangular(0.03),label='V_T_cal')
>>> uV_T_temp = ureal(0,0.006,label='V_T_temp')
>>> V_T = result( 18.64 + uV_T_cal + uV_T_temp, label='V_T' )
The CITAC Guide introduces a further multiplicative term \(R\) to represent repeatability errors (\(R \approx 1\))
In the GTC calculation this is represented by another uncertain number
>>> R = ureal(1.0,0.0005,label='R')
The uncertainty calculation
The calculation of \(c_\mathrm{NaOH}\) is now [4]:
>>> c_NaOH = R * (1000 * m_KHP * P_KHP)/(M_KHP * V_T)
>>> c_NaOH
ureal(0.102136159706...,0.000100500722124...,inf)
The contribution from different influences can be examined (and compared with Fig. A2.9 in the Guide)
>>> for i in rp.budget(c_NaOH,influences=[m_KHP,P_KHP,M_KHP,V_T,R]):
... print( " {}: {:G}".format(i.label,i.u) )
...
V_T: 7.47292E-05
R: 5.10681E-05
m_KHP: 3.21735E-05
P_KHP: 2.94842E-05
M_KHP: 1.88312E-06
The full uncertainty budget is
>>> for i in rp.budget(c_NaOH):
... print( " {}: {:G}".format(i.label,i.u) )
...
V_T_cal: 6.71088E-05
R: 5.10681E-05
V_T_temp: 3.28764E-05
P_KHP: 2.94842E-05
u_lin_tare: 2.27501E-05
u_lin_gross: 2.27501E-05
M_C: 1.84798E-06
Footnotes