Electronic Supplementary Material
Accreditation and Quality Assurance, Springer Verlag
Berlin and Heidelberg, 2013
Pitzer ion activities in mixed electrolytes for calibration of ion selective electrodes used in clinical
chemistry
Frank Bastkowskia)*, Petra Spitzera), Ralf Eberhardta), Beatrice Adela), Samuel Wunderlib), Daniel
Berdatb), Hanspeter Andresb), Olivier Brunschwigb), Michal Máriássyc), Roger Fehérd), Caspar Demuthd),
Fabiano Barbieri Gonzagae), Paulo Paschoal Borgese), Wiler Batista da Silva Juniore), Alena Vospělováf),
Martina Vičarováf), Sirinapha Srithongtimg)
a)
Physikalisch-Technische Bundesanstalt (PTB), Bundesallee 100, 38116 Braunschweig, Germany,
e-mail: frank.bastkowski@ptb.de, phone: +495315923323, fax: +49531592693323
b)
Federal Institute of Metrology (METAS), Lindenweg 50, 3003 Bern-Wabern, Switzerland
e-mail: samuel.wunderli@metas.ch, phone: +41583870383, fax: +41583870210
c)
Slovak Institute of Metrology (SMU), Karloveská 63, 842 55 Bratislava 4, Slovakia
e-mail: mariassy@smu.gov.sk, phone:+420260294522, fax: +420260294561
d)
Zurich University of Applied Sciences (ZHAW), Campus Grüental, 8820 Wädenswil, Switzerland
e-mail:caspar.demuth@zhaw.ch, phone: +41589345763, fax: +41 58 934 50 01
e)
National Institute of Metrology, Quality and Technology (INMETRO),Santa Alexandrina St, 416 Rio
Comprido, 20261-232 Rio de Janeiro, Brazil
e-mail: fbgonzaga@inmetro.gov.br, phone:+552126799134, fax: +552126799069
f)
Ceský metrologický institut (CMI), Okružní 31, Brno 638 00, Czech Republic
e-mail: avospelova@cmi.cz, phone: +420 545 555 322, fax: +420 545 555 183
g)
National Institute of Metrology (Thailand) (NIMT), 3/4-5 Moo 3 Tambol Klonghar Amphur Klong Luang
Pathumthani, 12120 Thailand
Environmental Research Training Center (ERTC),Technopolis. Klong 5, Klong Luang Pathumthani, 12120
Thailand
e-mail: sirina.sri@gmail.com, phone: +66025774197, fax: +66025774197
In the following, Pitzer´s equations for an aqueous electrolyte solution containing calcium-, magnesium-,
sodium-, potassium- and chloride ions, are shown. Moreover, all functions and parameters are given, which are
included in these equations[1]. Figure 7 gives an impression of the number of influences contributing to the
overall uncertainty of the ion activity coefficient of a cation, which is determined by the Pitzer equation.
Cations: Na+, K+, Mg2+, Ca2+
2
ln  M  z M
 f 
Na
m
a
 2  BMa  Z  CMa 
a 1
Na


mc   2   Mc 
ma  Mca 


c 1
a 1


Nc
+

N a 1
+

Na
 
ma  ma’  aa’M  z M 
a =1 a’ a 1
Nc Na
 m  m
c
c 1 a 1
Nn
+
m
n
n 1
 2  nM 
a
 Cca
Anion: Cl 
ln  X  z 2X  f  
Nc
 m  2  B
c
cX
 Z  CcX 
c 1
Nc


ma   2   Xa 
mc  Xac 


a 1
c 1


Na
+

N c 1 N c
+


mc  mc’  cc’ X  z X 
Nc Na
 m  m
c
a
 Cca
c 1 a 1
c =1 c’ = c +1
Nn
+
m
n
 2  nX 
n 1
Hückel-Term for electrostatic long-range interaction between solution species:
1


1
I2
2

 
f  A 

 ln 1  1.2  I 2  
1


2
1.2 
 1  1.2  I


Nc Na

mc  ma  B 'ca 
N c 1 N c
c 1 a 1

mc  mc’   'cc’ 
N a 1
Na
 m
a
 ma’   'aa’
a =1 a’  a 1
c =1 c’ = c +1
C-Term independent of ionic strength
CMX 

CMX
2  zM  z X
1
2

from Pitzer-Tables
C MX
Z
z
i
 mi
I
i
1
2
z
2
i
 mi
ionic strength
i
A Hückel-Parameter is one third the debye-Hückel limiting slope:


1
e2

A   2    N A  d w 1 / 2  
0


3
 4       ( H 2O)  k  T 
3/ 2
Am Hückel-Parameter
Am  A 
3
ln 10
Density of water d w
 t  a1 2  t  a2 
d w  a5  1 
 (cf. literature [2])
a3  t  a4  

a1  -3.983035  0.00067 C
a2  301.797 C
a3  522528.9C2
a4  69.34881 C
a5  999.974950 0.00084 kg  m-3
-2/6-
Relative permittivity of water  ( H 2O) (cf. literature [3])
 (H 2O)  c1+c2  t  273 .15C +c3  t  273 .15C 2
t: absolute temperature in °C
c1  249 .21
c2  0.79069 C 1
c3  0.72997103 C 2
B-functions (depending on ionic strength):



0    1  exp    I   2   exp    I

BMX
  MX
MX
1
MX
2




0    1  g   I   2   g   I
BMX   MX
MX
1
MX
2


1
2
B'MX   MX  g ' 1  I / I   MX  g '  2  I / I





For 1:1, 1:2 or 2:1 electrolytes, respectively: ( 1  2, 2  0,  (2)  0 )

0    1  exp  2  I

BMX
  MX
MX


0    1  g 2  I
BMX   MX
MX
B'MX

1  g ' 2  I / I
  MX


For 2:2 electrolytes: ( 1  1.4, 2  12,  (2)  0 )




0    1  exp  1.4  I   2   exp  12  I

BMX
  MX
MX
MX



0    1  g 1.4  I   2   g 12  I
BMX   MX
MX
MX


1
2
B'MX   MX  g ' 1.4  I / I   MX  g ' 12  I / I

x  i  I




argument for g- and g'-functions
1  2
for pairs in which cation or anion is univalent,  2  0
1  1.4 for higher valence pairs,  2  12


g x  2  1  1  x  e x / x2
 

x2 
g ' x   2  1  1  x    e  x  / x 2

 

2 
 

 0,  1,  2 from Pitzer-Tables (cf literature [4]),
mostly  2   0 is zero for pairs with one univalent ion
Temperature dependence of -parameters used:
1 1 
T 
 0   q1  q2      q3  ln    q4  T  T0   q5  T 2  T02
 T T0 
 T0 
q1
q2
q3
q4
q5
Na+Clˉ 0.0765 -777.03 -4.4706 0.008946 -3.3158E-6
K+Clˉ 0.04835 0
0
5.794E-4 0
Mg2+Clˉ 0.35235 0
0
-1.943E-4 0
Ca2+Clˉ 0.3159 0
0
-1.725E-4 0

-3/6-

1
 1  q6  q7  

q6
Na+Clˉ 0.2664
K+Clˉ 0.2122
Mg2+Clˉ 1.6815
Ca2+Clˉ 1.614
q7
0
0
0
0
T

T 
1
  q8  ln    q9  T  T0   q10  T 2  T02

T 
T0 
 0
q8
0
0
0
0
q9
6.1608E-5
10.71E-4
3.6525E-3
3.9E-3

q10
1.0715E-6
0
0
0
T0=298.15 K
 2 is considered to be zero in case of 1:1, 1:2 ions
Temperature dependence of C-parameters:
1 1 
T 

CMX
 q16  q17      q18  ln    q19  T  T0 
T
T
0

 T0 
q16
q17
q18
q19
Na+Clˉ 0.00127 33.317
0.09421 -4.655E-5
K+Clˉ -0.00084 0
0
-5.095E-5
Mg2+Clˉ 0.00519
0
0
-1.649333E-4
Ca2+Clˉ -0.00034 0
0
6.213E-5
CMX 

CMX
2  zM  z X
1
2

from Pitzer-Tables (cf. above)
CMX
Ion interaction approach: Theory and data correlation, Chapter 3, K.S. Pitzer, p.75-129
Activity coefficients in electrolyte solutions 2nd ed. (cf. literature [5])

Second virial coefficients  ij , ij , 'ij are dependent mainly of ionic strength:
Theta parameter values for cation-cation´ respectively anion-anion´ interactions
ij  ij  E ij I   I E 'ij I 
 ij  ij  E ij I 
'ij  E 'ij I 
ij   ij  I   'ij I 
ij from Pitzer-Tables
ij -values (cation-cation´ respectively anion-anion´):
Na+K+
Na+ Mg2+
Na+Ca2+
K+Mg2+
K+Ca2+
Mg2+Ca2+
-0.012
0.07
0.07
0.0
0.032
0.007
xMN  6  zM  z N  A  I arguments for J-functions (Integrals)
MN I  is a function of the ionic strength and electrolyte pair type and is nonzero only for non-symmetric
electrolytes (e.g.1-2 types).
E
-4/6-




zM  z N
 J 0 xMN   12  J 0 xMM   12  J 0 xNN 
4 I
E
z z
 MN
E
'MN I   M 2N  J1 xMN   12  J1 xMM   12  J1 xNN  
I
8 I
E
 MN I  
J1 xMN  is the derivative of the function J 0 xMN  with respect to xMN
J-integrals for
J 0 x   14  x  1 
J1 x   14  x 
1

x



 1  exp   y  e
0
x
 y 
  y 2 dy



 x

1  
x
 1  1   e  y   exp    e  y   y 2dy
x  
y

 y

0

xNaNa  6  1  1  A  I
xNaK  6  1  1  A  I
xNaMg  6 1 2  A  I
xNaCa  6  1  2  A  I
xKK  6  1  1  A  I
xKMg  6 1 2  A  I
xKCa  6 1 2  A  I
xMgMg  6  2  2  A  I
xMgCa  6  2  2  A  I
xCaCa  6  2  2  A  I
xClCl  6  1  1  A  I
Fig. 5
Cause and effect diagram for the Pitzer single ion activities  of a cation in a
complex electrolyte mixture (cf. Appendix). For each influence parameter a standard
uncertainty is attributed to the value. The uncertainties of the Pitzer parameters were
estimated type B uncertainties. All coefficients of the model equations (water density,
-5/6-
water relative permittivity with its temperature dependence, temperature dependence
of the Pitzer parameters including constants such as the Avogadro and Boltzmann
constant, etc.) were included with their uncertainties.
References
1. Harvie CE, Møller N, Weare JH (1984) The prediction of mineral solubilities in natural waters: The Na-KMg-Ca-H-CI-SO4-OH-HCO3-CO3-CO2-H2O system to high ionic strengths at 25°C Geochimica et
Cosmochimica Acta 48:723-751
2. M. Tanaka, G. Girard, R. Davis, A. Peuto, Bignell N (2001) Recommended table for the density of water
between 0 °C and 40 °C based on recent experimental reports. Metrologia 38:301-309
3. D.R. Lide (1995) CRC Handbook of Chemistry and Physics. 76th edn. CRC Press, Boca Raton, FL
4. Pitzer KS, Mayorga G (1973) Thermodynamics of Electrolytes. I I. Activity and Osmotic Coefficients for
Strong Electrolytes with One or Both Ions Univalent. J Phys Chem 77:2300-2308
5. Pitzer KS (1974) Thermodynamics of Electrolytes. III. Activity and Osmotic Coefficients for 2-2
Electrolytes. Journal of Solution Chemistry 3:539-546
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