Time-Dependent Determination of Unbiased Selectivity Coefficients

Article Cite This: Anal. Chem. 2017, 89, 13441−13448

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Time-Dependent Determination of Unbiased Selectivity Coefficients of Ion-Selective Electrodes for Multivalent Ions Elena Zdrachek and Eric Bakker* Department of Inorganic and Analytical Chemistry, University of Geneva, Quai Ernest-Ansermet 30, CH-1211 Geneva, Switzerland ABSTRACT: A new method for the determination of unbiased low selectivity coefficients for two of the most prevalent cases of multivalent ions (zi = 2, zj = 1 and zi = 1, zj = 2) was theoretically and experimentally substantiated. The method is based on eliminating the primary ion concentration near the membrane by extrapolating the linearized time dependencies of selectivity coefficients determined by the −1/3 separate solutions method (KPot or ij (SSM) as a function of t −1/6 t , depending on the charge combination of the two ions, to infinite time. The applicability of the method is demonstrated for ionophore-based Mg2+-, Ca2+-, and Na+-selective electrodes. It is shown that the high level of primary ion impurities in the salts of interfering ions can significantly limit the efficiency of the technique, as demonstrated with salts of different purity levels.

I

values of selectivity coefficients in the case of equally charged ions can be determined by analysis of the time-dependent behavior of selectivity coefficients determined by the traditional separate solutions method (SSM), which is considered one of the most frequently used20 among the half dozen methods recommended by IUPAC.1,2 According to Egorov’s method, the time-dependent potential change in a solution containing only the interfering ion of interest is monitored over time, giving time-dependent overestimated selectivity coefficient values. The time dependence of the primary ion concentration has its origin in diffusion theory, as the increasing diffusion layer thickness in the membrane results in an ever diminishing primary ion concentration at the sample side of the membrane. As a result, the selectivity coefficient plotted versus time raised to the power of −1/4 is extrapolated to zero (infinite time). This extrapolated value provides the unbiased selectivity coefficients value. Egorov’s extrapolation technique for the determination of unbiased selectivity coefficients has been evaluated and confirmed by Bakker with numerical simulations, where the protocol was further explored to membranes exhibiting strong inward and outward fluxes as well.21 Unfortunately, however, until today the application of this approach has been limited to cases of equally charged ions, which makes this otherwise attractive methodology of limited use in practice. To overcome this, modern ion-selective membrane response theory is used here to extend Egorov’s method to any divalent− monovalent ion combination. The results are experimentally confirmed with a number of ionophore-based membranes.

t is well-established that the selectivity coefficients over highly discriminating ions are usually overestimated when determined according to established IUPAC methods based on the Nikolsky−Eisenman equation1,2 and that they depend on specific measuring conditions, such as experimental time, measurement/conditioning history, and concentration of primary and/or interfering ions.3−7 The main reason for this behavior is an increased primary ion concentration at the sample side of the interface that has its origin in transmembrane transport from the inner reference solution that is typically triggered by ion-exchange processes at the sample− membrane interface.3,8−,12 Historically, a range of different approaches have been proposed to solve this problem, which include the complexation of the primary ions in the sample solution by adding a selective binding reagent,13 the adjustment of the inner reference solution composition to induce the strong inward primary ion flux,14 a careful preconditioning protocol to avoid initial contact with primary ions,15,16 or even the use of more advanced manners of instrumental control.17,18 At present, the modified separate solution method (MSSM)15,16 proposed by Bakker is still one of the most effective approaches to eliminate the influence of the primary ion on the electrode response in the interfering ion solution and to obtain unbiased selectivity coefficients. This method is based on avoiding the exposure of the ion-selective electrodes (ISEs) to the primary ion prior to the measurement of interfering ions. The key limitation of MSSM is that its protocol can be performed only once for any particular electrode and that it cannot be applied to commercially available electrodes as they arrive in a preconditioned state. An important step to estimate unbiased selectivity coefficients of “as received” ion-selective electrodes was recently introduced by Egorov’s group.19 It was shown that unbiased © 2017 American Chemical Society

Received: September 12, 2017 Accepted: November 14, 2017 Published: November 14, 2017 13441

DOI: 10.1021/acs.analchem.7b03726 Anal. Chem. 2017, 89, 13441−13448

Article

Analytical Chemistry

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EXPERIMENTAL SECTION Reagents. Potassium tetrakis[3,5-bis(trifluoromethyl)phenyl]borate (KTFPB), potassium tetrakis(4-chlorophenyl)borate (KTClPB), N,N′,N″-Tris [3-(heptylmethylamino)-3oxopropionyl]-8,8′-iminodioctylamine (Mg−IV), N,N-dicyclohexyl-N′,N′-dioctadecyl-3-oxapentanediamide (Ca−IV), 4-tertbutylcalix [4] arene-tetraacetic acid tetraethyl ester (Na−X), 2nitrophenyloctylether (NPOE), high molecular weight poly(vinyl chloride) (PVC), and tetrahydrofuran (THF) were of Selectophore grade (Sigma-Aldrich, Switzerland). Lithium acetate, sodium chloride (99.5%), potassium chloride (99.5%), magnesium chloride hexahydrate (99%), tetraethylammonium chloride (98%), calcium chloride dihydrate (99.5%), and calcium chloride tetrahydrate (99.995% Suprapur) were purchased from Sigma-Aldrich. Electrochemical Equipment and Electrode Preparation. Potentiometric measurements were carried out with a high impedance input 16-channel EMF monitor (Lawson Laboratories, Inc., Malvern, PA, U.S.A.) using a double-junction Ag/AgCl/3 M KCl/1 M LiOAc reference electrode (Metrohm Autolab, Utrecht, The Netherlands). The potentiometric measurements were based on solvent cast PVC membranes, composed of an ion-exchanger, an ionophore, PVC, and a plasticizer that were dissolved in 2 mL of THF and poured into a glass ring (22 mm ID) affixed onto a glass slide. The solution was allowed to evaporate overnight, giving a parent membrane. Each parent membrane was cut with a metallic hole puncher into disks of 8 mm diameter and mounted into Ostec electrode bodies (Oesch Sensor Technology, Sargans, Switzerland). The compositions of ISEs membranes are presented in Table 1.

Afterward, the selectivity coefficients were determined by SSM. For this, the electrodes were conditioned overnight in a 10−3 M primary ion solution, identical to the inner filling solution, before measurement. Then EMF values in interfering and primary ions solutions with different concentration levels were automatically recorded for 10 min with 1 s intervals, and selectivity coefficient values were calculated according to eq 1 each time. Activity coefficients were calculated according to ref 22.

Table 1. Membrane Compositions

where aaq j *is the activity of interfering ion j in sample m bulk, KPot is the selectivity coefficient, Daq ij i and Di

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THEORETICAL CONSIDERATIONS This work develops an approach for the estimation of unbiased selectivity coefficients of polymeric membrane electrodes for two of the most prevalent cases of multivalent ions: doubly charged primary ion/singly charged interfering ion and singly charged primary ion/doubly charged interfering ion. It is based on the combination of the recent discovery of Egorov’s group to employ the time dependence of overestimated selectivity coefficients determined by SSM for the calculation of true unbiased selectivity coefficients values for the case of equally charged ions19 and a simplification of the general equation describing mixed ion response behavior.23,24 It was previously established by Egorov’s group that a nonzero primary ion concentration in the near-boundary layer, aaq i (0), which causes the selectivity coefficient bias, can be diminished by extrapolating experimentally determined selectivity coefficients plotted against t−1/4 to a value of t−1/4 = 0 (t → ∞)19 a iaq (0)

PVCa

NPOEa

Mg− IVb

Ca− IVb

Na− Xb

2+

32.9

65.2

9.5

-

-

18.6

-

31.7

61.7

-

14.8

-

-

5.1

64.7

-

-

11.3

-

5.3

Mg ISE Ca2+ISE Na+-ISE

33.7

a

KTClPBb KTPFBb

−1

b

Mass percentage. Units are mmol kg . Ca−IV, Mg−IV, and Na−X are calcium, magnesium, and sodium ionophores, respectively (see Experimental Section).

δim(t ) =

(Ej − E i)z iF 2.303RT

+ log

c iaq(0) − c iaq * = q(c im * − c im(0))

caq i (0)

aj

(3)

(4)

cm i (0)

where and are the concentrations of primary ion i at the aqueous and membrane sides of the interface, m respectively, caq i * and ci * are the concentrations of the same ion in the bulk of the corresponding phases, and q is the permability ratio, which is defined as

ai z i/zj

πDimt

Following the same reasoning, the present system is treated by considering the ion-exchange reaction between two ions of different charge. When a membrane is exposed to a solution containing interfering ion salt, the surface polarization caused by transmembrane counterdiffusion fluxes is described as follows3,25

Selectivity Coefficient Determination. For comparison purposes, the unbiased selectivity coefficients were determined by MSSM.15,16 The electrodes were previously conditioned in 10−3 M interfering ion solution for 3−4 h (KCl for Mg2+selective electrode, Et4NCl for Ca2+-selective electrode, and CaCl2 for Na+-selective electrode), and the same solution was used as the inner reference solution. Then, EMF values in interfering and primary ions solutions, in this order, at different concentration levels were measured. Selectivity coefficients were calculated according to eq 1 using the EMF values for the highest measured ion activities corresponding to the Nernstian range of an electrode response log K ijPot =

(2)

solution are the diffusion coefficients of the primary ion i in the aqueous and membrane phases correspondingly, δaq i is the thickness of the aqueous diffusion layer (assumed to be constant with time), cm i * is the concentration of the ion i in the bulk of the membrane phase, and t is the duration of the measurement after introduction of a sample containing interfering ion only. This relationship was obtained on the basis of the consideration of a counterdiffusion process, assuming that the thickness of the diffusion layer in the membrane phase δm i (t) increases with time according to3

components ISE

⎛ a aq *K Potδ aqc m *(D m)1/2 ⎞1/2 j ij i i i ⎟ t −1/4 = ⎜⎜ ⎟ 1/2 aq π Di ⎝ ⎠

q=

(1) 13442

Dimδiaq Diaq δim(t )

(5) DOI: 10.1021/acs.analchem.7b03726 Anal. Chem. 2017, 89, 13441−13448

Article

Analytical Chemistry In the determination of selectivity coefficients by SSM, the electrode is immersed in a solution containing interfering ions only (caq i * = 0), which means that eq 4 can be rewritten as c iaq(0) = q(c im * − c im(0))

0

a iaq (0) ·10 z iF(Ei − EPB)/2.303RT + 0

(6)

z iF (Δαβ ϕi0 − E i0) 2.303RT

log c im(0) = log a iaq (0) +

The selectivity coefficient can be defined as log K ijPot =

(8)

and

E0i

⎛ ⎞ a iaq (0) 2.303RT ⎜ ⎟ log⎜ −z j / z i aq z iF (K ijPot)z j/ z i a jaq (0) ⎟⎠ ⎝ 1 − (a i (0))

μi0, α − μi0, β z iF

(9)

2.303RT 1 log m * z iF ci

(10)

It follows from eqs 7 and 8 that the concentration of the ion i partially exchanged with interfering ion at the membrane side of the phase boundary, cm i (0), is related to its concentration in the absence of such ion exchange, cm i *, as follows log

c im(0) c im *

= log

a iaq (0)

z iF + (E i0 − E PB) 2.303RT

a iaq (0) = (K ijPot)1/3 (qc im *a jaq (0))2/3 a iaq (0) = (K ijPot)2/3 (qc im *a jaq (0))1/3

(11)

a iaq (0) =

0

(12)

(DimK ijPot)1/3 (δiaqc im *a jaq *)2/3 π 1/3(Diaq )2/3

t −1/3 (19)

and for the case zi = 1, zj = 2 as

Fortunately, this relationship may be significantly simplified for relatively small levels of interferences, which corresponds to the present situation of low selectivity coefficient determination. Indeed, we have recently suggested a new approximation of potentiometric response in a mixed solution of ions of different charge for low levels of interference.27 According to ref 27, if the interference is small, EPB in the second term can be approximated by Ei, the potential measured in the absence of interference, and described by the Nernst equation as follows 2.303RT log a iaq (0) z iF

(18)

As the interfering ion j concentration in the bulk of the aqueous phase is assumed to be sufficiently high, we may assume that it does not change significantly in the nearboundary layer as a result of the ion exchange, and aaq j (0) = aaq j *. If it is considered that the diffusion layer thickness in the membrane phase changes according to eq 3, the time dependent change of the interfacial primary ion activity may be found for the case zi = 2, zj = 1 as

a iaq (0)· 10 z iF(Ei − EPB)/2.303RT + a jaq (0)· 10 z jF(Ej − EPB)/2.303RT = 1

E PB ≈ E i0 +

(17)

while the case zi = 1 and zj = 2 is obtained as

aq When substituting aaq i (0) for ci (0), eqs 6 and 11 can be aq combined to give a solution for ai (0), which is required for the development of the protocol. For this, an appropriate expression describing the phase boundary potential in the sample containing two ions of different charge is required, as it is well-known that the Nikolsky−Eisenman formalism is not applicable in the case of a mixed multivalent ion response. The general equation describing mixed ion response behavior may be represented as follows24 0

(16)

It was shown that this approximation can be successfully used for low levels of interference up to 10%,27 which suits the discussed case of low selectivity coefficient determination perfectly. By combining eq 16 with eqs 6 and 11, we now obtain an explicit solution for the primary ion activity in the nearboundary layer of the sample solution (position 0) if a membrane is exposed to a solution containing only interfering ion salt. It is written for the case zi = 2 and zj = 1 as

is defined as

E i0 = Δαβ ϕi0 +

(15)

2.303RT

E PB ≈ E i0+

where EPB is a phase boundary potential and Δβαϕ0i is the standard electrode potential, which is a direct function of the standard chemical potential differences (μ0,α and μ0,β i i ) for the ion i in membrane and aqueous phases Δαβ ϕi0 =

(Ej0 − E i0)z iF

This means that the selectivity coefficient should be constant for a particular electrode in the absence of experimental bias. Combining eqs 14 and 15, the latter may be solved for the phase boundary potential to give the simplified equation describing mixed ion response behavior

(7)

z iF (Δαβ ϕi0 − E PB) 2.303RT

(14)

3,5,7

At the same time the membrane concentration of the primary ion i before and after the ion-exchange process can be represented by the following expressions26 log c im * =

0

a jaq (0) ·(a iaq (0))−z j/ z i 10 z jF(Ej − Ei )/2.303RT ≈ 1

a iaq (0) =

(Dim)1/6 (K ijPot)2/3 (δiaqc im *a jaq *)1/3 π 1/6(Diaq )1/3

t −1/6 (20)

Thus, it follows from eqs 19 and 20 that one can eliminate the selectivity coefficient bias by approaching the situation when aaq i (0) → 0 by means of the extrapolation of experimental selectivity coefficients plotted against t−1/3 for the case zi = 2, zj = 1 or t−1/6 for the case zi = 1, zj = 2 to a value of t−1/3 (t−1/6) = 0 (t → ∞).

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RESULTS AND DISCUSSION As outlined in the Introduction and the Theoretical Considerations, the main reason for the overestimation of selectivity coefficients toward highly discriminated ions is

(13)

As a result, eq 12 is rearranged after inserting eq 13 and yields27 13443


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Figure 1. Selectivity coefficient determination by MSSM and SSM for (a) Mg2+-ISE and (b) Ca2+-ISE. Error bars are standard deviations (n = 3).

for the previously discussed cases of Mg2+ and Ca2+ ionselective electrodes, both MSSM and SSM protocols provides Nernstian response slopes for primary sodium ions. However, the calcium calibration curves exhibit a super Nernstian response at the concentrations above 0.01 M, see Figure 2a. It is likely that this higher than expected slope has its origins in the relatively high concentration of sodium ions contained in commercially available calcium chloride salts (CaCl2·2H2O, 99.5%), indicated as 10−2 wt % by the manufacturer. This high level of impurity is indeed expected to show a Hulanicki-type interference as discussed earlier in the literature.12,16 In this response mode, primary ions contribute partially to the potentiometric response, as their extraction into the membrane phase is thermodynamically preferred (ion-exchange selectivity) but still mass transport limited. To address this, a different calcium chloride salt (CaCl2·4H2O, 99.995% Suprapur) with a lower level of sodium ions impurities indicated as 5 × 10−4 wt % was explored in the same type of experiment. Indeed, Figure 2b demonstrates that the purer salt eliminates the super Nernstian response in the interfering ions solutions. A near Nernstian response is now obtained in a wide concentration range for both sodium and calcium ions in the MSSM

assumed to be an increased primary ion concentration at the sample side of the phase boundary of ion-selective membranes, as a result of an ion exchange at the interface, and is assumed to be coupled to transmembrane transport from the inner reference solution. One of the most efficient ways to eliminate this influence of the primary ion concentration increase is to change conventional SSM protocol for MSSM that avoids exposure to the primary ion prior to the measurements of interfering ions. Figure 1 shows responses of magnesium and calcium ionselective electrodes to potassium and tetraethylammonium cations, respectively, as obtained according to the MSSM and SSM procedures. Both methods yield Nernstian response slopes for primary ions, but only MSSM provides nearNernstian slopes toward interfering ions. The slope values of the Mg2+ and Ca2+ ion-selective electrode slopes toward K+ and Et4N+ decrease almost twice (from 55.8 to 26.1 mV and from 53.6 to 31.0 mV for Mg2+ and Ca2+ electrodes, respectively) while moving from the MSSM to the SSM measurements protocol. In Figure 2, an analogous experiment is shown with a sodium ion-selective electrode and calcium treated as interfering ion. As 13444


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Figure 2. Selectivity coefficient determination by MSSM and SSM for Na+-ISE using (a) CaCl2·2H2O (99.5%) and (b) CaCl2·4H2O (99.995% Suprapur) salts. Error bars are standard deviations (n = 3).

Table 2. Extrapolation Results ISE 2+

a

interfering ion +

Mg -ISE

K

Ca2+-ISE

Et4N+

Na+-ISE

Ca2+a Ca2+b

cj , M

log KPot ij (SSM), t = 10 min, (n = 3)

0.01 0.1 0.01 0.1 0.001 0.001 0.01

−1.30 −2.50 −1.97 −2.65 −4.77 −4.93 −5.37

± ± ± ± ± ± ±

0.06 0.13 0.02 0.06 0.08 0.37 0.24

log KPot ij (extrapol), (n = 3) −2.94 −2.94 −2.59 −2.87 −5.22 −5.58 −5.83

± ± ± ± ± ± ±

0.08 0.15 0.11 0.18 0.17 0.15 0.16

log KPot ij (MSSM), (n = 3) −2.83 ± 0.22 −2.95 ± 0.08 −5.90 ± 0.15

CaCl2· 2H2O (99.5%), bCaCl2· 4H2O (99.995% Suprapur).

that the purer calcium salts give significantly smaller (better) selectivity coefficients, as expected. Note that the theory underlying the methodology assumes that no primary ions are present in the sample bulk for the measurement of the interfering ion. At the next stage of our study, the applicability of the proposed extrapolation technique was explored. Figure 3 shows that the selectivity coefficients determined by SSM (log KPot ij (SSM)) decrease with increased interfering ion concentrations and the longer duration of the measurement. Indeed,

procedure. For the calcium chloride salt with more sodium impurities (Figure 2a) the unbiased selectivity coefficients were calculated from the three lowest measured concentrations that still showed a near-Nernstian response slope. The same concentration range was used for the linearization of time dependencies of the selectivity coefficients determined by SSM. For the purer calcium chloride salt (Figure 2b), the concentration range suitable for calculations could be extended from 0.001 to 0.01 M. The determined unbiased selectivity coefficients for both salts are presented in Table 2 and show 13445


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Figure 3. Time dependencies of selectivity coefficients determined by SSM for (a) Mg2+-ISE, (b) Ca2+-ISE, (c) Na-ISE using CaCl2·2H2O (99.5 wt %) salt, and (d) Na-ISE using CaCl2·4H2O (99.995 wt % Suprapur) salt.

selectivity coefficients determined beforehand by MSSM, and do not significantly depend on the concentration of the interfering ion. The extrapolation results also exhibit rather good reproducibility, as the confidence interval calculated for three parallel measurements does not exceed 0.2 logarithmic units. As evidenced by the data above, it is important to note that the presence of primary ion impurities in salts of interfering ions can limit the applicability of the proposed technique, as the linear dependence between KPot ij (SSM) and t−n can be violated. For example, it is apparent that for the calcium chloride salts containing higher levels of sodium ions −1/6 deviates (CaCl2·2H2O, 99.5%) the function KPot ij (SSM) − t −1/6 from linear behavior starting from 0.8 min , which results in an overestimation of the selectivity coefficient values (see Figure 4c). To minimize the influence of sodium ion impurities on the results of selectivity coefficient determination, only the upper part (shorter times) of the linearized function was used for extrapolation. The set of data points that was not used for extrapolation is marked with the orange color on the graph (Figure 4c). The deviation from linearity is readily apparent and may be corrected relatively easily.

the selectivity coefficient values decrease by 0.5−1.0 orders of magnitude with a 10-fold increase of interfering ion concentration (see Table 2). However, even in the case of the highest examined concentration level after 10 min measurement time, the selectivity coefficient values at that time exceed the unbiased ones determined before by MSSM (log KPot ij (MSSM)). The difference between unbiased selectivity coefficients determined by MSSM and the ones determined by SSM after waiting for 10 min is up to 1.5 orders of magnitude, depending on concentration (see Table 2). The time dependencies of the log KPot ij (SSM) values can be successfully linearized in accordance to the proposed theory. Figure 4 demonstrated that for a time interval between 2 and 10 min, all experimentally obtained KPot ij (SSM) values give good linearity versus t−1/3 for a doubly charged primary ion and singly charged interfering ion (zi = 2, zj = 1). Similarly, linearity is achieved for selectivity coefficients plotted versus t−1/6 in the case of a singly charged primary ion and doubly charged interfering ion (zi = 1, zj = 2), see again Figure 4. Thus, the extrapolation of these functions to t → ∞ (which corresponds to the condition of t−1/3 → 0 or t−1/6 → 0) should allow for the elimination of the selectivity coefficient bias, obtaining unbiased values of the selectivity coefficients. Indeed, as shown in Table 2, the extrapolated selectivity coefficients values (log KPot ij (extrapol)) correspond well to the unbiased

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CONCLUSIONS Low selectivity coefficients that are measured in practice in most cases are overestimated and depend on the measurement 13446


Article


Figure 4. Unbiased selectivity coefficient determination using the proposed extrapolation technique for (a) Mg2+-ISE, (b) Ca2+-ISE, (c) Na-ISE using CaCl2·2H2O (99.5 wt %) salt, and (d) Na-ISE using CaCl2·4H2O (99.995 wt % Suprapur) salt.

conditions owing to the primary ion flux into the sample solution and/or the primary ion impurities that are present in interfering ion salts. It is clear that this does not allow one to compare the results obtained in the different laboratories as well as performing the correct estimation of the influence of various membrane components or the change in composition of inner filling solution on the ISEs selectivity. By contrast, unbiased selectivity coefficients are constants for a particular electrode and can always serve as a necessary reference point. It is apparent that it is almost impossible to completely exclude the influence of primary ions originating from the impurities in the interfering ion salts. At the same time, as shown earlier by Egorov’s group for equally charged ions, the influence of an increase of primary ion concentration at the sample side of the interface due to transmembrane transport on the selectivity coefficients can be excluded by analysis of the time-dependent behavior of selectivity coefficients determined by SSM. A new method for determining unbiased low selectivity coefficients for two of the most prevalent cases of multivalent ions (zi = 2, zj = 1 and zi = 1, zj = 2) is demonstrated, thereby making the Egorov method applicable to most cases of practical relevance. It was established that the influence of the primary ion flux into the sample solution on selectivity coefficient values can be eliminated, and minimally biased selectivity coefficients can be obtained by extrapolating the linearized time depend-

encies of selectivity coefficients determined by SSM to t → ∞. This corresponds to the cases of t−1/3 → 0 for zi = 2, zj = 1 and t−1/6 → 0 for zi = 1, zj = 2. The applicability of the new method was demonstrated with ionophore-based Mg2+-, Ca2+-, and Na+selective electrodes. A satisfactory agreement between unbiased selectivity coefficients determined by MSSM and by the proposed extrapolation technique was achieved. The evident experimental advantage of the new technique over MSSM is the possibility of applying it repeatedly to electrodes that were previously in contact with the primary ion solution, as with commercially available probes. However, the proposed method is more sensitive to primary ion impurities in the interfering ion salts when compared to MSSM. Such a primary ion level will mark a minimum boundary concentration that will, of course, not extrapolate to zero with increasing time. MSSM is less problematic in this regard, as the low level impurities will be more readily absorbed by the membrane by ion-exchange, as the membrane is initially void of primary ion.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Eric Bakker: 0000-0001-8970-4343 13447


Article

Analytical Chemistry Author Contributions

The manuscript was written through contributions of all authors. All authors have given approval to the final version of the manuscript. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS The authors thank the Swiss National Science Foundation (SNF) and the University of Geneva for financial support of this study.

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REFERENCES

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Time-Dependent Determination of Unbiased Selectivity Coefficients

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