Development of the tracer-pulse method for adsorption studies of analyte mixtures in liquid chromatography utilizing mass spectrometric detection

Article abstract of DOI:10.1021/ac702399a

The tracer-pulse method provides the real adsorption data points directly from simple, straightforward calculations and is therefore a superior method for multicomponent adsorption isotherm determination in HPLC. Only one important problem has restricted

Full text of DOI:10.1021/ac702399a

Anal. Chem. 2008, 80, 2105-2112
Development of the Tracer-Pulse Method for
Adsorption Studies of Analyte Mixtures in Liquid
Chromatography Utilizing Mass Spectrometric
Detection
Jo1rgen Samuelsson,† Robert Arnell,‡ Jarle S. Diesen,§ Julius Tibbelin,§ Alexander Paptchikhine,§
Torgny Fornstedt,*,† and Per J. R. Sjo1berg|
Department of Physical and Analytical Chemistry, Uppsala University, BMC, Box 577, SE-751 23 Uppsala, Sweden,
AstraZeneca Process R&D, SE-151 85, So¨derta¨lje, Sweden, Department of Biochemistry and Organic Chemistry,
Uppsala University, BMC, Box 576, SE-751 23 Uppsala Sweden, and Department of Physical and Analytical Chemistry,
Uppsala University, BMC, Box 599, SE-751 24 Uppsala, Sweden
data (i.e., the real data points) from a mixture; most methods only
give the best parameter estimates.
The tracer-pulse method provides the real adsorption data
points directly from simple, straightforward calculations
and is therefore a superior method for multicomponent
adsorption isotherm determination in HPLC. Only one
important problem has restricted its use so far: the tracer
peaks are invisible using any conventional detection
principle. We present a solution to this problem with an
approach with a firm base in analytical chemistry, utilizing
stable isotopes and mass spectrometric detection. The
new approach was used for the determination of binary
adsorption isotherms, and a systematic investigation was
made of its main sources of error. With this modification,
the tracer method can be a prime choice for future
characterizations of multicomponent separation systems
and of competitive drug binding studies.
A classical method for determination of the adsorption iso-
therms is frontal analysis (FA), which is generally recognized in
a single component system to be the most accurate method. FA
has been used for binary and ternary mixtures.5 A prerequisite is
that the composition of the intermediate plateaus can be deter-
mined, which means that a fractionation and reinjection procedure
must be followed for systems with more than two components.1
A disadvantage is that high-efficiency columns are required for
ternary mixtures, to counteract the erosion of the intermediate
plateaus by kinetic effects.5
The tracer-pulse method (TP) and the perturbation peak
method (PP) belong to the so-called pulse methods.6 If a small
excess of molecules is injected into a column equilibrated with
an eluent containing the same solute molecules (a concentration
plateau), one single peak will appear in the chromatogram. In
reality, a total of three zones are created, one of them being the
displaced plateau molecules (i.e., the zone visualized as a peak)
and another later eluting zone being the injected molecules (tracer
peak). The latter zone will not be visualized as a peak since it has
a combined elution with the third zone, a deficiency zone of the
plateau molecules. This is the simplest case, taking place in a one
component system with a convex adsorption isotherm. For a
system showing a concave adsorption isotherm, the invisible tracer
peak should instead elute faster than the perturbation peak. The
tracer peak is impossible to detect unless the sample molecules
are labeled somehow so that they can be distinguished from the
plateau molecules. The perturbation peaks can be used for
adsorption isotherm determinations because the velocity is related
to the tangential slope of the adsorption isotherm at the actual
concentration plateau whereas the velocity of the tracer peak is
governed by the corresponding chord. However, as will be seen
below, the latter relation (TP) is much simpler and straightforward
for calculation of adsorption data than the former one (PP),
especially for competitive adsorption.
Adsorption isotherms describe the partitioning of solutes
between mobile and stationary phases at a given constant
temperature and are important for a deeper characterization of
both analytical and preparative separation systems.1,2 An analytical
example is the detailed characterization of drug-protein interac-
tions using immobilized protein stationary phases.3 Such systems
often contain complex multisite interactions with different affinities
and capacities and their adsorption isotherms must be determined
in a wide concentration range.4 However, today there exists no
good method for determination of the raw adsorption isotherm
* Corresponding author.
† Department of Physical and Analytical Chemistry, Uppsala University, BMC,
Box 577.
‡ AstraZeneca Process R&D.
§ Department of Biochemistry and Organic Chemistry, Uppsala University,
BMC, Box 576.
|
Department of Physical and Analytical Chemistry, Uppsala University, BMC,
Box 599.
(1) Guiochon, G.; Felinger, A.; Shirazi, D. G.; Katti, A. M. Fundamentals of
Preparative and Nonlinear Chromatography, 2nd ed.; Elsevier Academic
Press: San Diego, CA, 2006.
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(3) Go¨tmar, G.; Samuelsson, J.; Karlsson, A.; Fornstedt, T. J. Chromatogr., A
2007, 1156, 3-13.
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19-34.
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(6) Helfferich, F.; Peterson, D. L. Science 1963, 142, 661-662.
10.1021/ac702399a CCC: $40.75 © 2008 American Chemical Society
Published on Web 02/22/2008
Analytical Chemistry, Vol. 80, No. 6, March 15, 2008 2105

Both pulse methods were suggested in the early 60s when
gas chromatography started to be recognized not only as an
analytical tool but also as a powerful method for measuring
thermodynamic properties.6 The GC community focused on the
TP because of its more elegant and simple theoretical solution
and because of the great variety of on-line detectors available
allowing for detection of labeled molecules.7 The PP was consid-
ered a “less accurate” differentiate method5 and fell into oblivion.
The first experimental applications of the TP used radioactive
isotopes,8,9 but difficulties such as the handling of radioactive
solutes in the gas phase hindered the method to become
widespread. In the next decade, when mass spectrometry became
a standard detector for GC, TP was used with stable isotopes
detected with a quadrapole instrument.10,11 However, the new
application was introduced at a time when the interest for
measuring thermodynamic quantities in GC had declined. During
recent years in LC, the PP theory was extended to encompass
the multicomponent case and also validated for these adsorption
isotherm determinations.12,13
Recently, fundamental studies has been made of the TP in
HPLC.14-17 The hidden events in the column were visualized using
two different experimental strategies: (i) a radioactive labeled
approach and (ii) a method based on the use of two pure
enantiomers in a nonchiral separation system.14 A systematic
investigation of a similar phenomenon was made for frontal
analysis,15 and also very strange deformations of overloaded tracer
zones16 were visualized and systematically investigated. A tedious
chiral approach of TP was applied for determination of multicom-
ponent adsorption isotherms.17
The TP is the superior method for studies of competitive
adsorption but it has one serious problem, which has restricted
its widespread use in HPLC, how can we selectively detect only
a few injected molecules in a large population of identical ones?
A smart mass spectrometric solution might be a way to solve the
problem but has not been tried for the TP in LC yet. However,
time-of-flight secondary ion mass spectrometry (TOF-SIMS) has
been used for quantitative surface analysis and direct measure-
ment of adsorption isotherms.18,19 Mass spectrometry has also
been combined with frontal analysis for selective and rapid
screening of important inhibitor components in crude extracts.20,21
The aim of this study is threefold: (i) to develop a method for
implementation of mass spectrometric detection of stable isotopes
in the TP for LC, (ii) to systematically investigate the main sources
of error of the developed method, and (iii) to apply the method
for a multicomponent system.
THEORY
For a more thorough review of the theory involved, we refer
to previous work.16,17 In this study the competitive Langmuir
adsorption isotherm will be used:
aiCi
qi )
i ) 1,2,...,n
(1)
1 + biCi
∑
i
where qi and Ci are the stationary and mobile phase concentrations
of component i, ai is the distribution coefficient, and bi is the
association equilibrium constant.
Perturbation peaks, measured in the PP, originate from the
disturbance of the established solute equilibrium in the column.
In a two-component system, two perturbation peaks will be present
and the retention time of these peaks are
t0
2
∂q1 ∂q2
∂C1 ∂C2
tR )
2 + F
+
(
(
)
[
2
∂q1 ∂q2
∂q1 ∂q2
∂C2 ∂C1
F
-
+ 4
(2)
C)C0
|
]
(
x
)
∂C
∂C2
1
where t0 is the column hold-up time, F is the volumetric phase
ratio, C0 is the eluent solute concentration and qi(C1,C2) is a
competitive adsorption isotherm. It is important to note that it is
not possible to identify the perturbation peaks in a two-component
system, because both components contribute to retention times
and areas of each perturbation peak.
Tracer peaks consist of, in contrast to perturbation peaks, the
actual injected molecules and can thus be identified also in a
multicomponent system. In the TP, the injected solutes are labeled
so that each tracer peak can be followed by a selective detector.
In a two-component system, two tracer peaks will be present with
retention times according to
qi(C1, C2)
tR,i ) t0 1 + F
(3)
(
)
Ci
C)C0
(7) Kobayashi, R.; Chappelear, P. S.; Deans, H. A. Ind. Eng. Chem. 1967, 59
(10), 63-81.
(8) Gilmer, H. B.; Kobayashi, R. AIChE J. 1965, 11 (4), 702-705.
(9) Peterson, D. L.; Helfferich, F.; Carr, R. J. AIChE J. 1966, 12 (5), 903-905.
(10) Parcher, J., F.; Selim, M. I. Anal. Chem. 1979, 51, 2154-2156.
(11) Lin, P. J.; Parcher, J. F.; Hyver, K. J. J. Chem. Eng. Data 1984, 29, 420-
4239.
(12) Forsse´n, P.; Lindholm, J.; Fornstedt. T. J. Chromatogr., A 2003, 991, 31-
45.
(13) Lindholm, J.; Forsse´n, P.; Fornstedt, T. Anal. Chem. 2004, 76, 4856-4865.
(14) Samuelsson, J.; Forsse´n, P.; Stefansson, M.; Fornstedt, T. Anal. Chem. 2004,
76, 953-958.
(15) Samuelsson, J.; Fornstedt, T. J. Chromatogr., A 2006, 1114, 53-61.
(16) Samuelsson, J.; Arnell, R.; Fornstedt, T. Anal. Chem. 2006, 78, 2765-2771.
(17) Arnell, R.; Fornstedt, T. Anal. Chem. 2006, 78, 4615-4623.
(18) Vickers, P. E.; Castle, J. E.; Watts, J. F. Appl. Surf. Sci. 1999, 150, 244-
254.
(19) Abel, M.-L.; Chehimi, M. M.; Brown, A, M.; Leadley, S. R.; Watts, J. F. J.
Mater. Chem. 1995, 5 (6), 845-848.
EXPERIMENTAL SECTION
Apparatus. The chromatographic system was an Agilent 1100
from Agilent Technology (Palo Alto, CA) consisting of binary
pumps, autosampler, and a diode-array UV detector. The column,
an Eclipse XDB-C8 (4.6 mm × 150 mm, 5 µm) from Agilent
Technologies (Palo Alto, CA), was placed in a laboratory-
assembled column jacket and its temperature was controlled at
29.0 °C using a LAUDA type B circulating water bath (Ko¨ning-
shofen, Germany). All tubings in the chromatographic system
before the first flow-split was 0.13 mm PEEK, and the flow rate
was 0.70 mL/min. An API III plus triple quadrupole mass
spectrometer from PE-Sciex (Concord, ON, Canada) equipped
with an articulated IonSpray (pneumatically assisted electrospray,
ESI) interface was used.
(20) Zhu, L.; Chen, L.; Luo, H.; Xu, X. Anal. Chem. 2003, 75, 6388-6393.
(21) Fort, S.; Kim, H.-S.; Hindsgaul, O. J. Org. Chem. 2006, 71, 7146-7154.
2106 Analytical Chemistry, Vol. 80, No. 6, March 15, 2008

Figure 1. Illustration of the flow-splits and the makeup flows between the column and the mass spectrometer. Flow-split: (1) splits the flow
1:350 times, (2) 1:950 times, and (3) 1:10 times.
Chemicals. Acetonitrile, methanol (CHROMASOLV quality),
sodium acetate (>99%), uracil (>99%), and methyl- and ethyl
mandelate (>98%) were bought from Sigma-Aldrich. The water
used was from a Milli-Q water purification system ZMQS 5000Y
from Millipore (Molsheim, France).
reaction mixture was then diluted with Et2O (10 mL), washed with
NaHCO3 (aq) (3 × 5 mL) and subsequently with brine (1 × 5
mL), and dried over MgSO4. The organic phase was filtered, and
the solvent was removed under reduced pressure giving 0.51 g,
91% of MM* as transparent crystals.
1H NMR (500 MHz, CDCl3): δ 7.33-7.43 (m, 5H), 5.18 (s,
1H), 3.55 (br s, 1H). 13C NMR (125 MHz, CDCl3): δ 174.1, 138.2,
Solute Synthesis. All chemicals were used as received except
for methyl mandelate, which was recrystallized from a mixture
of n-pentane/Et2O (9/1), and ethyl mandelate, which was purified
on a silica column using freshly distilled n-pentane/ethyl acetate
(7/3) as an eluent. Toluene was dried over sodium metal with a
sodium benzophenone ketyl indicator prior to use. The reactions
were performed under an N2 atmosphere. The deuterium labeled
methyl mandelate (MM*) and ethyl mandelate (EM*) were
synthesized by a procedure modified from that of Toshikatsu et
al.22 1H NMR spectra were recorded at 500 MHz. Chemical shifts
are reported in ppm, relative to the residual solvent peak CDCl3
(7.26 ppm). 13C NMR spectra were recorded at 125 MHz, and
chemical shifts were reported relative to the solvent peak of CDCl3
(77.0 ppm). Mass spectra were measured at 70 eV (EI) and are
reported as m/z (relative intensity, %). Exact mass measurements
were made with an ESI-TOF mass spectrometer (Agilent Tech-
nologies series 1100 LC-MSD TOF mass spectrometer). IR
spectra were recorded on a FT-IR with an ATR accessory, and
signals are reported as ν˜ (cm-1, relative intensity).
1
128.6, 126.6, 72.9, 52.2 (h, JCD ) 22.3 Hz, intensity ratio 1:3:6:7:
6:3:1). FT-IR (neat): 3435, 3035, 2078, 1736, 1213, 1066. EI-MS:
169.1 (33.1), 152.1 (54.4), 107.1 (47.5), 77.1 (100). Exact mass (ESI)
m/z calcd for C9H7D3O3Na [M + Na+] ) 192.0716, found:
192.0682.
Synthesis of Ethyl-d5-mandelate (EM*). The procedure was
exactly as above except that 1.97 mmol of mandelic acid was used,
and methanol-d4 was exchanged to ethanol-d6 (0.5 mL). This
resulted in transparent crystals (0.32 g, 87%). 1H NMR (500 MHz,
CDCl3): δ 7.35-7.47 (5H), 5.18-5.20 (m, 1H), 3.60-3.62 (m, 1H).
13C NMR (125 MHz, CDCl3): δ 173.7, 138.4, 128.3, 126.5, 72.9,
1
1
61.3 (q, JCD ) 23.6 Hz, intensity ratio 1:2:3:2:1), 12.9 (h, JCD
)
19.4 Hz, intensity ratio 1:3:6:7:6:3:1). FT-IR (neat): 3437, 2914,
2245, 1725, 1184, 1059. EI-MS: 186.0 (17.3), 168.1 (29.5), 107.1
(48.7), 77.1 (100). Exact mass (ESI-MS) m/z calcd for C10H7D5O3-
Na [M + Na+] ) 208.0998, found: 208.0955.
Chromatographic Procedures. The eluent was 30/70 (v/v)
acetonitrile/water. The column hold-up time was estimated to 2.00
min with a RSD of 0.04% (n ) 5) by injecting uracil which is
generally recognized to be slightly retained. The lag time between
the UV detector and the mass spectrometer was determined by
injections of 20 µL of 7.5 mM of MM and EM to 0.365 min with
Synthesis of Methyl-d3-mandelate (MM*). To a stirring solution
of mandelic acid (0.50 g, 3.29 mmol) in a mixture of toluene (5
mL) and methanol-d4 (1 mL), boric acid (10 mol %) was added.
The reaction was stirred at ambient temperature for 24 h. The
(22) Toshikatsu, M.; Ishihara, K.; Yamamoto, H. Org. Lett. 2005, 7, 5047-5050.
Analytical Chemistry, Vol. 80, No. 6, March 15, 2008 2107

Figure 2. The relative error in the retention time versus eluent
concentration for the tracer peak (a, c) and the perturbation peak (b,
d), respectively, calculated for a 20 µL injection of 1, 7.5, 15, and 25
mM excesses (see legend). Two different Langmuir adsorption
isotherms were used (a, b) a ) 4.59 and b ) 2.31 M-1 and (c, d) a
) 4.59 and b ) 9.17 M-1. The number of theoretical plates was 7030.
Figure 3. The relative error in the adsorption isotherm is plotted
against the labeling effect as the error in the retention time compared
to the unlabeled substance. Unfilled symbols are for the a term and
filled symbols for the b term, and the line is the calculated error due
to the labeling effect. Two different Langmuir adsorption isotherms
were used: a ) 4.59 and b ) 2.31 M-1 (circles) and a ) 4.59 and b
) 9.17 M-1 (squares).
Table 1. The Error on the Estimated Adsorption
Isotherm Parameters Due to the Injection Volumea
tracer-pulse method
parameters error
perturbation peak method
mixture. The stock solutions were 1.02, 10.2, and 205 mM MM;
0.625, 6.25, and 125 mM EM; and 0.503 + 0.503, 5.03 + 5.03, and
101 + 101 mM mixture.
parameters
error
Cinj
b
a
b
b
a
b
[%]
[mM]
a
[1/M] [%]
[%]
a
[1/M] [%]
The plateau concentrations were constructed by connecting
the outlets of the flows of pure eluent (pump 1) and a stock
solution (pump 2) through a low-dead-volume PEEK tee before
entering the column. The percentage stock solutions used were
30, 50, 70, and 100% for low and mid stock solutions and 20, 40,
60, and 100% for high stock solutions. Determination of the
adsorption isotherms were conducted by injecting 20 µL of 7.5
mM labeled solute. The labeled solutes were dissolved in pure
eluent except when the high concentration stock solutions were
analyzed. Then it was dissolved in diluted high stock according
to 50% stock at the 20 and 40% plateaus, 75% stock at the 60%
plateau, and 100% stock at the 100% plateau.
1
4.585 2.309 -0.02 -0.11 4.584 2.310 -0.03 -0.07
4.583 2.305 -0.06 -0.28 4.571 2.308 -0.32 -0.14
4.580 2.303 -0.11 -0.35 4.555 2.304 -0.66 -0.30
4.577 2.298 -0.19 -0.57 4.535 2.301 -1.10 -0.44
7.5
15
25
true 4.586 2.311
1
4.586 2.311
4.585 9.167
0.00 -0.05 4.578 9.164 -0.16 -0.08
7.5
15
25
4.580 9.149 -0.12 -0.25 4.526 9.122 -1.30 -0.53
4.576 9.145 -0.21 -0.28 4.471 9.092 -2.49 -0.86
4.570 9.128 -0.34 -0.47 4.408 9.066 -3.88 -1.15
true 4.586 9.171
4.586 9.171
a The error is defined as (determined-true)/true adsorption isotherm
and is expressed as percentage. All conditions as in Figure 2.
All simulations were done using the equilibrium-dispersive
model, solved by using a modified Rouchon method.23 Injection
profiles were measured experimentally and were used as boundary
conditions in the simulations. Detector calibration was performed
by converting response to concentration by a third degree
polynomial function.
RSD of 6.4% (n ) 6). The labeling effect was determined by
injections of 7.5 mM labeled and unlabeled MM and EM, to -0.06
min with a RSD of 14% (n ) 3) and to -0.19 min with a RSD of
6.2% (n ) 3), respectively. The column efficiency was determined
to 7030 with an RSD of 1.8% (n ) 4). The UV detector was set at
260 nm for low and medium solute concentrations and 275 nm
for high solute concentrations.
RESULT AND DISCUSSION
Implementation of Mass Spectrometric Detection. The TP
has one major practical problem; the tracer zone cannot be
visualized as a peak by conventional detection principles. We have
previously used enantiomers and radioactive labeled compounds
to visualize the tracer peaks and thus to determine adsorption
isotherms with the TP.16 However, the enantiomeric approach is
very time-consuming and tedious and also not general (chiral
systems cannot be characterized). The radioactive labeling is more
general, but the detector is not as selective as the mass
spectrometer and the labeled compound could break down due
Samples were analyzed in positive ESI mode with a spray and
orifice potential of 3500 and 50 V, respectively. The nebulization
gas pressure (N2, boil-off from liquid nitrogen) was 40 psi. Data
were acquired with a dwell time of 100 ms in SIM mode. The
monitored ions, corresponding to the sodiated molecules, were
facilitated by the addition of 10 µL/min, (50% methanol) of a
sheath-flow containing 1 mM sodium acetate, where the monoiso-
topic and isotopic peaks were for MM 189, 190; MM* 192; EM
203, 204; and EM* 208.
The determination of adsorption isotherms were conducted
using three different stock solutions each (low, mid, and high
concentration) of single MM respective EM and of their 1:1
(23) Arnell, R.; Forsse´n, P.; Fornstedt, T. Comput. Chem. Eng. 2006, 30, 1381-
1391.
2108 Analytical Chemistry, Vol. 80, No. 6, March 15, 2008

was due to the lost sensitivity as the column effluent was diluted.
We therefore made a systematic investigation of the error due to
a large excess of the injected tracer (below). Another source of
error that we investigated systematically was the labeling effect;
i.e., the tracer does not have exactly the same properties as the
unlabeled molecule.
Error Due to the Excess Injected. The PP theory assume
that the injected sample that create the equilibrium disturbance
gives a negligible concentration deviation compared to the plateau
concentration, so that dq/dC does not change because of the
injected concentration deviation (cf. eq 2). In practice this is not
possible, and the PP experiences a more or less large error due
to the injected excess;13 below we will investigate if there exists
a similar error for the TP. In the TP, the injected sample also
contains a tracer, i.e., labeled molecules with identical properties
as the unlabeled ones. The chord of the adsorption isotherm
should not change due to the injected concentration deviation (see
eq 3). We previously showed qualitatively that if a large excess is
injected, the tracer peak will be deformed and have a nonrepre-
sentative retention time.16 It was also shown that if no excess is
injected, i.e., the total sample concentration (labeled + unlabeled)
is identical to the plateau concentration, no perturbation peak will
appear and the tracer signal has an ideal and identical retention
time independent of size of the tracer peak.16 The preparation of
ideal tracer samples, one for each plateau, is however a very
tedious procedure, so in practice small deviation to the eluent
would be preferred. In the present study we performed a more
quantitative analysis in order to investigate how the size of the
excess (concentration in mobile phase + excess) affects the
determination of adsorption isotherms and the retention time for
the perturbation and tracer peaks. For this purpose a series of
simulations were done with 20 µL injections containing an excess
of 1, 7.5, 15, and 25 mM on concentration plateaus between 0.01
and 200 mM. In the investigation, two different Langmuir adsorp-
tion isotherms were used with a ) 4.59 and b ) 2.31 M-1 and a
) 4.59 and b ) 9.17 M-1, corresponding to monolayer saturation
capacity of 1.98 and 0.5 M, respectively.
Figure 4. Parts a and c show the breakthrough of the eluent 40/40
mM of MM/EM and parts b and d show the elution of a 20 µL injection
of 7.5/7.5/50/50 mM of MM*/EM*/MM/EM on a concentration plateau
of 40/40 mM of MM/EM. The top row (a, b) is the UV-detector signal,
and the bottom row (c, d) shows the MS signals. Observe that the
MS baselines have been moved for easier visualization. TIC is the
total ion count.
to self-radiolysis. In addition, special care must be considered for
handling and disposal of radioactive material.
In this study, we developed a TP method based on using stable
isotopes measured with mass spectrometry (MS) to determine
adsorption isotherms of mixtures. The new approach has several
advantages compared to the previous ones: less tedious, more
general, and no special care need to be taken during handling of
the solutes. Still some problems needed to be considered:
saturation of the detector, MS-compatibility of the column effluent,
and long-term stability. The origin of these problems is that a wide
range in concentration, from very low to very high, is necessary
for proper isotherm determination when the purpose is to make
a complete census of all interactions in the separation system.
These problems were solved by dilution of the highly concentrated
column effluent in a stepwise manner, see Figure 1. The first split
reduced the flow to the mass spectrometer line down to around
2 µL/min while the rest of the flow is introduced into the UV
detector. After the first split, the effluent is diluted 476 times by
introducing 0.95 mL/min 50% (v/v) methanol/water. After the
dilution, the flow is once again reduced by a factor of around 950
down to around 1 µL/min. Finally, the effluent is diluted 11 times
by introducing 10 µL/min sheath-flow containing 1 mM sodium
acetate in 50% (v/v) methanol/water (sodium is introduced
because MM and EM is detected as sodium adducts). Thus, the
column effluent is totally diluted more than 5000 times which
allows us to use large solute concentrations in the eluent while
still operating in the linear dynamic range of the detector and an
addition decrease in ion suppression in the ion source. Further-
more, the mass entering the ion source is reduced more than
330 000 times, allowing us to run the experiments for longer time
without needing to clean the mass spectrometer system. Finally
we could optimize the separation and use high concentrations of
nonvolatile solutes and buffer components that are not MS-
compatible, because the compatibility could be solved postcolumn
using the splits.
In Figure 2, an error analysis is performed for the tracer (a
and c) and for the perturbation peak (b and d) for an adsorption
isotherm with qs ) 1.98 M (a and b) and with qs ) 0.5 M (c and
d). The relative retention time error ((tR-tR,ideal)/tR,ideal where tR,ideal
,
is calculated with eqs 2 or 3) was plotted against the bulk eluent
concentration. Observe that in these cases (convex type I
adsorption isotherms with injected excesses), the retention times
from the tracer and perturbation peak is always shorter compared
to the true retention time with infinitesimal excess. The opposite
would be true for type III (concave adsorption isotherm), and for
other types of adsorption isotherms (e.g., S-shaped) the trend
would be eluent concentration dependent. If we instead inject a
deficiency, the opposite trend would be noted for all different types
of adsorption isotherms.
The relative retention time error for both the tracer peak in
Figure 2 (a and c) and the perturbation peak (b and d) increases
with increasing injected excess. However, as the plateau concen-
tration increases, the error for the tracer peak declines rapidly
giving a much more favorable situation for the TP as compared
to the PP. We can also see that a lower monolayer capacity gives
a larger error for both methods; this is due to the larger loading
The disadvantage with the split is that it forces us to inject a
large excess of labeled solutes on low-plateau concentrations, this
Analytical Chemistry, Vol. 80, No. 6, March 15, 2008 2109

Figure 5. The adsorption isotherm data and the fitted competitive Langmuir model. The stationary phase concentration of (a) MM and (b) EM
is plotted versus the mobile phase concentrations. Symbols are experimental data.
factor and thus degree of nonlinearity experienced at the plateau
concentration. The reason for the similar error for the TP and PP
at low plateau concentrations is that the tracer zone has a more
or less combined elution with the perturbation peak. The error
decreases drastically for the TP with increasing concentration in
the plateau because the tracer peak propagates for a shorter period
of time in the perturbation zone and is less affected by the
concentration gradient created by the perturbation zone which
deforms the tracer zone.16 The higher the eluent concentration,
the larger is the difference in velocity between the zones and the
faster does the tracer zone leave the perturbation zone.
Several different practical problems exist when determining
adsorption isotherms with the PP. The first is to detect the peak;
here we need to inject a finite plateau concentration deviation,
leading to thermodynamic tailing. Gritti et al. reduced the error
in the determined retention time for the perturbation peak by
injecting a positive and a negative plateau deviation following each
other, so that the peaks had equal area and took the average of
the retention times from these peaks.24 Even if the error is reduced
considerably, this approach will not totally compensate for the
error because that would require that the tailing of the positive
and negative peak is identical but opposite to each other. This is
not necessarily true depending on the particular curvature of the
adsorption isotherm (which is unknown as the measure takes
place). Usually the detector sensitivity decreases with increasing
plateau concentration, so that larger absolute deviations compared
to the plateau concentration must be injected. For the TP, the
error decreases drastically with increasing plateau concentration
and is not really affected by increased injection excess at high
plateau concentration, which is a great advantage as compared
to the PP. Cavazzini et al. used the perturbation peak method to
determine the adsorption of MeOH in a straight phase system.25
The authors note that PP gave large errors in the low-concentra-
tion region, the solution was to use large perturbations, and use
the inverse method to find adsorption isotherm parameters to
estimate the retention time at extremely low plateau concentra-
tions. However, it should be mentioned that smaller excess could
be used for PP and TP if we do not need to split the column
effluent, resulting in smaller errors.
Lindholm et al. have studied the relative perturbation peak
retention error but reached slightly different results.13 The authors
investigated how the error changes with increasing eluent
concentration for blank injections, 100% and 50% excess compared
to the eluent. They found that the error in the linear range of the
adsorption isotherm is very small, has a maximum in the weakly
nonlinear range, and decreases slowly at higher concentration.
In the present study a more pragmatic approach is taken: we
instead use constant (not relative) excess and no such maximum
was found (see Figure 2). Our results are also in line with the
recent observation of Cavazzini et al. that the PP has large errors
in the low-concentration region.25
In Table 1, the error in the determined adsorption isotherm
parameters are presented. The error for the TP is generally much
lower as compared to the PP. The TP delivers good predictions
of the a term with an error below 0.4%. For the PP, the error is
up to around 10 times larger. The error in the b term is similar
for both methods for the high-capacity column, but as the capacity
of the column decreases, the error in the b term increases and
the PP delivers approximately twice as large errors compared to
the TP.
The maximum error experienced for the tracer peak experi-
ments in this study corresponded approximately to the situation
with an excess injected of 7.5 mM at low-concentration plateaus.
Error Due to the Labeling Effect. The labeled molecules
(MM*, EM*) are somewhat less retained than the unlabeled ones
(MM, EM), and this might affect the determined adsorption
isotherm. MM* eluted 0.06 min before MM, and EM* eluted 0.19
(24) Gritti, F.; Piatkowski, W.; Guiochon, G. J. Chromatogr., A 2003, 983, 51-
71.
(25) Cavazzini, A.; Nadalini, G.; Malanchin, V.; Costa, V.; Dondi, F.; Gasparrini,
F. Anal. Chem. 2007, 79, 3802-3809.
2110 Analytical Chemistry, Vol. 80, No. 6, March 15, 2008

could only find another paper dealing with this type of error, it
was when using the frontal analysis method in the staircase mode
where a serious source of error was found in the determination
of the volume between the T-connector (where the two pumps
meet) and the column.13 Two good but different methods gave
results that varied more than 5% which was not acceptable (see
Figure 4 in ref 13). Only one of the methods gave a value which
coincided with the PP method, and this value was therefore
assumed to be the correct one.
Experimental Determination of Competitive Adsorption
Isotherms. Because of the different mass of unlabeled and labeled
MM and EM, they could be detected by MS. The UV detector
gives the sum of all events in the chromatographic process. Figure
4a,c presents the breakthrough curve of 40/40 mM of MM/EM.
In parts b and d a 20 µL injection of 7.5/7.5/50/50 mM of MM*/
EM*/MM/EM is injected on the established concentration
plateau of 40/40 mM of MM/EM. Parts a and b show the UV
signal, and parts c and d show the MS signals.
In the UV track, two fronts can be seen with an intermediate
concentration plateau in between. The single ion monitoring (SIM)
tracks reveal that MM is displaced by the EM, and the intermedi-
ate plateau has a higher concentration as compared to MM in
the eluent. The displacement effect is a sign of competition inside
the column. As expected, the perturbation peaks elute before the
mass peaks (cf. Figure 4b,d). The perturbation and tracer peaks
cannot be distinguished in the total ion count (TIC), due to the
high ion current. However, in SIM mode the tracer peaks are
easily detected. All the MS signals were slightly noise reduced
due to the high-background noise.
Figure 6. Predicted band profiles for the 900 µL injection of (a) 50
mM of EM and MM and (b) 100 mM of MM and EM. The Langmuir
parameters used were a ) 2.49, b ) 0.906 M-1 (MM) and a ) 4.59,
b ) 2.31 M-1 (EM). The solid lines are calculated profiles, and the
symbols are experimental data.
min before the EM, which is 0.95% and 1.9% of the total retention
time of MM and EM, respectively. The labeling effect on EM will
therefore have a larger effect on the adsorption isotherm as
compared to MM. To estimate how large of an error a labeling
effect will have on the determined adsorption isotherm, simula-
tions of 20 µL excess injections of 7.5 mM on plateau concentra-
tions of eluents between 0.01 and 200 mM were performed by
changing the distribution coefficient to the desired retention time
and changing the equilibrium constant in such a way so that the
column saturation capacity (qs ) a/b) was constant. The Langmuir
models with parameter sets defined above were used; the labeling
error was allowed to vary between 0 and 5%. The acquired data
were fitted to the Langmuir model and are presented in Figure 3.
In Figure 3, one can see that the fitted distribution coefficient
is underestimated and that the underestimation increases pro-
portionally with increasing labeling effect; in fact the constant
coincides more or less with the assumed labeling effect (theoreti-
cal line in Figure 3). At a 2% error in the retention time, the
distribution coefficient is decreased by 2.5%. The error in the
equilibrium constant (b) is small and does not change with
increasing labeling error. The two different Langmuir adsorption
isotherms described above were used qs ) 1.98 M (circles) and
qs ) 0.5 M (squares) but no clear difference using these two
capacity values could be recognized (cf. Figure 3). The determined
labeling effect is in line with a recently presented general indirect
detection theory, where the retention time of the solute is
described by the probe’s equilibrium constant (in this case, the
plateau molecules) and the solute’s distribution coefficient (in this
case, the tracer).26
Adsorption isotherm parameters were fitted to the tracer-pulse
retention data. Nonlinear regression was performed using the
lsqnonlin algorithm as implemented in Matlab (MathWorks Inc.,
Nattick, MA). A global solution was sought by repeating the fitting
1000 times with random initial guesses spread over the whole
feasible solution space. The data fitted well to the Langmuir
adsorption isotherm (eq 1). Other models were tried as well, but
no statistically significant improvement was obtained. The Lang-
muir parameters were as follows: for MM a ) 2.49, b ) 0.906
M
-1 and for EM a ) 4.59, b ) 2.31 M-1. This results in a saturation
capacity of 2.75 and 1.98 M, respectively. The fitted competitive
3D-adsorption isotherms are plotted with the data in Figure 5: it
can be observed that the degree of nonlinearity is moderate even
at 100-200 mM concentrations of MM and EM.
Computer simulation band profiles were compared to experi-
mental chromatograms. Figure 6 shows the simulated and
experimental chromatograms corresponding to 900 µL injections
of (a) 50/50 mM of MM/EM and (b) 100 /100 mM of MM/EM.
The agreement is very convincing. A small deviation can be seen
at the diffusive rear of EM; the simulated signal reaches baseline
earlier than does the experimental signal. This is probably due
to an underestimation of the EM a term caused by the labeled
effect on the tracer.
The labeling effect could be decreased by substituting less
hydrogen to deuterium or use 13C instead, but it is recommended
to have at least three mass unit differences between the stable
isotopes and the unlabeled compound to be able to separate them
properly in the mass spectrometer.
CONCLUSION
The tracer-pulse method is a very good method for measuring
competitive adsorption isotherms since it delivers the real data
points on the adsorption isotherm (i.e., not only the parameter
estimates). Moreover, the TP theory is simple and straightforward
for any number of compounds in contrast to other methods for
It is difficult to find similar error data in the literature as a
comparison, i.e., studies on the systematic error of raw data. We
(26) Forsse´n, P.; Fornstedt, T. J. Chromatogr., A 2006, 1126, 268-275.
Analytical Chemistry, Vol. 80, No. 6, March 15, 2008 2111

n-component determinations such as the inverse method and
the perturbation method. In addition, in a multicomponent
situation, the tracer peaks can always be identified in contrast to
perturbation peaks. However, one important problem has re-
stricted its widespread use in HPLC: how to practically visualize
the tracer peaks.
the tracer pulse method is derived assuming that the tracer should
have exactly the same properties as the unlabeled compound; this
is, however, seldom the case. We found that the labeling does
not affect the determination of the equilibrium constant, but the
initial slope of the adsorption isotherm (distribution coefficient)
was more or less identical to the assumed distribution coefficient
of the labeled molecules.
We here present a solution to this problem, with a firm base
in analytical chemistry, by the implementation of stable isotopes
detected with mass spectrometry. This approach was used for
determining the competitive adsorption isotherms of a mixture
of methyl- and ethyl mandelate. The method development involved
the solving of many practical problems associated with highly
concentrated effluents and MS, by designing triple flow splits in
series for reducing the mass by more than 300 000 times and
diluting the effluent more than 5000 times before entering the
mass spectrometer inlet. Using these splits makes it possible to
use concentrated solutions of non MS-compatible mobile phases
and solutes, allowing a more general investigation to be conducted.
We systematically investigated the main sources of errors of
the method. One important source of error for a relating
technique, the perturbation method, is a too large equilibrium
disturbance made on the system. We found that the largest errors
in the retention times, due to an excess tracer injected, occurred
at low-plateau concentration and that the errors decreased very
drastically for the tracer peak, as compared to the perturbation
peak, with increasing plateau concentration. The error due to the
labeling effect was also systematically investigated. Theoretically
The model independent raw adsorption isotherm data delivered
by the tracer-pulse method gives the opportunity to derive the
number of different interactions that take place simultaneously
in a complex sample mixture and to calculate the energy involved
in each of these different types of interactions (e.g., hydrophobic,
electrostatic, polar). This will improve the fundamental under-
standing of the adsorption processes of mixtures. It will also be
possible to make accurate investigations in competitive drug
binding studies, with a complete census of all interactions (weak
as well as strong interactions).
ACKNOWLEDGMENT
This work was supported by a grant from the Swedish
Research Council (VR): “Fundamental Studies on Molecular
Interactions Aimed at Preparative Separations and Biospecific
Measurements” (T.F.).
Received for review November 22, 2007. Accepted
January 20, 2008.
AC702399A
2112 Analytical Chemistry, Vol. 80, No. 6, March 15, 2008