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1052 LCGC NORTH AMERICA VOLUME 30 NUMBER 12 DECEMBER 2012 www.chromatographyonline.com
T (C)
ln k
210
7
6
5
4
3
2
200 190 180 170 160 150 140
1/T (K-1)
0.00207 0.00217 0.00227 0.00237
David S. Jensen*, Thorsten Teutenberg, Jody Clark, and Matthew R. Linford**Department of Chemistry and Biochemistry, Brigham Young University, Provo, Utah; Institut fr Energie- und Umwelttechnik e. V., Duisburg, Germany; and Selerity Technologies Inc., Salt lake City, Utah. Direct correspondence to: mrlinford@chem.byu.edu.
Elevated Temperatures in Liquid Chromatography, Part III: A Closer Look at the van t Hoff Equation
Part I of this article series discussed some of the advantages of and
practical considerations for elevated temperature separations in
liquid chromatography (LC) (1). Part II reviewed some of the basic
thermodynamics of chromatography and elevated temperature
separations, which included a brief derivation and discussion of the
van t Hoff equation (2). This third and final part continues the
exploration of elevated temperatures in LC with a more detailed
discussion of the van t Hoff equation, exploring its usefulness and
relevance using various examples from the literature.
Van t Hoff plots can be a useful
and interesting part of data anal-
ysis for high-temperature liquid
chromatography (LC). Here, in part III
of this series, we take a closer look at the
van t Hoff equation using various exam-
ples from the literature.
Review of the
Advantages of Elevated-
Temperature Separations
Elevated temperatures offer a number of
benefits in LC. One such benefit is that
they facilitate retention mapping in which
the retention factor, k, is measured at dif-
ferent temperatures so that the values of k
over a range of temperatures can be pre-
dicted (3,4). Retention mapping can also
include the probing and predicting of k at
different mobile-phase compositions and
is widely used in method development
(3,5,6). Another benefit is that selectiv-
ity () may change with temperature,
which is important for retention map-
ping and is another parameter that can
be considered in method development
(3,5,6). Also, increasing temperatures
can improve sample throughput because
the van Deemter minima shifts to higher
flow rates; that is, the optimal efficiency
for a separation shifts to a higher mobile-
phase velocity (79). Finally, a decrease
in the organic modifier is possible due
to the change in the polarity of water at
increasing temperatures; water behaves
more like an organic solvent at elevated
temperatures. Furthermore, a more aque-
ous mobile phase is considered greener
because of a reduction in the amount of
organic modifier used (1014). Clearly,
there are good reasons for considering the
use of elevated temperatures in LC. Now,
lets discuss various aspects of high-tem-
perature separations in the context of the
van t Hoff equation.
Review of the
van t Hoff Equation
The van t Hoff equation is derived from
the following two basic thermodynamic
equations:
G0 = H0 TS0 [1]
and
G0 = -RT ln K [2]
where G0 is the Gibbs free energy,
H0 is the enthalpy of transfer, T is the
absolute temperature in kelvins, S0 is
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the entropy of transfer, R is the gas con-
stant, and K is the equilibrium constant.
When we equate these two equations
and solve for ln K we achieve the van t
Hoff equation:
ln K = - H0/RT + S0/R [3]
As discussed in part II (2), ln K = ln k,
where k is the retention factor and is
the phase ratio (VM/VS) the ratio of
the mobile-phase volume and stationary-
phase volume. By substituting k for K in
equation 3 we obtain the van t Hoff equa-
tion as it is commonly encountered in LC:
ln k = - H0/RT + S0/R ln [4]
Note that sometimes is used instead
of , where = 1/ = VS/VM. Thus, an
equivalent form of equation 4 is:
ln k = - H0/RT + S0/R + ln [5]
Unfortunately, both and are referred
to as the phase ratio.
The van t Hoff Equation
in Retention Mapping
Van t Hoff plots are often linear, which
makes them useful in retention map-
ping. For this purpose, a simpler, but
mathematically equivalent, version of
equation 4 can be used (3,15):
log k = A + B/T [6]
where plots of log k vs. 1/T are generated
under isocratic conditions (6,1618) and
A and B either have values as given by
equation 5 or may be viewed empirically.
While to some degree this semi-empirical
equation conceals the underlying ther-
modynamics, in most cases this is not the
primary concern. Of course, retention
mapping may be conveniently performed
using commercially available software.
When using temperature as a variable
to optimize a separation, chromatogra-
phers can create a series of van t Hoff
plots for various analytes to determine
the best temperature for the separation.
To demonstrate this optimization pro-
cess, Figure 1 shows a series of van t Hoff
plots for various drugs. In this example,
some of the drugs, such as in the circled
lines in the plot, show different slopes
and change elution order (reverse selec-
tivity) where their lines cross. Obviously,
the temperature at this crossing point
would be a very poor choice for separa-
tion conditions because the peaks would
coelute, that is, = k2/k1 = 1. Thus, in a
separation involving multiple analytes, a
series of van t Hoff plots can be used to
optimize the separation.
Thermodynamics of
Linear van t Hoff Plots
Linear van t Hoff plots such as those in
Figure 1 suggest that the retention mecha-
nisms for the analytes are constant; that
is, the values for H0, S0, and for the
analytes are constant over the temperature
range under consideration. Of course there
is the possibility that H0, S0, and are
mutually changing so that the net effect is
a linear relationship, but this will be dis-
cussed below. If the retention mechanism
is constant with temperature it may be
possible to compare the enthalpies (H0)
and entropies (S0) of similar analytes on
the same column, or of a single analyte
on different columns. It should be noted
again that the H0 transfer of the analyte
from the mobile phase to the stationary
phase can be derived from the slope of its
van t Hoff plot (see equation 4) (2). When
H0 is negative, which is typically the case
in reversed-phase chromatography, trans-
fer of the analyte from the mobile phase
to the stationary phase is favored and
exothermic. Clearly, the more negative
H0 is the more favorable the interaction
between the analyte and the stationary
phase is, which generally leads to larger
values of k. For example, in reversed-
phase chromatography, the H0 values
for a homologous series of increasingly
hydrophobic analytes such as the alkyl
benzenes with longer and longer alkyl
chains should steadily become more nega-
tive (exothermic). This effect is shown in
Figure 2: The alkyl benzenes with longer
alkyl chain lengths (more hydrophobic)
have larger slopes than those with shorter
chain lengths (less hydrophobic) (19).
Nonlinearities in
van t Hoff Plots
Because of Phase Transitions
As a corollary to the previous state-
ments, a nonlinear van t Hoff plot
shows that H0, S0, or are chang-
ing, that is, the retention mechanism
for the analyte is not constant over the
temperature range under consideration.
A possible explanation for a nonlinear
van t Hoff plot is a phase transition in
the stationary phase; at lower tempera-
tures the stationary phase will generally
be in a solid-like conformation and at
T (C)190
7
6
5
4
3
2
1
0
-1
0.00211
Aminohippuric acid Aminobenzoic acidCaffeineTheophyllineAminoantipyrine
ParacetamolPhenacetinAntipyrineHydroxyantipyrine
ln k
0.00231 0.002511/T (K-1)0.00271 0.00291 0.00311
170 150 130 110 90 70 50 30
Figure 1: Van t Hoff plots for test probes showing linear relationships between the natural logs of the retention factors vs. 1/T for these compounds. The inset shows the point at which the elution is reversed for aminoantipyrine and caffeine. From left to right, the data points correspond to 180, 150, 120, 90, 60, and 40 C. Adapted from reference 10 with permission.
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higher temperatures it will adopt a more
liquid-like conformation (13,20,21).
Thus, a nonlinear van t Hoff plot may
indicate that the thermodynamic inter-
actions between the analyte and sta-
tionary phase change when the station-
ary phase undergoes a phase transition.
The possibility of a phase transition in
a C18 phase seems reasonable because
long chain hydrocarbons have melting
points in the range of 2050 C, for
example, the melting point of octadec-
ane is approximately 28 C. For silica-
based C18 stationary phases, this phase
transition may occur in the range of
2050 C (2224). Of course, the melt-
ing transition of a C18 stationary phase
is much more complicated than the
simple melting of a pure hydrocarbon
because of the tethering of the chains
in the stationary phase, the density or
packing of the chains, and the presence
of other chemical groups in the film
such as endcapping agents.
Other phase transitions at higher
temperatures have also been reported.
For example, a phase transition around
100 C was found for a silica-based hybrid
C18 column (Figure 3). This transition was
attributed to a change in the conformation
of the stationary phase in the presence of
the mobile phase as a function of tem-
perature (19). This idea was substantiated
by solid-state nuclear magnetic resonance
(NMR) spectroscopy, which showed, over
a temperature range of 30150 C, that
the dry stationary phase did not undergo
any conformational changes. Differential
scanning calorimetry (DSC) was also per-
formed on the stationary phase under con-
ditions that mimicked typical LC mobile
phase conditions. DSC showed thermal
desorption of the mobile phase (70:30
wateracetonitrile) from the stationary
phase around the phase transition point
(circa 97 C), suggesting a change in the
conformation of the stationary phase when
the mobile phase was present.
The phase transition in Figure 3 pro-
duces two linear van t Hoff plots: region
I at lower temperatures (3297.3 C) and
region II at higher temperatures (97.3
200 C). Interestingly, H0 in region II
is approximately twice that of region I.
Coym and Dorsey (25) discussed this
possibility of an increase in H0 follow-
ing a phase transition:
It may seem odd that the enthalpy of transfer (retention) at high tem-perature is more favorable than at low temperature, because retention is greater at low temperature. The ther-modynamic quantity that governs retention is the free energy (G0), which has an entropy component (S0). Because of the change in hy-drogen bond structure of water with temperature, the entropy change as-sociated with retention changes with temperature. At lower temperatures, where the mobile phase is hydrogen bonded, there is a favorable entropy change upon retention. This is com-monly referred to as hydrophobic effect. However, at high tempera-tures, where there is little or no hy-drogen bonding, the entropy change would be expected to be much less. As a result, although the enthalpy (H0) of retention is more favorable at high temperature, it is outweighed by the entropic (S0) contribution.
Irregularities in van t Hoff
Plots Because of pH Effects
Unusual van t Hoff plots may be
observed in the separation of acids and
T (C)ln
k210
7
6
5
4
3
2
200 190 180 170 160 150 140
Toluene Ethyl benzene Propyl benzene Butyl benzene Amyl benzene
1/T (K-1)
0.00207 0.00217 0.00227 0.00237
T (C)
1/T (K-1)
ln k
175 125 75 25
3.5
2.5
1.5
0.5
0
0.00205 0.00245 0.00285 0.00325
Region I
Region II 32 C to 97.3 C
H = -4.2 kcal/mol
H = -8.0 kcal/mol
97.3 C to 200 C
1
2
3
Figure 2: Van t Hoff plots for a homologous series of alkyl benzenes. Increases in alkyl character result in larger slope values indicating more negative H0 values. From right to left, the data points correspond to 200, 190, 180, 170, 160, and 150 C. Adapted from reference 19 with permission.
Figure 3: Van t Hoff plot for toluene demonstrating curvilinear behavior around 100 C. Adapted from reference 19 with permission.
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bases (26) because a change in the tem-
perature, and therefore polarity, of the
mobile phase can affect the pKa and pKb
values of weak acids and bases. Obvi-
ously, a change in the ionization state of
an analyte or buffer in a mobile phase
can alter retention (27), and selectiv-
ity changes associated with temperature
changes are larger for polar and ioniz-
able analytes than for nonpolar analytes
(2832). In particular, an analytes pKa
value can shift approximately 0.03 pKa
units per degree Celsius (33,34). These
effects are complex and are usually
strongly manifested when pH pKa; that
is, where both the weak acid and conju-
gate base have appreciable concentrations
(29). The success of these types of separa-
tions depends on the nature of the buf-
fer and analyte (30). Two examples are
shown in Figures 4 and 5. Figure 4 shows
an increase in retention of protriptyline,
a tricyclic antidepressant, with increasing
temperature on two different columns.
Figure 5 also shows analytes that exhibit
(unusual) negative slopes in their van t
Hoff plots. As a side note, temperatures
can also affect large molecules for
example, proteins may undergo confor-
mational changes with temperature (31).
Confirming the Linearity
of van t Hoff Plots and
Evaluating Changes in Entropy
Chester and Coym (35) explored the
possibility of changing during a van t
Hoff analysis and noted that, at least in
theory, a change in could compensate
for changes in H0 or S0, leading to an
(apparently) linear van t Hoff relation-
ship. This statement is consistent with
some of the concerns raised by Gritti and
Guichon (36,37); that is, different com-
pensating or canceling factors may lead
to the linearity often observed in van t
Hoff plots. To eliminate this possibility,
Chester and Coym (35) noted a slightly
more advanced use of van t Hoff analy-
sis in which one plots ln vs. 1/T, where
is the selectivity (k2/k1) between two
analytes. This relationship is obtained
from equation 4 by subtracting the van t
Hoff relationship for the second analyte
from the van t Hoff relationship for the
first, leading to the following equation:
ln k2 - ln k1 = ln(k2/k1) =
ln = (1/RT )(H02 H01) + (1/R)
(S02 S01) [7]
which can also be expressed as
l n = (1/RT ) H 02,1 + (1/R)
S02,1 [8]
If the individual van t Hoff relation-
ships for the first and second analytes
are linear, and the van t Hoff relation-
ship in equation 7 or 8 for their selectiv-
ity is also linear, then there is a higher
probability that H0, S0, and are
constant (or at least not substantially
changing) over the temperature range
in question. Thus, if one wishes to use a
van t Hoff analysis to extract H0 and
S0 values for an analyte, it would prob-
ably be advisable to apply this additional
check on the data. If the resulting plot
of equation 7 or 8 is linear, it would add
credence to any claim that meaningful
thermodynamic information could be
extracted from the analysis. In addition,
if the plot of ln vs. 1/T is nonlinear
then the stationary phase may undergo a
conformational change in the tempera-
ture range studied (3841).
As a corollary to these last points,
using b1 = S01/R ln for the first
analyte and b2 = S02/R ln for the
second analyte under the same condi-
tions or same column, we can calculate
the difference in entropies of transfer
for the two analytes as S02,1 = S02
S01 = R(b2 b1), where this latter term
is the gas constant multiplied by the dif-
ference between the y-intercepts of the
two van t Hoff plots for the two ana-
lytes. Note that the phase ratio, which
we often do not know, has canceled,
leaving us with the difference between
the two entropies of transfer. This anal-
ysis can be useful on a series of com-
pounds, where they are all compared to
one member of the series.
T (C) T (C)
1/T (K-1) 1/T (K-1)
ln k
ln k
75
(a) (b)
65 55 45 35 25 75 65 55 45 35 251.7 0.75
0.65
0.55
0.45
0.35
0.25
1.6
1.5
1.4
1.3
1.2
1.1
10.0029 0.0030 0.0031 0.0032 0.0033 0.0029 0.0030 0.0031 0.0032 0.0033
T (C) T (C)
1/T (K-1) 1/T (K-1)
ln k
ln k
55
(a) (b)
2
1.5
0.5
-0.5
-1.5
-1
0
1
2
1.5
0.5
-0.5
-1.5
-1
0
1
50 45 40 35 30 25 20 55 50 45 40 35 30 25 20
0.00305 0.00315 0.00325 0.003250.00335 0.003350.00305 0.00305
Figure 4: Van t Hoff plots of protriptyline obtained at pH 7.8. Column: (a) Inertsil ODS 3V, (b) X-Terra RP18. Flow rate: 1.0 mL/min. Temperature increases from right to left. Adapted from reference 33 with permission.
Figure 5: Van t Hoff plots of acidic and basics analytes. (a) Phosphate buffer pH(25 C) = 8.10, and (b) tris + HCl buffer pH (25 C) = 8.09; mobile phase contains 50% (v/v) methanol. Analytes: = 2,4-dichlorophenol, = 2,6-dichlorophenol, = ben-zylamine, = benzyldimethylamine. Adapted from reference 29 with permission.
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Concerns About the
van t Hoff Equation
A careful study of the van t Hoff equa-tion and its use in elevated-temperature LC suggests that there is some question regarding its fundamental accuracy (36,37). For example, the van t Hoff relationship assumes that the stationary phase is homogeneous, which is clearly not the case in general, a stationary phase will contain different types of sites, which will have different affini-ties for a given analyte. The van t Hoff equation also assumes that both the stationary phase and the mobile phase remain constant as a function of tem-perature. Neither will be entirely true. The adsorption and absorption (parti-tioning) of mobile-phase components in the stationary phase, which will alter the properties of the stationary phase, will vary with temperature, and the mobile phase will also change with temperature; for example, the static per-mittivity (dielectric constant) of water will change with temperature. Perhaps a measured view of these concerns is to acknowledge their validity, while also noting that in many circumstances it appears that these effects are not so extreme that useful information cannot be obtained by van t Hoff analysis.
Conclusions
Van t Hoff plots can be a useful and interesting part of data analysis for high temperature LC. They are a valuable tool in an empirical sense for retention mod-eling, and thermodynamic data may be extracted from them. They can reveal phase transitions in stationary phases, and the changes in pKa values of analytes with temperature. Plots of ln vs. 1/T can help confirm that H0 and S0 are constant with temperature. It should be understood that the underlying assump-tions of the van t Hoff equation are not entirely correct.
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David S. Jensen and Matthew R. Lin-ford are with the Department of Chem-istry and Biochemistry at Brigham Young University in Provo, Utah. Direct correspon-dence to: mrlinford@chem.byu.edu.
Thorsten Teutenberg is with the Institut fr Energie- und Umwelttechnik e. V. in Duisburg, Germany.
Jody Clark is with Selerity Technologies Inc., in Salt Lake City, Utah.
For more information on this topic,
please visit
www.chromatographyonline.com/Linford
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