van't hoff

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: [email protected].

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

ES160069_LCGC1212_1052.pgs 11.28.2012 23:01 ADV blackyellowmagentacyan


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.


DECEMBER 2012 LCGC NORTH AMERICA VOLUME 30 NUMBER 12 1055www.chromatographyonline.com

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.



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.


DECEMBER 2012 LCGC NORTH AMERICA VOLUME 30 NUMBER 12 1057www.chromatographyonline.com

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.

References

(1) D.S. Jensen, T. Teutenberg, J. Clark, and

M.R. Linford, LCGC North Am. 30(9),

850862 (2012).

(2) D.S. Jensen, T. Teutenberg, J. Clark, and

M.R. Linford, LCGC North Am. 30(11),

992998 (2012).

(3) J.W. Dolan, J. Chromatogr., A 965(12),

195205 (2002).

(4) S. Wiese, T. Teutenberg, and T.C. Schmidt, J.

Chromatogr., A 1218(39), 68986906 (2011).

(5) J.W. Dolan, L.R. Snyder, N.M. Djordjevic,

D.W. Hill, D.L. Saunders, L. Van Heukelem,

and T.J. Waeghe, J. Chromatogr., A 803(12),

131 (1998).

(6) R.G. Wolcott, J.W. Dolan, and L.R. Snyder,

J. Chromatogr., A 869(12), 325 (2000).

(7) G. Vanhoenacker, F. David, and P. Sandra,

Chromatogr. Today 3(3), 1416 (2010).

(8) Y. Xiang, B. Yan, B. Yue, C.V. McNeff,

P.W. Carr, and M.L. Lee, J. Chromatogr., A

983(12), 83-89 (2003).

(9) G. Vanhoenacker and P. Sandra, Anal. Bio-

anal. Chem. 390(1), 245248 (2008).

(10) A.M. Edge, S. Shillingford, C. Smith, R.

Payne, and I.D. Wilson, J. Chromatogr., A

1132(12), 206210 (2006).

(11) C.M. Hong and C. Horvth, J. Chromatogr.,

A 788(12), 5161 (1997).

(12) J.V. Tran, P. Molander, T. Greibrokk, and

E. Lundanes, J. Sep. Sci. 24(12), 930940

(2001).

(13) S. Heinisch and J.-L. Rocca, J. Chromatogr.,

A 1216(4), 642658 (2009).

(14) J. Bowermast and H.M. McNair, J. Chro-

matogr. Sci. 22, 165170 (1984).

(15) L.R. Snyder, J.J. Kirkland, and J.W. Dolan,

Introduction to Modern Liquid Chromatog-

raphy (Wiley, Hoboken, New Jersey, 2010).

(16) P.L. Zhu, L.R. Snyder, J.W. Dolan, N.M.

Djordjevic, D.W. Hill, L.C. Sander, and T.J.

Waeghe, J. Chromatogr., A 756(12), 2139

(1996).

(17) P.L. Zhu, J.W. Dolan, and L.R. Snyder, J.

Chromatogr., A 756(12), 4150 (1996).

(18) R.G. Wolcott, J.W. Dolan, L.R. Snyder, S.R.

Bakalyar, M.A. Arnold, and J.A. Nichols, J.

Chromatogr., A 869(12), 211230 (2000).

(19) Y. Liu, N. Grinberg, K.C. Thompson, R.M.

Wenslow, U.D. Neue, D. Morrison, T.H.

Walter, J E. OGara, and K.D. Wyndham,

Anal. Chim. Acta 554(12), 144151 (2005).

(20) J.F. Wheeler, T.L. Beck, S.J. Klatte, L.A.

Cole, and J.G. Dorsey, J. Chromatogr., A

656(12), 317333 (1993).

(21) K.B. Sentell and A.N. Henderson, Anal.

Chim. Acta 246(1), 139149 (1991).

(22) L.A. Cole and J.G. Dorsey, Anal. Chem.

64(13), 13171323 (1992).

(23) L.A. Cole, J.G. Dorsey, and K.A. Dill, Anal.

Chem. 64(13), 13241327 (1992).

(24) D. Morel and J. Serpinet, J. Chromatogr., A

214(2), 202208 (1981).

(25) J.W. Coym and J.G. Dorsey, J. Chromatogr.,

A 1035(1), 2329 (2004).

(26) T. Teutenberg, Anal. Chim. Acta 643(12),

112 (2009).

(27) B. Hemmateenejad, J. Chemom. 19(1112),

657667 (2005).

(28) S. Pous-Torres, J.R. Torres-Lapasi, J.J.

Baeza-Baeza, and M.C. Garca-lvarez-

Coque, J. Chromatogr., A 1163(12), 4962

(2007).

(29) C.B. Castells, L.G. Gagliardi, C. Rfols,

M. Ross, and E. Bosch, J. Chromatogr., A

1042(12), 2336 (2004).

(30) L.G. Gagliardi, C.B. Castells, C. Rfols,

M. Ross, and E. Bosch, J. Chromatogr., A

1077(2), 159169 (2005).

(31) R.C. Chloupek, W.S. Hancock, B.A. Mar-

chylo, J.J. Kirkland, B.E. Boyes, and L.R. Sny-

der, J. Chromatogr., A 686(1), 4559 (1994).

(32) L.G. Gagliardi, C.B. Castells, C. Rfols, M.

Ross, and E. Bosch, Anal. Chem. 78(16),

58585867 (2006).

(33) S.M.C. Buckenmaier, D.V. McCalley, and

M.R. Euerby, J. Chromatogr., A 1060(12),

117126 (2004).

(34) S.M.C. Buckenmaier, D.V. McCalley, and

M R. Euerby, J. Chromatogr., A 1026(12),

251259 (2004).

(35) T.L. Chester and J.W. Coym, J. Chromatogr.

A 1003(12), 101111 (2003).

(36) F. Gritti and G. Guiochon, J. Chromatogr., A

1099(12), 142 (2005).

(37) F. Gritti and G. Guiochon, Anal. Chem.

78(13), 46424653 (2006).

(38) F. Wang, T. OBrien, T. Dowling, G. Bicker,

and J. Wyvratt, J. Chromatogr., A 958(12),

6977 (2002).

(39) T. OBrien, L. Crocker, R. Thompson, K.

Thompson, P.H. Toma, D.A. Conlon, B.

Feibush, C. Moeder, G. Bicker, and N. Grin-

berg, Anal. Chem. 69(11), 19992007 (1997).

(40) E. Papadopoulou-Mourkidou, Anal. Chem.

61(10), 11491151 (1989).

(41) R.K. Gilpin, S.E. Ehtesham, and R.B. Greg-

ory, Anal. Chem. 63(24), 28252828 (1991).

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: [email protected].

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


Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.

Documents

van't hoff