Comparative Study on the Thermodynamic Retention Models in HPLC

J. I. Sz~nt6* / T. Veress

Institute for Forensic Science, P. O. Box 314/4, H-1903 Budapest, Hungary

Key Words

Liquid chromatography Solubility parameter model Thermodynamics Partition model Solvophobic model Retention Selectivity Phase characteristics

S u m m a r y

A series of model substances with known solubility parameter were chromatographed and from the temper- ature dependence of the capacity factor, some of the thermodynamic parameters influencing solute retention were determined. A linear relationship was derived between the enthalpy density and the solute solubility parameter from which a graphical method was introduced for the determination of phase characteristics.

Comparing the predicted and measured capacity factor values it has been found that the predicted values are very sensitive to the literature data selected for the computation; however, by using the van der Waals molecular volume in the calculation significantly lower deviation was found from the measured data. Two equations are given for the prediction of selectivity and as the mathematical criteria of the validity of the used thermodynamic models. The results show that the predicted selectivity values are similar to the measured data using given initial parameters in the computation. However, the unreliability of the literature data makes the application of the retention models diff icult.

Introduction

During the evolution of the theory of HPLC, partition [1-3J, solvophobic [4 -6 ] , solubility [7-12] , kinetic "basketball" [13] and other models have been developed for the prediction of the capacity factors. It is, however, evident, that chromatographers still have difficulties, due to the limitation in the theoretical approaches. In this paper we attempt to compare some of these physical (mathemati- cal) models in order to confront them with the chromato- graphically measured data.

Dedicated to Professor J. F. K. Huber on the occasion of his 60th birthday.

596

Theo ry

The thermodynamic description of the solute behaviour in HPLC developed by J. F. K. Huber is based on the partition (distribution) model. Here, the retention time is related to the limiting value of the concentration ratio of the solute in the stationary and mobile phases, i.e., to the partition coefficient, Kio, at infinite dilution [1-3 ] . Ac- cording to this concept, the retention time of solute i, tRi, can be expressed as

tRi = to (1 + Kio q) (1)

where t o is the retention time of a non-retained component and q represents the so-called column constant which is the ratio of the volumes of the stationary and mobile phases in the column. The capacity factor, k', is given as

kl = ( t R i - t 0 ) / t o (2)

It should be noted that Huber used the symbol of K to describe the capacity factor; here, we utilize the inter- nationally accepted symbol of k'.

From eqs. (1) to (2) one can write that

kl = Kio q (3)

In the solvophobic theory developed by Horv~th and co- workers [ 4 -6 ] a similar expression was derived for the capacity factor:

k '= Kr (4)

tn eq. (4) K is the thermodynamic equilibrium constant of the reversible binding of the solute to the functional groups at the stationary phase surface which is equivalent to the partition coefficient of eqs. (1) and (3); ~ is the column constant in reversed-phase liquid chromatography which in turn is equivalent to the column constant q in eq. (1). The capacity factor is related to the column con- stant and the free energy change of interaction, AG, nor- malized to the experimental temperature [4]:

Ink '= I n r AG/RT (5)

where T is the absolute temperature and R is the gas constant.

The solubility parameter model of Schoenmakers [14, 15] is related to the partition concept (eq. (1)). This model gives the following expression for the capacity factor:

Ink ,= (vi/RT) [(6 i - 6m) 2 - - ( 6 i - - 6s) 2 ] + Inr (6)

Chromatographia Vol. 20, No. 10, October 1985 Originals

0009-5893/85/10 0596-05 $ 03.00/0 �9 1985 Friedr. Vieweg & Sohn Verlagsgesellschaft mbH

where vi is the molar volume of solute i, 6 is the solubi l i ty parameter of the solute (i), mobile (m) and stationary (s) phase, and ~b is the column constant as earlier. I t should be noted that Schoenmakers et al. used the term ns/nm as the column constant. I t fol lows from eqs. (5) and (6) that

- RT ( I n k ' - In@) = z~G (7)

or

- RT ( I n k ' - In~) = vi [ ( 8 i - (~s)2 - ( 6 i - 6m )2 ] (8)

For the partial molar free energy change we have previously defined the function ~i [16].

This function can be obtained by rewriting eq. (7):

{I) i = Ah i - TAsi (9)

or

~i = - RT (Ink'- Inq~) (10)

where subscript i refers to the solute.

Taking into consideration that according to Hildebrand's theory the enthalpy of solute i in phase f, hif, is given by

hif = vi ((~i- (~f)2 (11}

eqs. (7) and (8) seem not to be equivalent. Therefore eq. (8) should be corrected with an additional entropy term. The corrected expression may be wri t ten as

(~i = Vi [ ( 6 i - (~s)2 -- ( 6 i - 6m )2 ] -- TASi. (12)

Therefore

Z&hi = vi [ (8 i_ 8s)2 _ (8 i _ 8m )2] (13)

or, with some rearrangements,

Ahi/v i = 28 i (8 m -- 8 s) + (82 + 82) . (14)

Using the substitution

X i = Asi/R + In~ (15)

from eqs. (9) and (10) we obtain:

Ink ' = - Ah i /RT + X i. (16)

Furthermore, according to the solubi l i ty concept the capacity factor wi l l change with the alteration of the mobile phase strength by adjusting the solubi l i ty para- meter from 8ml to ~m2 and the change of capacity factor, in logarithmic form (AIn k'), can be writ ten as follows:

/\ In k' = v i /RT [(8ml + 6m2) - 26i ] (6rn1-6m2) (17)

+ (•1 - ~'2 }

Finally, the selectivity, O~ji , is defined as: I t

eii = ki /k i (18)

Its logarithmic form can be derived from eq. (16) as

In0zji = - ( A h j - Ahi ) /RT + (X j - )k i ) (19)

or, in a simplif ied form:

In~ji = Q1 + Q2 (20)

where the Q values, Q1 and Q2, are the enthalpy and entropy part of the selectivity, respectively, A similar

expression can be writ ten on the basis of the solubi l i ty theory:

Inozji = Q~ + Q~ (21)

where

Q1 - (Sm- 6s)/RT[(Sm + 6s ) ( v j - vi) (22)

- 2 ( v j6 j - Vi(~i) ]

and

Q~ = (;k;* - X*) (23)

The stars refer to the values calculated on the basis of the solubi l i ty parameter model.

E x p e r i m e n t a l

Apparatus

The HPLC measurements were carried out using a Hewlett- Packard Model 1084B liquid chromatograph equipped wi th a Hewlett-Packard 79875 variable wavelength UV detector. 250ram x 4.6ram i.d. Nucleosil 10 C-18 columns (Chrom- pack) were used.

Reagents

Methanol and benzene were HPLC grade purchased from E. Merck. The other chemicals were reagent grade obtained from Reanal (Budapest, Hungary).

Al l measurements were performed isocratically in the temperature range of 40-75~ generally with a f low rate of 2 ml/min.

Resul ts and D iscuss ion

A series of model substances of known solubi l i ty parameter term were chromatographed at different temperatures and the van't Hoff plots (Ink I vs. l /T ) were prepared from which both /kh i and ~'i values can be determined according to eq. (16). The data obtained are collected in Table I; the van der Waals molar volumes (Vwi) were calculated according to Bondi's method [17}. In the van't Hoff plots no considerable deviation was observed from linearity which indicates that the enthalpy changes were constant under the experimental conditions.

Plotting enthalpy density (Ah/v) versus solute solubi l i ty parameter we have introduced a graphical method for the determination of the phase characteristics (8 s and 6m). The relationship between the enthalpy density and the solubi l i ty term is presented in Fig. 1. The l inearity of the plots in Fig. 1 shows an agreement with eq. (14). From the slope and intercept of the plots, 8s and ~m can be computed: according to eq. (14) the slope and the intercept are equal to (62 + 8 2 ) and 2(5 m - 8 s) respectively. Thus two independent equations are available for the two un- knowns.

Chromatographia Vol. 20. No. 10, October 1985 Originals 597

Table I. Chromatographically measured and calculated data*

Solute

No. Compound

1 aniline 2 nitrobenzene 3 benzene 4 toluene 5 ethylbenzene 6 n-propylbenzene 7 n-butylbenzene

1 aniline 2 nitrobenzene 3 benzene 4 toluene 5 ethylbenzene 6 n-propy~benzene 7 n-butylbenzene

91.15 102.28 88.91

106.3 122.46 139.44 156.05

91.15 102.28

88.91 106.3 122.46 139.44 156.05

Vwi

56.38 62.64 48.36 59.51 6974 79,97 90.20

56.38 62.64 48.36 59.51 69.74 79.97 90.20

Mobile phase composition CH3OH-H20

(v/v)

70 : 30

65 : 35

k; lnki

1.36 0.307 1.95 0.668 2.38 0.867 3.21 1.166 4.24 1,445 6.02 1.795 8.81 2.176

1.41 O.344 2.25 0.811 2.81 1.033 4.06 1.401 5.73 1.746 8.76 2.170

13.81 2.625

* v i = molar volume [mol/cm 3] Vwi van der Waals volume [mol/cm 3] Ah i = partial molar enthalpy change [cal/mol]

h i = see eq. (15) r = correlation coefficient The correlation coefficient refers to eq. (16) The value of k i was measured at 40~

Table II. Solubil i ty parameter (6) values [(cal/cm3) 1/2] taken from the literature

Mobile phase constituents

Methanol Water

14,48 23.14 14.50 23.53 15.85 25.52 12.9 21

Solutes

Nitro- Aniline Benzene Toluene

benzene

9.15 8.90 10.86 9.16 8.93

9.71 9,35 11.1 9.2 8.9

11.73

Ethyl- n-Propyl- benzene benzene

8.80 8.65 8.84 8.64

~h i

- 887 - 1420 - 1404 - 1876 - 2261 - 2792 - 3366

- 1031 - 1803 - 1789 - 2338 - 2810 - 3414 - 4045

- 1 , 1 2 - 1 . 6 1

- 1 . 4 0

- ] .86 - 2.20 - 2,70 - 3.25

- 1 . 3 1

- 2 . 0 9

- 1 . 8 5

- 2.36 - 2.78 - 3.33 - 3 . 8 9

0,986 0.998 0,998 0.999 0.999 0.999 0.999

0.998 0.999 0.998 0.999 0.999 0.999 0.999

Ref. n-Butyl- benzene

[18] 8.58 [12]

I7] I19]

i.\.6 40 \ ?

\ 4 \ 3

~-n 30 E u

=5 6 <3 %-5~4

2 0 " ~ 3

10

8 9 1 0

j" [ea,lem3Jll2 F i 9 . 1

Plot of enthalpy density (Ah/v w and Ah/v) vs. the solubil ity para- meter (8 }.

Mobile phase: 65:35 (v/v) methanol-water. Curve I.: Z~h/v w vs. 6; Curve I1.: Ah/v vs. 6.

Solutes: I aniline, 2 nitrobenzene, 3 benzene, 4 toluene, 5 ethyl- benzene, 6 n-propylbenzene

In add i t iona l to this graphical method, the so lub i l i t y

parameter of the mob i le phase m i x t u r e (Sin) can also be

calcu lated f r om the f o l l o w i n g expression [16 ] :

8 m = ~ ~i(~i (24) i

where v i is the vo lume f rac t ion o f the mob i le phase con-

st i tuents and (~i represents data avai lable in the l i terature.

Tab le II presents selected so lub i l i t y parameter values

f r o m the l i terature. The scatter ing of the l i terature data

is obv ious even in the case of the mob i le phase const i-

tuents ((~CH30 H and ~H20) . Consequent ly , the mob i le

phase so lub i l i t y parameter ca lcu la ted f r om eq, (24) w i l l

depend on the selected l i terature data. Tab le I I I (co lumn

A) conta ins the mob i le phase so lub i l i t y parameters cal-

cu la ted f r om eq. (24) using the in i t ia l so lub i l i t y values

presented in Tab le II.

Tab le III compares the calcu lated so lub i l i t y parameters of

the mob i le phase (6rn) w i t h terms ob ta ined by the graph-

ical me thod descr ibed above. There are d i f f e ren t values of

dev ia t ion in Tab le III depending on the select ion o f the

in i t ia l l i terature data fo r the ca lcu la t ion o f $m- As can be

also seen f r om Tab le I I I s ign i f i cant ly lower dev ia t ion was

f o u n d by using p l o t I in the graphical method, i.e., by using

van der Waals molecu lar vo lumes in the graphical presenta-

t ion .

5 9 8 Chromatographia Vol. 20, No. t0, October 1985 Originals

Table III. Comparison of the solubility parameter of the mobile phase* (6m, (cal/cm3)l/2) calculated using eq. (24} and the values of the solubility parameter of the stationary (&s) and mobile phase (&m) established from Fig. 1.

calculated from eq. (24)

A

17.61 17.66 19.23 15.74

6 m G s Deviation, %

established from plot I

in Fig. I

15.37

established from plot II

in Fig. 1

13.69

established from plot I

in Fig. I

D

10.39

established from plot II

in Fig. 1

9.77

Avs. B (B = 100%)

14.6 14.9 25.1

2.4

Avs. C (C = 100%)

28.6 29.0 40.0 1 5.0

65:35 (v/v) methanol-water

From the fact that (~m is sensitive to the initial data a- vailable in the literature, i t follows that the prediction of the capacity factor wil l also depend in all cases on the selected literature values of the solubil i ty parameters when we substitute 6m values in eq. (12).

In a large computation series we have actually found that the calculated capacity factors are indeed very sensitive to the selected literature data. In some cases the calculated capacity factor values were in an acceptable agreement with the measured chromatographic data given in Table I. Unfortunately, however, using other literature data, the deviation between the predicted and measured capacity factors became quite unacceptable. Obviously the weakness in the congruency is regarded first of all to be due to the scattering of the values given in the literature: values re- ported by different authors are seldom similar or even comparable.

In the theoretical part we have given two mathematical models (based on two different thermodynamic models) for the selectivity, eq. (20) and eq. (21), in logarithmic form. In order to compare these models let us asume that both eq. (20) and eq. (21) lead to the same selectivity value of a given solute pair. In this case it is mathematical- ly true that

Q1 + Q2 = Q~ + Q~ (25)

However if the two models are valid also in the thermo- dynamic sense then the two enthalpy (Q1) and entropy (Q2) terms of eqs. (20) and (21) are equal; i.e.:

Q1

and

Q2

=Q~' or Q~'-Q1 =0 (26)

=Q~ or Q~-Q2=0 (27)

Thus the deviation of Q* vs. Q can be a measure of the validity of the models to be compared. At the same time the satisfaction of eqs. (26) and (27) shows that the selec- t iv i ty values predicted by eqs. (20) and (21) are identical with the chromatographically measured data.

In our experiments Q1 and Q2 were estimated from the van't Hoff plots, Q~ by means of the solubil i ty concept (eq. (22)) and Q~ from eq. (21) using the measured capaci- ty factors for the calculation, since in our assumption

mentioned above, ej~ must be equal to OZji. Table IV com- pares the selectivity calculated for adjacent peaks.

The Q values are also listed in Table IV. They contain the errors of the capacity factor measurements, as well as the errors of the selected literature data in the computation of mobile phase polarity (solubil i ty parameter). In our case the deviation of Q* vs. Q does not refer to a con- siderable disharmony between the thermodynamic models compared above, since the deviation values can be at- tributed to the errors of measurements and computations for most of the peak pairs understudied. On the other hand, the thermodynamic models used for data processing during establishing the data collected in Table IV are approximately equivalent for the prediction of selectivity, since, as mentioned above, in the case of not too large differences in the compared Q values the predicted selec- t iv i ty values are nearly similar to the measured chromato- graphic data. However, the Q values wil l be changed depend- ing on the selection of the initial literature data. Therefore the unreliabil i ty of the literature data makes it d i f f icul t to build up a real picture of any retention model in HPLC.

Nevertheless, the capacity factor and the selectivity are thermodynamic terms. Therefore refining the measuring techniques of the initial parameters used in the math- ematical models, and refining the reproducibil i ty of the measured chromatographic data wil l certainly reanimate the HPLC thermodynamic models.

Acknowledgements

The authors are deeply indepted to Profs. L. de Galan and H. A. H. Bil l iet for helpful discussion and comments.

List of Symbols

z~G free energy change h enthalpy Ah partial molar enthalpy change k' capacity factor K thermodynamic equil ibrium constant (equivalent to

Kio) Kio partit ion coefficient at inf inite di lut ion (equivalent

to K)

Chromatographia Vol. 20, No. 10, October 1985 Originals 599

Table IV. Comparison of the selectivity calculated for adjacent peaks, a'b

Peak pair v i Ah ;~i 8i In~ Q1 Q~ Q2 Q~

aniline 56.38 - 1 0 3 1 - 1.311 10.3 0.462 1.245 0.623 - 0.783 - 0.161

nitrobenzene 62.64 - 1 8 0 3 - 2.094 10

nitrobenzene 6264 - 1 8 0 3 - 2.094 10 0.222 -0 .023 -0 .054 0.247 0.276

benzene 48.38 - 1789 - 1.847 9.16

benzene 48.36 - 1789 -1 .847 9.16 0368 0.885 0.988 - 0.516 - 0,620

toluene 59.51 - 2338 - 2.363 8.93

toluene 59.51 - 2338 - 2.363 8 93 0.345 0.761 0.838 0.418 0.493

ethylbenzene 69.74 2810 - 2.781 8.84

ethytbenzene 69.74 - 2810 - 2.781 8_84 0.425 0.974 1022 - 0,549 0.597

propylbenzene 79.97 - 3414 - 3.330 8,64

propylbenzene 79.97 - 3414 - 3.330 8,64 0.455 1.018 0,875 - 0.561 - 0,420

butylbenzene 90,20 - 4045 - 3.891 8.58

Deviation of

Q{ vs. Q1 Q~ vs. Q2 % %

- 50.0 79.9

134.8 11.7

11.6 20.2

10.1 17.9

4.9 8.7

- 14 ,0 - 2 5 . 1

a vi = molar volume [cm3/molJ; z&h = partial molar enthalpy change lcai/motJ; 2~ i = entropy change, 5 i = solubiJity parameter l(cal/cm 3) 1/2]; ~x = selectivity (= ki/ki); Q1 = enthalpy part of selectivity; Q2= entropy part of selectivity; asterisk refers to the values calculated on the basis of the solubility parameter model.

b mobile phase: 65:35 (v/v) methanol-water. Solubility parameters: methanol: 12,9 (cal/cm3) 1/2 I191, water: 21 (cal/cm3) 1/2 [19], Calculated values for the mobile phase: 15.74; for the stationary phase: 10.39.

ns/nm q Q

Q1 Q2 R As

to

tRi T v

Vw

K

X

t) r

co lumn constant (equ iva lent t o q and r

co l umn constant (equ iva lent to ns/nm and r

e i ther en tha lpy or e n t r o p y par t of se lect iv i ty

en tha lpy par t of se lec t iv i ty

e n t r o p y par t o f se lect iv i ty

gas constant

part ia l mo lar e n t r o p y change

vo id vo lume

re tent ion t ime

absolute tempera tu re

molar vo lume

v o n d e r Waals molar vo lume

se lec t iv i ty

so lub i l i t y parameter

capac i ty fac to r (symbol used by Huber)

e n t r o p y change and co lumn constant con ta in ing

aux i l i a ry parameter

vo lume f rac t ion

co l umn constant (equiva lent to ns/n m and q)

part ia l mo lar free energy change

S u b s c r i p t s

f mob i le phase

i solute or mob i le phase const i tuents

j, i adjacent peaks

m mob i le phase ( m l , m2 = mob i le phases w i t h di f fer-

ent polar i t ies)

s s ta t ionary phase

S u p e r s c r i p t

* quant i t ies ca lcu lated on the basis o f the so lub i l i t y

parameter model .

References

[1] J,F.K. Huber, in "Advances in Chromatography 1970", A. Zlarkis, ed., University of Houston, Houston, Texas, 1970; p. 348.

[2] J.F.K. Huber, J. Chromatogr. Sci. 9, 72 (1971). [3] J.F.K. Huber, C.A.M. Meijers, J .A.R.J . Hulsman, Anal.

Chem. 44, 111 i1972). [4] Cs. Horvdth, W. Melander, Internet. Lab. 11 (Nov./Dec.,

1978). [5] Cs. Horv~th, W. Melander, I. Moln~r, J. Chromatogr. 125,

129 (1976). [6] Cs. Horvdth, W. Melander, I. Moln~r, Anal. Chem. 49, 142

(1977). [7] R. Tijssen, H. A. H. Billiet, P. J. Schoenmakers , J. Chromato-

gr. Sci. 122, 185 (1976). [8] P.J. Schoenmakers, H.A.H. Billiet, R: Tijssen, L. de Galan,

J. Chromatogr. 149, 519 (1978). [9l P.J. Schoenmakers, H.A.H. Billiet, L, deGalan, J. Chro-

matogr. 218, 261 (1981). [10] P.J. Schoenmakers, H.A.H. Billiet, L. de Galan. Chromato-

graphia 15, 387 (1982). [11] H. A. H. Billiet, P. J. Schoenmakers, L. de Galan, J. Chro-

matogr. 218, 443 (1981). [12] T.L. Hafkenscheid, E. Tomlinson, J. Chromatogr. 264, 47

(1983). [13] F.E. Regnier, Advances in Liquid Chromatography, 4th

Annual American Eastern-Europian Symposium on Liquid Chromatography, September 10--14 1984, Szeged, Hungary.

[14] P.J. Schoenmakers, H.A.H. Billiet, L. de Galan. Chromato- graphia 15, 205 (1982).

[15] P.J. Schoenmakers, Dissertation, University of Delft, 1982. 116] J.I. Szdnt~, Chromatographia 17, 27 (t983), [17] A. Bondi, J. Phys. Chem. 68, 441 (1964). [18] W. Holzm~iller, K. Altenburg, "'Physik der Kunststoffe", Aka-

demie Verlag, Berlin, 1961; p. 141. [19] L.R. Snyder, J.J. Kirkland, Introduction to Modern Liquid

Chromatography, John Wiley & Sons, New York, 1974; Hungarian translation: MLiszaki K~nyvkiad6, Budapest, 1979; p. 142.

Received: Dec. 28, 1984 Revised manuscript received: Feb. 14, 1985 Accepted: Feb. 19, 1985 A

600 Chromatographia Vol. 20, No. 10, October 1985 Originals