Thermodynamics of adsorption of binary aqueous organicliquid mixtures on a RPLC adsorbent
Fabrice Gritti a,b, Georges Guiochon a,b,∗a Department of Chemistry, University of Tennessee, Knoxville, TN 37996-1600, USA
b Division of Chemical Sciences, Oak Ridge National Laboratory, Oak Ridge, TN 37831-6120, USA
Received 23 January 2007; received in revised form 29 March 2007; accepted 4 April 2007Available online 19 April 2007
bstract
The surface excess adsorption isotherms of organic solvents commonly used in RPLC with water as co-eluent or organic modifiers (methanol,thanol, 2-propanol, acetonitrile and tetrahydrofuran) were measured on a porous silica surface derivatized with chlorotrimethylsilane (C1-silicaith 3.92 �mol C1 groups per m2 of SiO2), using the dynamic minor disturbance method. The 5 �m diameter particles were packed in a 150 mm4.6 mm column. The isotherm data were derived from signals resulting from small perturbations of the equilibrium between the aqueous–organic
olutions and the adsorbent surface. The partial molar surface area of the adsorbed components were assumed to be the same as those of the pureomponents. The difference σ − σ∗
i between the surface tensions of the adsorbed mixtures and that of the pure liquids was measured as a functionf the organic modifier molar fraction. A simple and unique convention for the position of the Gibbs dividing surface was proposed to delimithe Gibbs’s adsorbed phase and the bulk liquid phase. The activity coefficients of the organic modifiers and of water and their thermodynamic
quilibrium constants between the two phases were measured. The strong non-ideal behavior of the adsorbed phase is mostly accounted for byhe surface heterogeneity. Some regions of the surface (bonded –Si(CH3)3 moieties) preferentially adsorb the organic compound while the regionslose to unreacted silanols preferentially adsorb water.
Understanding the thermodynamics of equilibrium betweensolution and a solid adsorbent [1–13] is essential to assess theerformance of a porous material in terms of molecular recogni-ion. An adsorbent wet by a solution will adsorb selectively oner a few components of the solution. This selectivity dependsn the composition of the liquid. A direct application of thishenomenon is the separation of samples in reversed-phase liq-id chromatography (RPLC). A pulse of sample, dissolved inhe mobile phase, percolates through a bed of porous, chemi-ally modified silica. In order to achieve the elution of all the
ample components in a reasonable time and/or to modify theelectivity of the separation, the mobile phase is a solution ofweak solvent (e.g., water in RPLC, hexane in NPLC) and a
trong solvent (e.g., methanol, acetonitrile or tetrahydrofuran inPLC, isopropanol in NPLC). The progressive increase of theoncentration of the strong solvent (gradient elution chromatog-aphy) is a common application of the influence of a change inhe mobile composition on selective adsorption.
Methanol, ethanol, 2-propanol, acetonitrile and tetrahydro-uran are the organic modifiers most frequently used in RPLC.ased on the observation that elution times are systematically
maller with acetonitrile than with methanol, chromatographersave classified methanol as a weaker eluent than acetonitrile.his decision may not be justified. Some have advanced aslternative explanations, the higher solubility of most sampleomponents in acetonitrile than in methanol, others the strongerdsorption of acetonitrile on RPLC packing materials and itsore effective competition with the analyte components for
dsorption on the stationary phase. Not being solidly basedn thermodynamics, these explanations are fragile. The solu-ility of a compound is not directly related to its distributionetween the bulk liquid and the adsorbed phase. A complete
The chemical potentials of the component i in the adsorbeda l
6 F. Gritti, G. Guiochon / J. C
nderstanding of the adsorption of organic compounds requiresse of the fundamental concept of the Gibbs surface excessegarding the adsorption of a liquid mixture at a solid–liquidnterface.
The goal of this paper is an investigation based on the Gibbsurface excess of the adsorption properties of five classicalPLC organic solvents (methanol, ethanol, 2-propanol, acetoni-
rile and tetrahydrofuran) from their aqueous solutions onto ailica surface that has been derivatized with chlorodimethyl-ilane (i.e., fully endcapped). This work is a first step in anttempt to better understand retention mechanisms in RPLC, i.e.he adsorption of solutes from a binary solution onto an imper-
eable solid surface. The impermeability of the solid surfaceo liquid solutions used in RPLC was demonstrated by Kazake-ich et al. [29], based on the comparison between the specificore volume of porous silica particles measured by chromatog-aphy and by low temperature nitrogen adsorption. In this firstttempt, we do not take the dissolved analyte into considerationut focus on the thermodynamics of adsorption of the binaryobile phase (water + organicsolvent). The surface excess of
he organic compound is measured for the whole possible com-osition range of the mixture (0–100%). The simplest adsorbent,silica surface covered with a monolayer of trimethylsilane
roups, –Si(CH3)3, is used. The equilibrium between the solu-ion and the adsorbent surface is perturbed locally by injectionf a small pulse of organic modifier (linear perturbation), andhe elution signal is detected. This is the minor disturbance
ethod. The heterogeneity of the adsorbent surface, the non-deal behavior of the adsorbed phase and the exact value ofhe thermodynamic equilibrium constant K between the bulkolution and the adsorbed phase will be discussed on the singleasis of the Gibbs surface excess isotherms obtained and of theonvention used to define the position of the Gibbs’ dividingurface. We present first the derivation of the activity coeffi-ients in the adsorbed phase, followed by a discussion of theethods used to derive numerical estimates valid in the case
tudied.
. Theory
.1. The Gibbs-Duhem relationship
The Gibbs-Duhem relationship is inherent to the constructionf the extensive state function G. To derive it, the Gibbs-elmholtz enthalpy is written for both the adsorbed phase
noteda) and the bulk solution (notedl).The definition of the state function G is:
= U + PV − TS − σA (1)
here U is the internal energy of the system, P its pressure, V itsolume, T its temperature, S its entropy, σ the interfacial tensionetween the solution and the adsorbent that are in contact and A
s the adsorbent surface area. According to the first and secondrinciples of thermodynamics, any infinitesimal and reversibleransformation causing independent increments of entropy dS,olume dV , surface area dA and number of mole dni, leads to
pb
togr. A 1155 (2007) 85–99
he following increment of the internal energy, U:
U = T dS − P dV + σ dA +∑
i
μi dni (2)
here μi is the chemical potential of component i in the solu-ion. The corresponding infinitesimal increment of the Gibbsree energy, G (Eq. (1)), writes:
G = (T dS − P dV + σ dA +∑
i
μi dni) + (P dV + V dP)
− (T dS + S dT ) − (σ dA + A dσ)
= −S dT + V dP − A dσ +∑
i
μi dni (3)
Integration of Eq. (2) at constant temperature T, under con-tant pressure P and at constant surface tension σ, and chemicalotential μi, relates the change in internal energy to the increasesn the entropy, the volume, the surface area and the number of
olecules of component i in the system from 0 to S, 0 to V, 0 toand 0 to ni, respectively.1
− 0 = U = T
∫ S
0dS − P
∫ V
0dV + σ
∫ A
0dA
+∑
i
μi
∫ ni
0dni = TS − PV + σA +
∑i
niμi
(4)
Combination of Eqs. (1) and (4) gives:
=∑
i
niμi (5)
nd after differentiation of Eq. (5)
G =∑
i
ni dμi +∑
i
μi dni (6)
Finally, the Gibbs-Duhem relationship is obtained by com-aring Eqs. (3) and (6),
Adσ = S dT − V dP +∑
i
ni dμi (7)
Eq. (7) is the classical Gibbs-Duhem relationship that applieso a solution adsorbed on an adsorbent surface. This same rela-ionship can be applied to a bulk solution that is not under thenfluence of the potential field of the adsorbent surface. It is thenritten:
= S dT − V dP +∑
i
ni dμi (8)
.2. Chemical potentials in the bulk solution and in thedsorbed liquid phase
hase and in the bulk solution, μi and μi, respectively, shoulde derived consistently. In the bulk phase, the chemical potential
1 The internal energy is obviously zero in the absence of matter.
hroma
dcva(s(
μ
wuv
ppT
μ
t
μ
wbm
ptibti
μ
wtaoei
μ
2
b
t
μ
o
−o
γ
npfntbsa
2
sfmcp
n
w
hiafitt(pAwtvt
mao
F. Gritti, G. Guiochon / J. C
epends on three independent intensive parameters, P, T and theomposition of the solution, xl
i. For the pure component i, theariation of its chemical potential μ∗
i with pressure and temper-ture can be derived from Eq. (8). At constant temperature TdT = 0), the variation of the chemical potential with the pres-ure P is given by integration of the Gibbs-Duhem relationshipEq. (8)):
l,∗i (T, P) − μ
l,∗i (T, P0) =
∫ P
P0vl,∗i (T, P) dP (9a)
here μl,∗i (T, P0) is the chemical potential of the pure liq-
id component i at the reference atmospheric pressure P0 andl,∗i (T, P) is its molar volume.
If we assume that the pressure under which the adsorptionroblem is studied is not very different from the atmosphericressure, v
l,∗i can be considered as independent of the pressure.
hen:
l,∗i (T, P) = μ
l,∗i (T, P0) + v
l,∗i (P − P0) (9b)
The chemical potential μli(T, P, xl
i) of component i in a mix-ure of molar composition xl
i is:
li(T, P, xl
i) = μl,∗i (T, P) + RT ln[γl
i (xli)x
li]
= μl,∗i (T, P0) + v
l,∗i (P − P0) + RT ln[γl
i (xli)x
li]
(10)
here γli (x
li) is the activity coefficient of compound i in the
ulk mixture. The activity coefficients generally depends on theixture composition.In the adsorbed layer of adsorbed solution, the chemical
otential of compound i depends on four intensive parameters,he temperature T, the pressure P, the composition xa
i and thenterfacial tension σ. Repeating the same calculation as for theulk liquid phase but considering now Eq. (7) instead of Eq. (8),he general expression of the chemical potential of component in the adsorbed solution, at constant pressure P is:
ai (T, P, σ, xa
i ) = μa,∗i (T, P, σ∗
i ) − a∗i
ti(σ − σ∗
i )
+ RT ln[γai (xa
i )xai ] (11)
here σ∗i is the interfacial tension between the pure liquid i and
he solid surface. a∗i is the molar surface area of compound i,
lone, on the adsorbent surface and ti is the average numberf adsorbed monolayers of pure compound i. Thermodynamicquilibrium between the adsorbed and bulk pure liquid phasemposes that:
a,∗i (T, P, σ∗
i ) = μl,∗i (T, P) = μ
l,∗i (T, P0) + v
l,∗i (P − P0)
(12)
.3. Solid–liquid thermodynamic equilibrium
Combining Eqs. (10)–(12), the equilibrium relationshipetween the adsorbed and the bulk liquid phases is given by a
togr. A 1155 (2007) 85–99 87
he equality of their respective chemical potential in each phase:
ai (T, P, σ, xa
i ) = μli(T, P, xl
i) (13a)
r
a∗i
ti(σ − σ∗
i ) + RT ln[γai (xa
i )xai ] = RT ln[γl
i (xli)x
li] (13b)
r
li (x
li)x
li = γa
i (xai )xa
i exp
[−a∗
i
ti
σ − σ∗i
RT
](13c)
Eq. (13c) represents the fundamental equilibrium thermody-amic relationship between the adsorbed and the bulk liquidhases. Its application requires the knowledge of the molar sur-ace area a∗
i (T ) of the compound in the adsorbed monolayer. Theumber of adsorbed monolayers, ti, is a priori unknown. Theerm σ − σ∗
i is the free energy of immersion into the solutiony reference to immersion into the pure liquid. It can be mea-ured experimentally, based on the measurement of the excessdsorbed amount of component i onto the adsorbent surface.
.4. Excess amount adsorbed and adsorbed phase structure
The surface in contact with the bulk liquid phase attractsome components of the solution. The composition of the liquid,rom the adsorbent surface to the bulk, varies in an unknownanner with the distance to the surface. The excess amount of
omponent i, nei , in the adsorbed phase with respect to the bulk
hase in equilibrium (composition xli) writes [14]:
ei = (xa
i − xli)∑
i
nai (14)
here nai is the adsorbed amount of component i.
Note that nei is unique and can be measured. On the other
and, the total number of mole in the adsorbed phase,∑
inai ,
s completely arbitrary and so is the molar fraction in thedsorbed phase, xa
i , because the real physical concentration pro-le above the surface is unknown. However, the thermodynamic
reatment of the adsorption behavior requires a physical delimi-ation between two apparently immiscible phases in equilibriumadsorbed and bulk solutions) with their own chemical com-ositions (xa
i and xli, respectively). Fig. 1 illustrates this point.
ccording to the choice for the maximum distance za abovehich the liquid is considered as identical to the bulk solution,
he number of moles in the adsorbed phase differs. za can be con-eniently expressed as a multiple t of the adsorbed monolayerhickness.
In the case in which the partial molar surface areas of theixture components do not depend on the composition xa
i butre equal to the molar surface areas of the pure components, a∗
i ,ne can write:
At∑ =∑
a∗i x
ai (15a)
i
nai i
For a binary mixture, the molar fractions can be expressed asfunction of the unique, experimental excess amount adsorbed
88 F. Gritti, G. Guiochon / J. Chroma
Fig. 1. Representation of the true excess amount adsorbed of component i(hatched area) relatively to the bulk concentration xl
i and the equivalent adsorbedphase represented by the different rectangles, which correspond to different val-ut
a
x
ampts
2
s1
wg
RTd(γl
1xl1)
γl1x
l2
2
t
K
g
K
n
t
2
dt
K
wmbb
n
wp
pm
n
w
ttcomponent 1 can directly be accounted for by Eq. (23). t ischosen by convention. Comparison between the experimentaldata and the results of Eq. (23) needs the determination of three
es of the number of adsorbed monolayer t. Note the interdependence betweenand xa
i .
nd of the parameter t:
a1 = Atxl
1 + a∗2n
e1
At + (a∗2 − a∗
1)ne1
(15b)
Knowing the molar surface areas of the pure components, a∗1
nd a∗2, and the surface area of the adsorbent A, it is possible to
easure the molar compositions xai of the adsorbed phase. The
arameter t is directly related to the arbitrary choice made forhe position of the Gibbs dividing surface above the adsorbenturface (see explanation later, Fig. 1).
.5. Case of a binary mixture
In this work, we study the adsorption of a binary mixture on aolid surface. According to Eq. (14), the amounts of componentsand 2 adsorbed at equilibrium are
ne1 = −ne
2
na1 = ne
1 + xl1
∑i
nai
na2 = ne
2 + xl2
∑i
nai
(16)
The Gibbs-Duhem relationships, Eqs. (7) and (8), combinedith the equilibrium between adsorbed and bulk liquid mixturesive, at constant T and P:
Eq.(8) : 0 = xl1 dμl
1 + xl2 dμl
2 ⇔ dμl2 = −xl
1
xl2
dμl1 = −
Eq.(7) : −A dσ =(
ne1 + xl
1
∑i
nai
)dμl
1 +(
−ne1 + xl
2
∑i
nai
)
togr. A 1155 (2007) 85–99
.5.1. Homogeneous surfaceThe selectivity of an adsorbent for component 1 with respect
o component 2 is defined as:
1/2 = xa1x
l2
xl1x
a2
= α (18)
Combining Eq. (18) with Eq. (13c) for components 1 and 2ives:
1/2 = γl1γ
a2
γa1 γl
2
exp
[a∗
1
t1
σ − σ∗1
RT− a∗
2
t2
σ − σ∗2
RT
](19)
Combining Eqs. (14), (15) and (18) gives:
e1 = At(K1/2 − 1)xl
1xl2
K1/2a∗1x
l1 + a∗
2xl2
(20)
Obviously, in Eq. (20), K1/2 depends on the composition ofhe mixture studied. It is not a constant.
.5.2. Heterogeneous surfaceLet assume that the surface area A of the adsorbent can be
ivided into N different patches, Aj . On each patch j, the selec-ivity K1/2,j is written:
1/2,j = xa1,jx
l2
xl1x
a2,j
(21)
here xa1,j and xa
2,j are the molar fraction of the adsorbateolecules 1 and 2 adsorbed on the patch j. The excess num-
er of moles of component 1 in the adsorbed phase is giveny
e1 =
j=N∑j=1
ne1,j =
j=N∑j=1
naj (xa
1,j − xl1) (22)
here naj = na
1,j + na2,j . A similar relationship applies to com-
onent 2.If we assume that there are t monolayers in the adsorbed
hase on each patch j of surface area Aj , the overall excessole numbers of adsorbate 1 are
e1 =
j=N∑j=1
Ajt(K1/2,j − 1)xl1(1 − xl
1)
K1/2,ja∗1x
l1 + a∗
2(1 − xl1)
(23)
ith A =∑jAj
In the simple case in which the surface A is divided intowo types of patches (N = 2) and the adsorbed and bulk solu-ions are ideal, the measurable excess amount adsorbed of
−xl1
xl2
dμl1 ⇔ − A
RTdσ = ne
1
γl1x
l1x
l2
d(γl1x
l1)
(17)
F. Gritti, G. Guiochon / J. Chromatogr. A 1155 (2007) 85–99 89
Table 1Activity coefficient in the bulk estimated from the UNIFAC method
a The activity coefficient given for the eluent whose volume fraction is zero c
ndependent parameters, the surface area A1 and the two ther-odynamic equilibrium constants K1/2,1 and K1/2,2.
.5.3. Determination of the activity coefficients in thedsorbed phase
The activity coefficients of the components in the bulk liq-id phase (γl
i ) can easily be estimated using the UNIFAC groupethod [15,16]. The details of the calculation are given in ref-
rence [17] for the calculation of the activity of a ternary liquidixture of methanol, water and phenol. The precision of theNIFAC method is only fair (< 10%) but it has a wide applica-
ion range [18]. Table 1 lists the activity coefficients of the fiverganic solvents studied in their aqueous solutions, as functionsf the volume fraction of the organic solvent.
The calculation of the activity coefficients in the adsorbedhase do not require the knowledge of details regarding the het-rogeneity of the adsorbent surface. The required informationncludes the activity coefficients in the bulk phase (as afore-
entioned), the overall surface area A of the adsorbent, theolar surface areas of the pure liquid components a∗
1 and a∗2
n the same adsorbent, the average number t of monolayersdsorbed and the changes in interfacial tension σ − σ∗
1 and− σ∗
2 between the liquid solution and the pure liquids. Com-ining Eqs. (18) and (13c), it is possible to derive the logarithmf the ratio between γa
1 and γa2 :
nγa
1
γa2
= lnγl
1
γl2
− ln K1/2 + a∗1
t1
σ − σ∗1
RT− a∗
2
t2
σ − σ∗2
RT(24)
From the definition of K1/2 (Eq. (18)), the structure of thedsorbed phase (Eq. (15)) and the excess amount adsorbed (Eq.14)), one can express K1/2 as a function of the measurable
Combining Eqs. (28) and (30), the activity coefficients in thedsorbed phase are calculated as follows:
n γa1 = ge + xa
2 ln γa1 − xa
2 ln γa2
= xa2 ln
γa1
γa2
+∫ xa
1
0ln
γa1
γa2
dxa1 (31a)
n γa2 = ge + xa
1 ln γa2 − xa
1 ln γa1
= −xa1 ln
γa1
γa2
+∫ xa
1
0ln
γa1
γa2
dxa1 (31b)
.5.4. Estimate of the molar surface areas a∗i
In order to calculate the molar surface area of water, methanol,thanol, 2-propanol, acetonitrile and tetrahydrofuran, one needsstandard reference. This standard reference can be given by theolar surface area of nitrogen, N2 on a silica surface modifiedith reaction with a dimethylalkyl silane. Amati and Kovats
19,20] have determined the space requirement of a single N2olecule on a series of modified silica surfaces. They showed
hat this space was of the order of 20 ± 1 A2 on hydrophobicurface but of only 16.2 ± 0.1 A2 on neat silica surfaces.
Based on the van der Waals atomic radius of nitrogen andhe van der Waals bond length in the nitrogen molecule, the
inimum space requirements for an adsorbed molecule of nitro-
en would be only 11.0 A2. The difference with the 20 A2
a factor × 1.82) is explained by the steric void between thedsorbed molecules and by surface motions. It is now possi-le to estimate the molecular surface area of any other liquid
msdt
able 3hysico-chemical characteristic of the C1-silica column
a Measured from BET experiments with space requirements of N2 = 16.2 A2.b Measured from elemental analysis.
a The densities are considered at a temperature of 20 ◦C, except nitrogen at196 ◦C.
nowing its density and that of nitrogen. Accordingly,
∗i = aN2
(ρl
N2
ρli
Mi
MN2
)2/3
(32)
able 2 lists the space requirements of the liquids men-ioned above, based on the density of liquid nitrogen at 77 K0.807 g/cm3).
. Experimental
.1. Chemicals
The mobile phases used in this work were mixtures ofethanol, ethanol, isopropanol, acetonitrile or tetrahydrofuran
nd water. The volume fractions were 0 (pure water), 0.5, 1, 5,0, 20, 30, 40, 50, 60, 70, 80, 90, 95, 99, 99.5 and 100 (purerganic solvent). Water, methanol, isopropanol, acetonitrile andetrahydrofuran were all HPLC grade and purchased from Fishercientific (Fair Lawn, NJ, USA). Ethanol was absolute (200roof) and purchased from the same manufacturer.
.2. Columns
The column used in this work was packed with silica end-apped with trimethylchlorosilane (TMS). It was generouslyffered by the manufacturer (Waters, Mildford, MA, USA). Theolumn tube dimensions are 150 mm ×4.6 mm. The characteris-ics of the bare silica and of the modified silica measured by the
anufacturer are given in Table 3. BET was used to measure thepecific surface area, the total pore volume and the average poreiameter. The C1surface coverage was calculated according tohe carbon content measured by elemental analysis.
re Specific surfacearea (m2/g)a
C endcapping(%)b
C1surface coverage(�mol/m2)
349 0 0.00235 4.48 3.92
hromatogr. A 1155 (2007) 85–99 91
g
m
Tqa
3
AmeUdWtaffdraslr1pvt
3
smabt
V
fe
V
n
Ffisc
3m
mowi4Tl
x
waoofw[i
c
3p
wtw
F. Gritti, G. Guiochon / J. C
For 1 g of unbonded silica, the mass of tethered C1 chains isiven by:
his mass is important to know because it allows to report anyuantity measured on the C1-silica column to the unit surfacerea of the neat silica.
.3. Apparatus
The injections were acquired using a Hewlett-Packard (Palolto, CA, USA) HP 1090 liquid chromatograph. This instru-ent includes a multi-solvent delivery system (tank volumes, 1 L
ach), an auto-sampler with a 25 �L sample loop, a diode-arrayV-detector, a RI-detector (HP1047), a column thermostat and aata station. Compressed nitrogen and helium bottles (Nationalelders, Charlotte, NC, USA) are connected to the instrument
o allow the continuous operations of the pump, the auto-samplernd the solvent sparging. The extra-column volumes is 0.041 mLrom the auto-sampler needle seat to the UV cell. It is 0.211 mLrom the auto-sampler needle seat to the RI cell. All the retentionata were corrected for these contributions. The flow-rate accu-acy was controlled by pumping the pure mobile phase at 295 Knd 1 mL/min during 50 min, from each pump head, succes-ively, into a volumetric glass of 50 mL. The relative error wasess than 0.25%, so that we can estimate the long-term accu-acy of the flow-rate at less than 3 �L/min at flow rates aroundmL/min. All measurements were carried out at a constant tem-erature of 295 K, fixed by the laboratory air-conditioner. Theariation of the ambient temperature during the acquisition ofhe peak profiles never exceeded ±0.5 K.
.4. Minor disturbance method
The excess amount of organic modifier adsorbed was mea-ured using the minor disturbance method [21,22]. For a binaryixture, the perturbation of the equilibrium between the bulk
nd the adsorbed phases generates a single signal that is detectedy the refractive index detector. The elution volume of this per-urbation (VR) is related to the excess amount adsorbed by:
R(cl1) = VM +
(dne
1
dcl1
)(34)
The column thermodynamic hold-up volume is determinedrom the integration of Eq. (34) between 0 and c
l,∗1 . Since the
xcesses are zero for these two extreme concentrations:
M =∫ c
l,∗1
0 VR(cl1) dcl
1
cl,∗1
(35)
The excess amount adsorbed is given by:
e1(cl
1) =∫ cl
1
0(VR(cl
1) − VM) dcl1 (36)
s[ea
ig. 2. Experimental contraction factor α at room temperature (T = 295 K) forve aqueous–organic binary liquid mixtures vs. the molar fraction of the organicolvent. Data taken from references [23–27]. Note that the correction for theontraction volume never exceeds 4%.
.5. Measurement of the concentration of the organicodifier
The different mobile phases used in the minor disturbanceethod measurements were prepared by mixing known volumes
f water and organic modifier. The sum of these two volumesas 250 mL. The volume fraction of the organic modifier, φ1,
s known experimentally (0%, 0.5%, 1%, 5%, 10%, 20%, 30%,0%, 50%, 60%, 70%, 80%, 90%, 95%, 99%, 99.5% and 100%).he mole fraction of organic modifier xl
1 was calculated as fol-ows:
l1 = 1
1 + (1 − φ1/φ1)(ρ2/ρ1)(M1/M2)(37)
here ρ1 and ρ2 are the densities of the organic modifiernd water, respectively, and M1 and M2 their molar mass. Inrder to determine the concentration (cl
1 in mol/mL) of therganic modifier, one needs to know the volume contractionactor α (α < 1, see Fig. 2) upon preparation of the mixturesater–methanol [23], water–ethanol [24], water–2-propanol
25], water–acetonitrile [26] and water–tetrahydrofuran [27]. cl1
s calculated as follows:
l1 = φ1
α
ρ1
M1(38)
.6. Measurement of the external porosity by exclusion ofolystyrene standards
The measurement of the external porosity of the C1-columnas made from Inverse Size Exclusion Chromatography. Injec-
ion of four polystyrene standards of sufficiently high moleculareight (MW = 90, 000, 400,000, 575,000 and 900,000) corre-
ponding to the exclusion branch of the ISEC were performed28,29]. The extrapolation of the linear correlation between thelution volumes and the cubic root of the molecular weight tomass of zero gives the interparticle volume. The polystyrene
92 F. Gritti, G. Guiochon / J. Chroma
Fig. 3. Plot of the retention volume of polystyrene standards in pure THF as afl(
ss
4
4
owMiatsvTs
m
lo
A
iitcr
c
utamub(
V
brbsoutvmte
utottsCofovv
uT
4
t
unction of the cubic root of their molecular weight. The extrapolation of theinear trend to zero gives the interparticle volume in the chromatographic columnsee more details in Section 3).
tandards were dissolved into THF. The experimental results arehown in Fig. 3.
. Results and discussion
.1. Surface area of the C1-silica adsorbent
One important column characteristic is the surface area, A,f the adsorbent packed in it. It is a priori unknown becausee ignored the mass of adsorbent inside the column. From theDM measurements, the average value of the void volume VM
nside the column measured with methanol, ethanol, 2-propanol,cetonitrile and tetrahydrofuran is equal to 1.979. The interpar-icle volume Vex measured by the injection of the polystyrenetandards is equal to 0.995 mL. Accordingly, the total mesoporeolume Vp is obtained from the difference between VM and Vex.he mass of adsorbent mads is then derived from the knownpecific total pore volume Vs
p measured by BET:
ads = VM − Vex
Vsp
= 1.979 − 0.995
0.69= 1.43 g (39)
The adsorbent surface A inside the column, corrected for thearger space requirement of one adsorbed molecule of nitrogenn hydrophobic surface, is:
= 1.43 × 235 × 1.235 = 415 m2
The factor 1.235 in this equation accounts for the differencen space requirement of one molecule of nitrogen at 77 K which
s larger when it is adsorbed on a hydrophobic surface (20.0 A2)han when adsorbed on neat silica (16.2 A2) [19,20]. The spe-ific surface area of 235 m2/g were calculated based on a spaceequirement of nitrogen of 16.2 A2.
fhtm
togr. A 1155 (2007) 85–99
The mass of silica and of the bonded C1chains inside theolumn are:
msilica = 1.43
1.09987= 1.30 g and
mC1 = mads − msilica = 0.13 g
We can check the validity of the experimental hold-up vol-me VM measured from the MDM method. The condition thathe column tube volume is the sum of the volumes VM, Vsilicand VC1 should be verified. The density of the neat silica waseasured by pycnometry (ρsilica = 2.12 g/cm3) and the molec-
lar volume of the attached C1 chain –Si(CH3)3 is assumed toe 75 A3/molecule, as measured by Kazakevich and co-workersρC1 = 1.61 g/cm3) [29]:
C = πr2i L = 2.493 cm3 = VM + msilica
ρsilica+ mC1
ρC1
= 1.979 + 1.30
2.12+ 0.13
1.61= 2.673 cm3
The rather large difference observed (+7%) may be explainedy the fact that the MDM method do not provide the true geomet-ical void volume of the column. If the value of VM determinedy the MDM holds, the specific pore volume of the adsorbenthould be 0.93 mL/g, a value close to the specific pore volumef the neat silica. Instead, we measured the column void vol-me by pycnometry using dichloromethane and methanol ashe two solvents. The volume measured was V0 = 1.863 mL, aalue significantly smaller than that obtained from the MDMethod (−7%). The specific pore volume is in this case equal
o 0.67 mL/g, a value that is in better agreement with the BETxperimental value of 0.69 mL/g.
This seems to demonstrate that the correct value for the hold-p volume might be better given by pycnometry rather than byhe MDM method. The MDM seems to give an overestimatef the column void volume. More data, however, are requiredo confirm this conclusion. For instance, in the literature [29],he values measured for the hold-up volume of a similar C1-ilica column by pycnometry were 1.913 and 1.917 mL (MeCN-H2Cl2and MeCN-THF pairs of solvent). Measured from MDMr from a labeled pure component injection, these volumes wereound to be 1.956, 1.978, 2.015 and 1.975 mL, e.g. a systematicverestimate of about + 3.5%. In this work, we consider thealue VM = 1.874 mL, which leads exactly to the experimentalalue of Vs
p = 0.69 mL/g.The masses of the adsorbent, silica and C1chains in the col-
mn are then mads = 1.33 g, msilica = 1.18 g and mC1 = 0.15 g.he surface area A inside the column is estimated at 386 m2.
.2. Number of adsorbed monolayer t: the convention
As explained in the theoretical part, the delimitation betweenhe adsorbed and the bulk phases requires an arbitrary choice
or the position of the Gibbs dividing surface. Once this surfaceas been chosen, the total number of adsorbed molecules andhe molar fractions in the adsorbed phase can be calculated. This
eans that the interfacial tension σ and the activity coefficient
F. Gritti, G. Guiochon / J. Chromatogr. A 1155 (2007) 85–99 93
F . Them .
itacb
t
(e
hotppmatastfo
(
(
(
I
v
t
aiwfd
ig. 4. Definition of the choice for the position of the Gibbs dividing surfaceodifier, which exhibits a plateau (dna
1 = 0) for a particular molar composition
n the adsorbed phase, γai , will be referred to this choice. Placing
he Gibbs dividing surface amounts to choosing the number ofdsorbed monolayers t. To be consistent over the entire molaromposition investigated, t should remain a constant and shoulde independent of the bulk composition. Accordingly,
1 = t2 = t (40)
This condition simplifies the Gibbs-Duhem relationship (Eq.27)) and the activity coefficients in the adsorbed phase canasily be derived through Eqs. (31a) and (31b).
One needs to make a choice for the value of t. Fig. 4 illustratesow t is arbitrarily chosen. Any choice for t, whether it is smallerr larger than 1, integer or non-integer, will be acceptable from ahermodynamic viewpoint. Our choice is based on the inflectionoint present on the plot of the excess amount adsorbed for aarticular composition different from 0 and 1. Physically, thiseans that , at this composition, the variation of the total amount
dsorbed reaches a minimum. In a case of an homogeneous sta-ionary phase, this inflection point occurs at xl
1 = 1 and the totalmount adsorbed does not vary (dna
1 = 0) because the surface isaturated with the pure component. For heterogeneous surfaces,his inflection point is located at a composition, which differsrom one. Three cases can be treated according to the numberf adsorbed monolayers t chosen:
1) For the smallest values of t, the derivative of the total amountadsorbed na
1 versus the bulk composition xl1 at the inflection
point is negative. This situation is unrealistic because theadsorption isotherm should be an increasing function of thebulk composition. However, there would be no objectionfrom a thermodynamic point of view.
2) For the highest values of t, the same derivative is positive.This corresponds to the actual situation. However, we haveno clue of deciding what would be exactly this positivederivative.
ttbt
choice is made on the representation of the total number of mole of organic
3) For a particular value of t, this derivative is equal to zero.By convention, we will choose this unbiased condition todetermine the unique, arbitrary value of t. It is importantto keep in mind that this particular value of t do not nec-essarily reflect the true number of adsorbed monolayers. Itcorresponds to a strict minimum.
From the definition of the total amount adsorbed,
dna1
dxl1
= d(Atxa1/x
a1(a∗
1 − a∗2) + a∗
2)
dxl1
= Ata∗2
[xa1(a∗
1 − a∗2) + a∗
2]2
dxa1
dxl1
= 0 ⇔ dxa1
dxl1
= 0 (41)
From Eqs. (16), (15a) and (39), we have at the inflection pointin the above case 3:
dna1
dxl1
= dne1
dxl1
+ At
xa1(a∗
1 − a∗2) + a∗
2= 0 (42)
Combining Eqs. (40) and (13c), according to the above con-ention, the choice for the value of t is given by:
= − 1
A
([dne
1
dxl1
]I
(xl1a
∗1 + [1 − xl
1]a∗2) + (a∗
2 − a∗1)[ne
1]I
)
(43)
Fig. 5 A–E show the numbers of layer t calculatedccording to Eq. (41) for methanol–water, ethanol–water,sopropanol–water, acetonitrile–water and tetrahydrofuran–ater mixtures. t is equal to 0.61, 1.48, 2.23, 2.63 and 2.24
or methanol, ethanol, isopropanol, acetonitrile and tetrahy-rofuran, respectively. From a qualitative point of view, the
rue average number of adsorbed monolayer increases whenhe hydrophobicity of the alcohol increases, e.g. with the car-on number. It would not be too much distant from the realityo affirm that the adsorbed methanol–water system forms a
94 F. Gritti, G. Guiochon / J. Chromatogr. A 1155 (2007) 85–99
Fig. 5. Experimental excess number of mole of five adsorbed organic solvent from water measured by the minor disturbance method vs. the molar fraction of theorganic compound in the bulk mixture xl
1. The coordinates (abscissa, ordinate and slope) at the inflection point I are given to allow the calculation of the numberof adsorbed monolayer T consistently with the convention described in Fig. 4 and Eq. (41). (A) Methanol, (B) ethanol, (C) 2-propanol, (D) acetonitrile and (E)tetrahydrofuran.
hromatogr. A 1155 (2007) 85–99 95
slt
4
sitidat
fcfsco
n
rs
Table 4Best fitting parameters (ε, KCH3 , KOH) of Eq. (44) to the experimental excessamount
ingle monolayer, adsorbed ethanol–water system two mono-ayers and adsorbed isopropanol–water, acetonitrile–water andetrahydrofuran–water systems three monolayers.
.3. Surface heterogeneity
Fig. 5 A–E clearly demonstrate the heterogeneity of the C1-ilica surface. Indeed, if the surface was homogeneous, thenflection point I in these plots would be observed at the composi-ion xl
1=1. The molar compositions at which the excess isotherms zero are experimentally observed for a single molar fractionifferent from zero and 1. It is about 0.75, 0.37, 0.32, 0.47nd 0.53 with methanol, ethanol, 2-propanol, acetonitrile andetrahydrofuran.
As a first assumption, one can assume that the adsorbent sur-ace is made of two distinct patches. One represents the surfaceovered by the trimethylsilane groups, the complementary sur-ace being the remaining accessible unreacted silanols after theurface derivatization. According to Eq. (23), one can define twoonstant K1/2, one for the surface –Si(CH3)3groups, KCH3 , thether for the surface Si–OH groups, KOH:
e1 = At
(ε
(KCH3 − 1)xl1(1 − xl
1)
KCH3a∗1x
l1 + a∗
2(1 − xl1)
+ [1 − ε](KOH − 1)xl
1(1 − xl1)
KOHa∗1x
l1 + a∗
2(1 − xl1)
)(44)
The value of A(= 386 m2) was measured according to theesults in Section 4.1. t is fixed according to the convention cho-en for the position of the Gibbs dividing surface (see previous
beao
ig. 6. Fit of the excess number of mole of methanol, ethanol, 2-propanol, acetonitrileixtures. Note the disagreement between the experiment and the model.
ection). The parameters to be estimated are ε (surface fractionccupied with –Si(CH3)3 groups), KCH3 and KOH. Performinghe fit of the experimental values of the excess amount adsorbede1 to Eq. (44) makes sense only if the estimated parametersemain constant with the molar fraction in the liquid phase xl
1.bviously, according to the definition of the parameter K1/2
Eq. (18)), there is no reason for KCH3 and KOH to be indepen-ent of xl
1. The only case for which this hypothesis is exactlyrue is when the molecular sizes of the adsorbate molecules arehe same (a∗
2 = a∗1 = a∗) and the liquid and adsorbed phases
ehave ideally (γl1 = γl
2 = γa1 = γa
2 = 1). The constant K1/2rites then,
1/2 = exp
(a∗
t
σ∗2 − σ∗
1
RT
)(45)
The results of the fit are shown in Table 4. They make sense
ut qualitatively, only, and the comparison between the differ-nt organic modifiers is not trustworthy. The constant KCH3
nd KOH are well larger and smaller than 1, respectively. Inther words, the organic modifier and water are preferentially
and tetrahydrofuran to Eq. (44), which assumes ideal bulk and adsorbed liquid
9 hromatogr. A 1155 (2007) 85–99
arttaatpppavbtiptomis
4
fdonpfirwlftfnaioiwtitot
sOito2os
Fig. 7. Activity coefficients of the components in the binary aqueous–organicld[
daS
t
[
ma
K
6 F. Gritti, G. Guiochon / J. C
dsorbed on the trimethylsilyl groups and on the silanol groups,espectively. Note that the average surface fraction occupied byhe hydrophobic group is about 60% which is consistent withhe 3.92 �mol/m2 surface concentration of Si(CH3)3 and thebout 8 �mol/m2 of initial surface concentration of silanols. Theverage values of KCH3 and KOH are about 15 and 0.20, respec-ively. As expected, one is larger than 1 on the hydrophobicart of the surface and the second is smaller than one on theolar silanol surface. However, one cannot attribute much morehysical information based on the results of this fit because thessumption of identical molecular size and mixture ideality isery unlikely. Evidence is given in Fig. 6 where the agreementetween the experiment and the best fit is obviously poor. Inhe next section, the non-ideality of the adsorbed phase will benvestigated and the true surface molecular areas of the com-ound will be conserved (Table 1). The activity coefficients inhe adsorbed phase will be determined experimentally basedn the known activity coefficients in the bulk (UNIFAC groupethod, ±10%, Fig. 7 A and B) and the convention defined
n the previous section for the position of the Gibbs dividingurface.
.4. Non-ideality of the water–organic adsorbed phase
The convention for the position of the Gibbs dividing sur-ace was discussed in the previous Section 4.2. It has lead to theetermination of the number t of adsorbed monolayers for eachrganic modifier in the adsorbed phase when the variation of theumber of mole of organic modifier adsorbed at the inflectionoint I is strictly zero. The procedure to derive the activity coef-cients γa
1 and γa2 is given in the theory Section 2.5.3. First, the
atio of the activity coefficient of the organic solvent to that ofater is determined according to Eq. (24). It is informative to
ook at the variation of the surface tensionσ − σ∗2 when the molar
raction of the organic solvent increases from 0 (pure water, σ∗2 )
o 1 (pure organic solvent, σ∗1 ). These plots are given in Fig. 8
or the five organic modifiers used in this work. As the carbonumber increases in the alcohol compounds (methanol, ethanolnd 2-propanol), the molar fraction for which the surface tensions minimum decreases (0.75, 0.40 and 0.30). Also, the variationf the surface tension between pure water and pure organic mod-fier, σ∗
1 − σ∗2 , are −16, +1 and +11 mN/m. The adsorption of
ater on the C1-silica surface is significant because, for instance,he surface tension between the C1-silica surface and pure waters less than the surface tension measured with pure ethanol (abouthe same −1 mN/m) and pure 2-propanol (−11 mN/m). Note,n the other hand, that pure methanol, pure acetonitrile and pureetrahydrofuran stabilize the surface energy of the surface.
The activity coefficients of the two adsorbed components arehown in Fig. 9 for the five aqueous organic binary mixtures.bviously, they demonstrate that the adsorbed phase is all but
deal. The thermodynamic consistency of our treatment can beested by the measured value of the activity coefficient of the pure
rganic modifier when xl
1 = 1. It is found for methanol, ethanol,-propanol, acetonitrile and tetrahydrofuran activity coefficientsf 1.020, 1.000, 0.987, 0.995 and 0.996, respectively. The verymall difference compared to the expected value of 1 is simply f
iquid mixtures (bulk phase) vs. the molar fraction of the organic component. Theata were estimated from the UNIFAC group contribution method at T = 295 K18]: (A) organic component and (B) water component.
ue to the fact that the integration of Eqs. (26a), (26b), (31a)nd (31b) is based on a limited number of data point (17 points).till, the thermodynamic consistency is excellent.
Let now consider the following molecular exchange betweenhe adsorbed and the bulk phase:
ORGANIC]l+ [H2O]a ⇔ [ORGANIC]a + [H2O]l K(T )
(46)
Basically, this equilibrium describes the exchange ofolecules of organic modifier and water from one phase to
nother. The equilibrium constant writes:
(T ) = γa1 xa
1γl2x
l2
a a l l= exp
(−a∗
2
t
σ − σ∗2
RT+ a∗
1
t
σ − σ∗1
RT
)(47)
γ2 x2γ1x1
The phase rule or the degree of freedom of the system is:
= N − p + 3
F. Gritti, G. Guiochon / J. Chromatogr. A 1155 (2007) 85–99 97
F ganicw
wntf
F
ig. 8. Variation of the surface tension σ − σ∗2 vs. the molar fraction of the or
ater.
here N is the number of independent components (= 2), p theumber of phases in the system (= 2) and the term 3 accounts forhe three intensive variables, pressure P, temperature T and sur-ace tension σ. If the pressure and temperature are fixed (average
codx
ig. 9. Measurement of the activity coefficients of the binary adsorbed liquid mixtures
modifier by reference to the surface tension exerted by the adsorbent on pure
olumn pressure drop and T = 295 K), there remains one degreef freedom to define completely the equilibrium state. This lastegree of freedom is the molar composition of the bulk xl
1. Oncel1 is chosen, the equilibrium constant can be calculated.
onto the C1-silica adsorbent consistent with the convention described in Fig. 4.
98 F. Gritti, G. Guiochon / J. Chromatogr. A 1155 (2007) 85–99
F ase am
wuiosossi0p0ths
stRwTw
K
eb
rp
K
e
5
ddvfsastpgdtca
ig. 10. Measurement of the equilibrium constant K(xl1) between the bulk ph
ixtures studied and consistent with the convention described in Fig. 4.
Fig. 10 shows the variation of the equilibrium constant Kith the molar fraction of the organic modifier in the bulk liq-id phase. Obviously, the thermodynamic equilibrium constants dependent on the definition of the adsorbed phase, hencen the parameter t. It is interesting to note the average inten-ity of the equilibrium constant as a function of the naturef the organic modifier. Surprisingly, probably because of itsmall molecular size and easy access to the residual surfaceilanols, the equilibrium constant K for water-methanol mixtures the highest and varied between 2 and 4. It decreases between.75 and 1, and between 0.55 and 0.90, with ethanol and 2-ropanol, respectively. K varied between 1 and 1.3, and between.95 and 1.35 with acetonitrile and tetrahydrofuran, respec-ively. Acetonitrile–water and tetrahydrofuran–water mixturesave comparable overall adsorption behavior on the C1-silicaurface.
In chromatography, one often measures the equilibrium con-tant related to the distribution of a retained analyte betweenhe adsorbed and bulk phases when it is infinitely diluted. InP-HPLC, water is always considered as the weakest solventsith respect to organic solvents such as those used in this study.he distribution of the organic modifier infinitely diluted in pureater is given by the equilibrium constant K∞
1 :
∞1 (T ) = γa
1 xa1
γl xl= exp
(a∗
1
t
σ∗2 − σ∗
1
RT
)(48)
1 1
The thermodynamic equilibrium constants of methanol,thanol, 2-propanol, acetonitrile and 2-propanol between theulk and adsorbed phases are 3.91, 0.97, 0.68, 1.30 and 1.34,
Iti
nd the adsorbed phase onto the C1-silica adsorbent for the five binary liquid
espectively. Inversely, the equilibrium constant of water in theure organic modifier is:
∞2 (T ) = γa
2 xa2
γl2x
l2
= exp
(a∗
2
t
σ∗2 − σ∗
1
RT
)(49)
K∞2 (T ) is 2.24, 0.99, 0.86, 1.14 and 1.11 with pure methanol,
thanol, 2-propanol, acetonitrile and THF, respectively.
. Conclusion
The less arbitrary convention for the position of the Gibbsividing surface above the adsorbent surface is the one thatefines it as the position for which there is an extremum of theariation of the total number of mole of organic modifier as aunction of the distance to the surface. Assuming that the molarurface area of the mixture components adsorbed on the surfacere the same as those of the pure components simplifies con-iderably the expression of the Gibbs-Duhem relationships inhe Gibbs’ adsorbed phase. The activity coefficients of the com-onents in the bulk phase were estimated using the UNIFACroup contribution method, which is fairly accurate. Then, theerivation of the activity coefficients of the two components inhe binary adsorbed phase is straightforward and the equilibriumonstants between bulk binary solutions of various compositionsnd the adsorbed layer can be derived exactly.
This work has numerous implications in chromatography.t may be used to illustrate the heterogeneous character ofhe surface of conventional chromatographic adsorbents, whichs consistent with the simultaneous presence on their surface
hroma
oauicl
latibwitmtacpuls
opsstRm
A
t1a
RCrN
R
[[[
[
[[[
[[
[[[[[
[[[26] K. Hickey, W.E. Waghorne, J. Chem. Eng. Data 46 (2001) 851.
F. Gritti, G. Guiochon / J. C
f residual silanols (–Si–OH, � 4.0 �mol/m2) and of tetheredlkylsilane groups (–Si(CH3)3, � 4.0 �mol/m2). Based on thenique convention for the position of the Gibbs dividing surface,t can provide exact values of the thermodynamic equilibriumonstant K, which governs the equilibrium between the bulkiquid and the adsorbed liquid phase.
Obviously, this work could be extended to the study of ternaryiquid mixtures. Most commonly used mobile phases in RPLCre made of two solvents (as studied in this work). They are usedo elute mixtures. In analytical applications, this third components at infinite dilution. However, its interactions with the adsor-ent surface are much stronger than those of organic solvents,hich explains why analytes are retained. One important issue
n chromatography is to evaluate the dependence of the reten-ion of an analyte on the organic solvent concentration (xl
1). Theethod described could be extended to three-components sys-
em, with one component being infinitely diluted, and wouldllow the derivation of the exact thermodynamic distributiononstants of the analytes between the adsorbent and the liquidhase. For preparative purposes, large sample concentrations aresed and it would be interesting to evaluate the impact of the ana-yte concentration on the adsorption of binary aqueous–organicolvents.
Another extension of this work would permit the comparisonf various RPLC adsorbents, which can differ by their surfaceroperties. The nature of the alkyl chain bonded to silica, theirurface density, the presence or absence of endcapping of thetationary phases are all important column parameters that affecthe adsorption of the binary solutions used as mobile phases inPLC. These parameters and their influence on the retentionechanism of analytes could be the topic of new investigations.
cknowledgments
This work was supported in part by grant CHE-06-08659 ofhe National Science Foundation, by Grant DE-FG05-88-ER-3869 of the US Department of Energy, and by the cooperativegreement between the University of Tennessee and the Oak
[[[
togr. A 1155 (2007) 85–99 99
idge National Laboratory. We thank John O’Gara (Watersorp., Milford, MA, USA) for the synthesis of the packing mate-
ials used in this study. We thank Marianna Kele and Uwe Dietereue for their fruitful discussions about this work.
eferences
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