Thermodynamics of adsorption at the aqueous–air interface

Journal of Colloid and Interface Science 337 (2009) 39–45

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Thermodynamics of adsorption at the aqueous–air interface Risto Pajarre *, Pertti Koukkari VTT Technical Research Centre of Finland, Biologinkuja 7, Espoo, P.O. Box 1000, FIN-02044 VTT, Finland

a r t i c l e

i n f o

Article history: Received 30 March 2009 Accepted 7 May 2009 Available online 18 May 2009 Keywords: Adsorption Air–water interface Monolayer model Thermodynamic modeling Surface tension

a b s t r a c t A thermodynamic model is presented for the aqueous–air interface. For common organic solutes a relationship between the activity coefficients in the bulk solution and on the surface layer is deduced by using the concept of a surface monolayer. This relation has enabled development of an equation expressing the air–water surface adsorption coefficient as a function of bulk activity coefficients, surface tensions, molar volumes and vapour pressures of the adsorbing species. The data for 51 compounds, including alkanes, cycloalkanes, aromatics, halogenated hydrocarbons, alcohols, ethers and esters, and ketones were used in the study. Ó 2009 Elsevier Inc. All rights reserved.

species is transferred from gas to the aqueous solvent surface (while maintaining the original surface area) can be written as

1. Introduction Air–water surface adsorption is an area of considerable interest both in physical and environmental sciences as the interfacial properties greatly affect distributions and mass transfer rates of chemicals in any systems where gaseous and aqueous phases coexist, whether they are of environmental, industrial or biological nature. A key parameter often determined by experimental methods or whose prediction is aimed for by various semi-empirical models is the water surface adsorption coefficient (Ki/a) (e.g. [1–3]). Thermodynamic studies based on the idea that the surface properties can be predicted using models in which the topmost molecular layer of a liquid is modelled as a separate phase with its thermodynamic properties different but related to those of the bulk liquid have been found successful in describing adsorption behaviour in high temperature metallurgical systems [4,5]. The purpose of this study was to find a consistent relationship between the activity coefficients in the bulk and on the surface layer at the infinite dilution limit and consequently to predict values of adsorption coefficient based on other known thermodynamic data.

The water surface adsorption coefficient (Ki/a) is defined as

Ci

AAds AAds s s b Solv $ Ads þ Solv ASolv ASolv

ðIÞ

The factor AAds/ASolv is needed because in general the molar surface areas of the adsorbing substance and solvent are not equal. The equilibrium coefficient for the process can be written as

K eq

!   AAds DG0Adsorption asAds abSolv ASolv ¼   AAds ¼ exp  RT agAds asSolv ASolv ¼ exp

AAds 0;s AAds 0;b 0;s l0;g Ads þ ASolv lSolv  lAds  ASolv lSolv

RT

! ð2Þ

If one makes the monolayer assumption [34] that the surface effects are restricted to the single molecular layer closest to the surface, the standard state chemical potential at the surface becomes 0;b l0;s þ Ai r i i ¼ li

ð3Þ

where ri is the surface tension of the pure substance i. Keq can then be written as !   DG0;vap p0 AAds ðrSolv  rAds Þ Ads þ AAds ðrSolv  rAds Þ K eq ¼ exp ¼ vap exp RT RT pAds

2. Theory

K i=a ¼

g

Ads þ

ð1Þ

ð4Þ

where Ci is the adsorbed amount as mol/m2 and Ca the concentration in air as mol/m3. The chemical process where the adsorbing

The adsorbed amount (or relative surface excess [35]) in the monolayer model can be written as

Ca

xb

* Corresponding author. Fax: +358 20 722 7026. E-mail address: Risto.Pajarre@vtt.fi (R. Pajarre). 0021-9797/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jcis.2009.05.023

CAds ¼

nsAds  nsSolv xbAds

Solv

A

!

ns !0 Ads

nsAds xbAds  A ASolv

ð5Þ

40

R. Pajarre, P. Koukkari / Journal of Colloid and Interface Science 337 (2009) 39–45

Nomenclature A total surface area (m2) molar surface area of species X (m2/mol) AX molecule of the adsorbing species in phase or layer a Adsa aaX activity of species X in phase or layer a Ca concentration of the adsorbing species in air (mol/m3) DG0Adsorption standard Gibbs energy change for the adsorption reaction (J) thermodynamic equilibrium coefficient of an adsorption Keq reaction adsorption coefficient (m) Ki/a Avogadro constant (mol1) Na a amount of species X in phase or layer a(mol) nX

Vg VX V crit X

gas volume (m3) molar volume of species X (m3/mol) critical molar volume of species X (m3/mol)

V Paq X

Paquette molar volume of species X (m3/mol)

p0

standard state pressure (1 bar)

rX

pvap Ads

equilibrium vapour pressure of the adsorbing species (bar) gas constant (J mol1 K1) solvent molecule in phase or layer a temperature (K)

Superscripts b bulk liquid phase g gas phase s surface layer

R Solva T

Approaching the infinite dilution limit for the adsorbing species we have

K eq

!

xðadsÞ!0

xsAds c1;s Ads ¼ g xAds ðp=p0 Þ

nsAds A

ASolv c1;s Ads g

nAds RT V g p0

¼ K i=a

ASolv c1;s c1;s p Ads p0 þ xbAds ngAds 0 RT Ads RT

Applying the thermodynamic equilibrium relationship between molar fractions in liquid solution and in ideal gas vapour

p  xgAds vap b Ads pAds

c

ASolv c1;s c1;s Ads p0 Ads p0 þ 1;b RT cAds pvap Ads

c

¼

ASolv K i=a 1 þ vap 1;b RT p c Ads Ads

!1

1 pvap Ads

exp

  AAds ðrSolv  rAds Þ RT

ð9Þ

Eq. (9) requires knowledge of molar surface areas of the adsorbing species and the solvent. In this paper two different assumptions were experimented: (1) Direct estimation of molar surface areas based on molar volumes of the substances [36]:

ð10Þ

(2) Use of Paquette volumes from an anisotropic surface model [37] instead of molar volumes in the previous equation leading to 4=15

V crit i

 6=15 V crit i

l0;X a

infinite dilution activity coefficient of the adsorbing species in phase or layer a standard state chemical potential for species X in phase or layer a (J/mol) surface tension of the pure liquid substance X (N/m)

adsorbing molecules can be considered spherical in estimating the molar surface areas. Also, it is assumed that the molar surface areas are the same at infinite dilution as they would be for the pure organic liquid.

3. Correlation between surface and bulk activity coefficients

4. Interpretation and refining of the linear correlation

2.1. Molar surface areas

Ai ¼ 1:091  Na1=3 V i

a c1; Ads

ð8Þ

If the surface infinite dilution activity coefficients would be known, Eq. (9) could also be solved for Ki/a.

Ai ¼ 1:091  Na1=3 V 2=3 i

Ci

ð7Þ

From Eqs. (4) and (8) we finally obtain 1;s Ads

reduced coordination number adsorbed amount at the air–water interface (mol/m2)

X

The thermodynamic data (at 25 °C) used is presented in Table 1. All the experimental adsorption coefficient values are from Ref. [1] and infinite dilution activity coefficient values from Ref. [38]. The surface activity coefficients calculated from Eq. (9) with the two surface area assumptions are presented in Figs. 1 and 2.

we get

K eq ¼ K i=a

mole fraction of species X in phase or layer a

b

Vg

ð6Þ

xbAds ¼

xa

ð11Þ

where is the critical molar volume of substance i. It should be noted that both of Eqs. (10) and (11) effectively assume that the

Surface activity coefficient models used in metallurgy and materials science often assume that the activity coefficients have the same compositional dependency as those in the bulk but are scaled by some factor less than unity [5,34,41]. This is intuitively reasonable based on the reduced co-ordination number, and therefore reduced number of interactions, for the atoms and molecules on the surface layer. In classical treatises of surface thermodynamics [42] the reduced co-ordination number is typically assumed to have a value of 0.75, corresponding to the case of closely packed spheres, where six of the nearest neighbours of each sphere are in the same layer and three on both the layer above and below it. In more practically oriented studies of surface thermodynamics of mixtures [41,43,44] the effective co-ordination number has been considered an empirical parameter (that can include the effect of surface relaxation and other phenomena) with values typically varying from about 0.75 to values slightly above unity. The activity coefficient trends deduced above, especially the one using directly bulk molar volumes, do not seem to go through the origin, which would be desirable in order to reach an easily applicable and thermodynamically consistent surface activity coefficient model from a known bulk activity. It was decided to test how the scaling behaviour would change if it was assumed that the activity coefficients would include a simple Flory–Huggins like molar volume dependent combinatorial contribution that would not be scaled with the reduced co-ordination number, leading to Eq. (12)

41

R. Pajarre, P. Koukkari / Journal of Colloid and Interface Science 337 (2009) 39–45 Table 1 Thermodynamic data used in the modelling.

n-Pentane n-Hexane n-Heptane n-Octane n-Nonane n-Decane 2,2,4-Trimethylpentane 1-Nonene Cyclohexane Cyclooctane Benzene Toluene p-Xylene Ethylbenzene Isoprpylbenzene Styrene Biphenyl Naphtalene Nitrobenzene Thiophene Ethanol 1-Propanol 2-Propanol 2-Methyl-1-propanol Acetone 3-Methylbutan-2-one 2-Butanone Pentanal Cyclopentanone Diethyl ether Diisopropyl ether Di-n-propyl ether Methylphenyl ether 1,4-Dioxane Methyl formate Ethyl formate Methyl acetate Ethyl acetate Isobutyl acetate Dichloromethane Chloroform 1-Chlorobutane 1-Bromobutane 1,1,2,2-Tetarachloroethane 1,1,1-Trichloroethane 1,2-dichloroethane Chlorobenzene Iodobenzene Bromobenzene 1,3-Dichlorobenzene 1,4-Dichlorobenzene Water

log(Kia) (experimental)

;b lnðCaAds Þ

Vi (m3/mol)

Reference

V circt ðm3 =molÞ i

Reference

ri (N/m)

Reference

P vap ðbarÞ Ads

Reference

Notes

7.22 6.9 6.6 6.23 5.91 5.58 6.47 5.72 6.97 6.25 6.32 5.93 5.57 5.58 5.39 5.56 4.1 4.69 3.99 6.43 4.15 3.83 3.85 3.61 4.78 4.42 4.6 4.68 3.77 5.3 4.83 4.81 4.89 3.88 5.55 6.08 5.06 4.59 4.16 6.63 6.4 6.35 6.07 5.22 6.54 6.08 5.9 4.71 5.73 5.57 5.57 1.807E05

11.487 12.941 14.565 16.051 17.632 19.198 14.820 15.654 11.306 13.579 7.832 9.165 10.497 10.471 11.629 9.798 13.042 11.101 8.169 7.346 1.329 2.613 2.063 3.886 2.000 4.431 3.275 5.395 3.374 4.496 6.469 7.275 8.326 1.693 2.741 3.857 3.118 4.168 6.739 5.448 6.678 8.939 9.435 8.095 8.656 6.450 9.499 10.893 9.984 11.100 11.533 [22]

1.175E04 1.316E04 1.475E04 1.635E04 1.797E04 1.959E04 1.661E04 1.741E04 1.080E04 1.348E04 8.940E05 1.069E04 1.238E04 1.230E04 1.401E04 1.156E04 1.496E04 1.185E04 1.027E04 7.950E05 5.869E05 7.515E05 7.690E05 9.295E05 7.400E05 1.069E04 9.016E05 1.052E04 8.912E05 1.047E04 1.409E04 1.373E04 1.092E04 8.571E05 6.165E05 8.026E05 8.017E05 9.779E05 1.334E04 6.358E05 8.069E05 1.044E04 1.073E04 1.058E04 1.004E04 7.943E05 1.017E04 1.118E04 1.050E04 1.141E04 1.008E04 5.600E05

[6] [6] [6] [6] [6] [7] [8] [6] [6] [6] [6] [6] [6] [6] [6] [6] [9] [10] [11] [12] [6] [6] [6] [6] [6] [6] [6] [13] [6] [13] [13] [13] [13] [14] [13] [13] [13] [13] [13] [13] [6] [13] [13] [6] [6] [6] [13] [13] [13] [13] [13] [15]

3.110E04 3.680E04 4.280E04 4.920E04 5.550E04 6.240E04 4.680E04 5.260E04 3.080E04 4.100E04 2.560E04 3.160E04 3.780E04 3.740E04 4.270E04 3.520E04 5.020E04 4.070E04 3.490E04 2.190E04 1.680E04 2.180E04 2.220E04 2.740E04 2.090E04 3.080E04 2.670E04 3.130E04 2.580E04 2.800E04 3.860E04 3.820E04 3.410E04 2.380E04 1.720E04 2.290E04 2.280E04 2.860E04 4.010E04 1.770E04 2.390E04 3.000E04 3.190E04 3.250E04 2.810E04 2.200E04 3.080E04 3.510E04 3.240E04 3.510E04 3.510E04 0.0720

[15] [15] [15] [15] [16] [16] [15] [16] [15] [17] [15] [15] [15] [16] [18] [18] [18] [15] [18] [19] [15] [16] [16] [16] [15] [16] [15] [16] [18] [15] [16] [18] [20] [15] [15] [15] [15] [15] [16] [18] [15] [18] [18] [18] [18] [18] [15] [15] [15] [18] [18] [39]

0.0155 0.0179 0.0197 0.0211 0.0224 0.0234 0.0202 0.0226 0.0247 0.0293 0.0282 0.0279 0.0280 0.0287 0.0277 0.0315 0.0392 0.0401 0.0434 0.0321 0.0220 0.0233 0.0209 0.0225 0.0240 0.0246 0.0240 0.0254 0.0328 0.0167 0.0173 0.0200 0.0351 0.0328 0.0244 0.0232 0.0247 0.0234 0.0231 0.0272 0.0267 0.0216 0.0259 0.0356 0.0251 0.0319 0.0330 0.0387 0.0352 0.0354 0.0325

[21] [21] [21] [21] [21] [21] [21] [21] [21] [21] [21] [21] [21] [21] [21] [22] [21] [21] [21] [12] [21] [21] [21] [21] [21] [23] [21] [21] [21] [21] [21] [21] [21] [21] [21] [21] [21] [21] [21] [22] [21] [21] [21] [21] [21] [22] [21] [21] [21] [21] [22]

6.84E01 2.02E01 6.10E02 1.87E02 5.81E03 1.82E03 6.57E02 7.24E03 1.30E01 7.52E03 1.27E01 3.84E02 1.18E02 1.27E02 6.17E03 8.64E03 1.30E05 1.12E04 3.77E04 1.08E01 7.87E02 2.81E02 5.77E02 1.54E02 3.08E01 6.72E02 1.21E01 4.87E02 1.51E02 7.16E01 2.01E01 8.37E02 4.56E03 4.98E02 8.34E01 3.25E01 2.88E01 1.26E01 2.34E02 5.80E01 2.63E01 1.36E01 5.26E02 5.58E03 1.66E01 1.05E01 1.60E02 1.32E03 5.52E03 2.65E03 2.41E03

[24] [24] [24] [24] [24] [24] [25] [6] [6] [6] [6] [6] [6] [6] [6] [6] [26] [27] [28] [29] [6] [6] [6] [6] [30] [6] [6] [31] [6] [25] [25] [25] [22] [22] [22] [22] [22] [22] [32] [6] [6] [22] [22] [6] [6] [6] [22] [22] [22] [22] [22]

a a a a a a a a a a a a a a a a a, b a, b a, d a

e

a e

c

a a a e e a, e a, e

a

Activity coefficient based on solubility measurements. Solid at 25 °C. Liquid surface tension and molar volumes extrapolated to 25 °C. Tabulated vapour pressure values (over solid) have been adjusted for calculations to correspond to those over supercooled liquid based on melting point temperature and heat of fusion [33,27]. c Critical volume predicted based on the Marrero–Gani group contribution method [40]. d Surface tension value extrapolated to 25 °C. e Vapour pressure extrapolated to 25 °C. b

ln c1;s Ads  ln ðV Ads =V Solv Þ  ð1  V Ads =V Solv Þ h i  ln ðV Ads =V Solv Þ  ð1  V Ads =V Solv Þ ¼ b ln cbulk;1 Ads

ð12Þ

where b is the reduced coordination number on the surface layer. Activity coefficient data have been plotted in the form of Eq. (12) for the two alternative surface area cases in Figs. 3 and 4. In the first case, the bulk molar volumes were used in Eq. (12), while in the second case bulk volumes were applied for the bulk and Paquette volumes for the surface layer. In the second case Eq. (12) becomes

    Paq Paq Paq Paq ln c1;s Ads  ln V Ads =V Solv  1  V Ads =V Solv h i ¼ b ln cbulk;1  ln ðV Ads =V Solv Þ  ð1  V Ads =V Solv Þ Ads

ð12bÞ

In Figs. 3 and 4 the shown regression line is the best least squares fit that goes through the origin. With both molar volume assumptions the inclusion of the Flory–Huggins adjustment to the activity coefficients resulted in a better fit with one adjustable parameter regression line than was originally obtained using two adjustable parameters. Overall, the model using Paquette volumes gives a marginally better fit, with the difference being most noticeable with the

R. Pajarre, P. Koukkari / Journal of Colloid and Interface Science 337 (2009) 39–45

ln(γ ∞Ads (surface)) according to the model

42

13 11 y = 0.6333x - 0.3393 2

9

R = 0.9599

7 5 3 1 -1

0

2

4

6

8

10

12

14

16

18

20

ln( γ ∞Ads(bulk)) experimental Fig. 1. Activity coefficients for surface areas derived directly from molar surface areas using Eq. (10).

ln(γ ∞Ads (surface)) according to the model

16 14 y = 0.7218x + 0.544

12

2

R = 0.9772

10 8 6 4 2 0 0

2

4

6

8

10

ln(γ ∞Ads (bulk))

12

14

16

18

20

experimental

Fig. 2. Activity coefficients for surface areas derived using Paquette volumes (Eq. (11)).

ln(γ ∞Ads (surf)) - ln(VAds /VSolv) - (1-VAds /VSolv)

20

y = 0.709x

18

2

R = 0.9865

16 14 12 10 8 6 4 2 0 0

5

10

15

20

25

ln(γ ∞Ads)(bulk)) - ln(VAds /VSolv) - (1-VAds /VSolv) Fig. 3. Activity trends plotted using molar volume -based surface areas (Eq. (10)).

30

43

R. Pajarre, P. Koukkari / Journal of Colloid and Interface Science 337 (2009) 39–45

ln(γ ∞Ads (surface)) - ln(VAds /VSolv) - (1-VAds /VSolv )

25

y = 0.8283x 2 R = 0.9872

20

15

10

5

0 0

5

10

15

20

25

30

ln(γ ∞ Ads (bulk)) - ln(VAds /VSolv) - (1-VAds /VSolv) Fig. 4. Activity trends plotted using surface areas based on Paquette volumes (Eqs. (11) and (12b)).

compounds having relatively small activity coefficient values. It should be noted that while the linear correlation fit is almost equally good for both surface layer molar volume assumptions, there is a significant difference in the resulting value for the reduced coordination number.

The correlation was applied together with the bulk activity coefficient, surface tension, vapour pressure, and molar volume data to Eq. (14)

K i=a ¼

5. Estimation of water surface adsorption coefficient based on other thermodynamic data

þ ln

V Paq Ads V Paq Solv

  ! V Ads V Ads  1 V Solv V Solv ! Paq V Ads



! þ 1

ð13Þ

V Paq Solv

vap ASolv c1;s Ads pAds

exp

! AAds ðrSolv  rAds Þ RT  1;b RT ASolv pvap Ads cAds

-3 -3.5 -4

Model Log( Ki/a )

-4.5 -5 -5.5 '

-6 -6.5 -7 -7.5 -8 -8

-7.5

-7

-6.5

-6

ð14Þ

As the comparison is made within the same data set which was used to derive the activity coefficient relation in the first place, this is not a truly predictive test. However, the activity coefficient correlation has only one adjustable parameter, and based on Fig. 4, a rather small subset of the used data would have produced closely the same value for that parameter. Comparison of the correlated and experimental adsorption coefficient values are given in Fig. 5 and Table 2. The two outliers with more than 0.4 log units difference between the correlated and actual log K values are ethyl formate

The final single adjustable parameter correlation between activity coefficients in the bulk and surface layers to be applied together with Paquette volumes with the surface layer is 1;b ln c1;s Ads ¼ 0:8283 ln cAds  ln

RT

-5.5

-5

-4.5

-4

-3.5

-3

Experimental Log(K i/a ) Fig. 5. Comparison of experimental adsorption coefficients and those calculated from Eq. (14) applying the correlated surface activity coefficient values.

R. Pajarre, P. Koukkari / Journal of Colloid and Interface Science 337 (2009) 39–45

44

Table 2 Comparison of experimental and estimated adsorption coefficients.

n-Pentane n-Hexane n-Heptane n-Octane n-Nonane n-Decane 2,2,4-Trimethylpentane 1-Nonene Cyclohexane Cyclooctane Benzene Toluene p-Xylene Ethylbenzene Isoprpylbenzene Styrene Biphenyl Naphtalene Nitrobenzene Thiophene Ethanol 1-Propanol 2-Propanol 2-Methyl-1-propanol Acetone 3-Methylbutan-2-one

log(Ki/a) (experimental)

log(Ki/a) (estimated)

7.22 6.9 6.6 6.23 5.91 5.58 6.47 5.72 6.97 6.25 6.32 5.93 5.57 5.58 5.39 5.56 4.1 4.69 3.99 6.43 4.15 3.83 3.85 3.61 4.78 4.42

7.29 7.00 6.76 6.46 6.22 5.94 6.68 5.81 7.02 6.24 6.28 5.84 5.43 5.52 5.24 5.40 3.87 4.70 4.14 6.45 4.09 3.78 3.76 3.53 4.69 4.25

2-Butanone Pentanal Cyclopentanone Diethyl ether Diisopropyl ether Di-n-propyl ether Methylphenyl ether 1,4-Dioxane Methyl formate Ethyl formate Methyl acetate Ethyl acetate Isobutyl acetate Dichloromethane Chloroform 1-Chlorobutane 1-Bromobutane 1,1,2,2-Tetarachloroethane 1,1,1-Trichloroethane 1,2-Dichloroethane Chlorobenzene Iodobenzene Bromobenzene 1,3-Dichlorobenzene 1,4-Dichlorobenzene

and iodobenzene, which were outliers also in all activity coefficient correlations. The second term in Eq. (14) is for all the substances discussed here small enough to be ignored. This enables presenting the adsorption coefficient in the form

log K i=a ¼ log

RT vap ASolv 1;s Ads pAds

c

þ

AAds ðrSolv  rAds Þ logðeÞ RT

! ð15Þ

which readily enables estimation of the effect of changes in the various experimental parameters of the model on the predicted adsorption coefficient. A change of 0.1 log units is caused by an approximately 20% relative change in vapour pressure or in the (surface) activity coefficient value, or an absolute change of 570 J/ mol in the value of AAdsrAds. 6. Conclusions In this paper a thermodynamic model for adsorption on the aqueous–air interface was developed leading to an expression for the adsorption coefficient as a function of surface area, surface tension, vapour pressure, and infinite dilution activity coefficient on the surface in the monolayer surface approximation. The experimental data set used indicates that the infinite dilution surface activity coefficients can be estimated based on the corresponding bulk values assuming that the energetic interactions at the surface are scaled by the amount given by a reduced surface coordination number, same for all organic solvents, while an entropic Flory– Huggins like contribution would be assumed not to scale with the coordination number. The presented model describes successfully the adsorption behaviour of a wide range of organic compounds while using only one adjustable universal parameter (the reduced coordination number). The thermodynamic model allows more extensive usage of bulk liquid thermodynamic values for the estimation of adsorption coefficients than has been possible so far. It also offers further confirmation to the idea that the monolayer approach is a fruitful method for investigating properties of liquid–gas interfaces.

log(Ki/a) (experimental)

log(Ki/a) (estimated)

4.6 4.68 3.77 5.3 4.83 4.81 4.89 3.88 5.55 6.08 5.06 4.59 4.16 6.63 6.4 6.35 6.07 5.22 6.54 6.08 5.9 4.71 5.73 5.57 5.57

4.33 4.47 3.96 5.08 4.51 4.62 4.86 4.00 5.68 5.20 4.96 4.51 3.92 6.57 6.21 6.07 5.94 4.99 6.38 6.10 5.90 5.41 5.64 5.60 5.50

Acknowledgments We thank the Finnish Funding Agency for Technology and Innovation (TEKES) for its financial support.

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