Study on water evaporation through 1-alkanol monolayers by the thermogravimetry method

time using1-alkanolss under

tion amongcalculatedemperature.cludes thewhen theeous phasehe activation

Journal of Colloid and Interface Science 272 (2004) 472–479www.elsevier.com/locate/jcis

Study on water evaporation through 1-alkanol monolayers by thethermogravimetry method

Muhammad Rusdi and Yoshikiyo Moroi∗

Chemistry and Physics of Condensed Matter, Graduate School of Sciences, Kyushu University-Ropponmatsu, Ropponmatsu 4-2-1, Chuo-ku,Fukuoka 810-8560, Japan

Received 17 June 2003; accepted 6 January 2004

Abstract

The influence of 1-alkanol monolayers on the rate of water evaporation has been studied by measuring water loss per unitthermogravimetry. The evaporation rate of water from the surface covered by an insoluble monolayer for each of four saturated(C13OH, C15OH, C17OH, and C19OH) was measured as a function of temperature and alkyl chain length, where the monolayer waequilibrium spreading pressure. The evaporation rate decreased with increasing alkyl chain length or increasing molecular interac1-alkanol molecules in the insoluble monolayer. Using the Arrhenius equation, the activation energy for the water evaporation wasfrom the temperature dependence of the evaporation rate, which showed that the activation energy decreased with increasing tOn the other hand, the activation energy increased with increasing alkyl chain length, which indicates that the activation energy inenergy to cross the insoluble monolayer at the air/water interface. This energy increased almost linearly with alkyl chain length,length is longer than a dodecyl group. This means that water molecules need more energy to escape from the liquid to the gasacross a membrane of longer 1-alkanols, which becomes more evident at lower temperatures. The temperature dependence of tenergy was slightly larger for longer 1-alkanols than for shorter ones. 2004 Elsevier Inc. All rights reserved.

Keywords: Water evaporation; 1-Alkanols; Thermogravimetry; Activation energy of evaporation; Transport across insoluble monolayers

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1. Introduction

The first recorded scientific study on the evaporationof water from a surface covered by an insoluble molayer appeared nearly 80 years ago in an experimenHedestrand [1] for a spreading monolayer on a substand he found no measurable effect. The Hedestrand rwas criticized by Adam [2] and Rideal [3] from the viewpoint that the layer of stagnant air over the water surfhad a much greater effect on the evaporation rate thaninsoluble film. In a subsequent experiment, therefore, Rieliminated this stagnant layer by reducing the pressure athe surface. This was the first experiment for the evaration rate measurement considering the stagnant gaslayer. Sebba and Briscoe [4] developed a device to ctrol the gas flow rate for measuring the water evaporarate across an insoluble film, but they did not analyze

* Corresponding author.E-mail address: [email protected] (Y. Moroi).

0021-9797/$ – see front matter 2004 Elsevier Inc. All rights reserved.doi:10.1016/j.jcis.2004.01.014

,t

s

gas flow in the vapor phase and hence could not calcuthe evaporation rate values. Navon and Fenn [5,6] analthe operation for vaporization and provided an equationcalculating the evaporation resistance. A similar technwas employed by Walker [7]. At the same time, Archer aLa Mer further developed a model which had been pposed by Langmuir and Schaefer, and the model was fumodified by Barnes. The Langmuir and Schaefer modeldeeply reviewed in [8]. This model was used to measureevaporation rate of water by suspending, over the surfacwater in a film balance trough, a flat container with a perable bottom supporting a solid desiccant (calcium chloriAfter the modified method of Langmuir and Schaefer wproposed, almost no study on water evaporation acrosinsoluble monolayer has been reported in relation to theduction of vapor pressure above the monolayer, althothere were a few papers on the evaporation across asorbed film [9,10]. In addition, the reproducibility of thevaporation rate was quite low in many published papon the evaporation rate; the experimental points showlot of scatter. Therefore, a more reliable method would

http://www.elsevier.com/locate/jcis

M. Rusdi, Y. Moroi / Journal of Colloid and Interface Science 272 (2004) 472–479 473

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indispensable for a more quantitative analysis and for ather understanding of the evaporation rate of water. Prevstudies above have used an indirect method for the ratesurement in the sense that they used the weight changedesiccant above the surface. The method proposed herectly measures the evaporation rate by tracing the weloss of water with time. This paper aims to measureanalyze more precisely the evaporation rate of water fthe surface covered by an insoluble 1-alkanol monolayeevaluate the effects of the alkanol chain length and of tperature on the activation energy of the evaporation,finally to determine the critical alkyl chain length of thalkanol which can significantly reduce the water evaporarate.

2. Experimental

2.1. Material

1-Tridecanol (C13OH) and 1-pentadecanol (C15OH) wereobtained from Kishida Kagaku. 1-Heptadecanol (C17OH)and 1-nonadecanol (C19OH) were purchased from TokyKasei Kogyo. The purification of those materials was mas follow; C13OH was recrystallized several times from tmolten material (without solvent), C15OH and C19OH werepurified by repeated recrystallizations from ethanol sotions, while C17OH, a kind gift from Dr. Shibata of KyushUniversity, was used as received, and its purity was mthan 99%. The purities of these 1-alkanols were checby elemental analysis; the observed and calculated vawere in satisfactory agreement within experimental e(±0.2%).

2.2. Method

The apparatus was based upon a conventional thegravimetric analysis (Fig. 1, Rigaku Thermo Plus 2), whthe sample pan had a large area of 0.739 cm2 to make theedge effect as small as possible. The apparatus can dthe changes of both weight and temperature with timemultaneously, where the temperature was controlled wi±0.1 ◦C throughout a run except for an initial fluctuatioA characteristic of the present experimental setup is todry air flowing through a furnace tube at a constant teperature. The flow of dry air is able to reduce the stagngaseous layer formed just above surface to some exMoisture in the flowing air from a small air pump was copletely removed by passing it first through concentratedfuric acid and second by keeping it over dried silica gThen, the dry air was passed two times through a filtepore size 0.22 µm (Millipore, SLGV025LS) in order to rmove dust. The flow rate was controlled by a flow mewith a needle bulb [11]. The experimental reproducibilwas such that the traces of the repeated weight decreasetime could be superimposed for the same sample.

-ai-

-

t

.

h

Fig. 1. Schematic illustration of the modified apparatus for thermogravtry.

A 150 µl portion of 1-alkanol-saturated aqueous solutwas pipetted into the pan, and then, a tiny solid particle ofalkanol was placed on the surface of the solution. This methat the 1-alkanol-saturated solution was used as a subin order to avoid further dissolution of the 1-alkanols inthe aqueous phase. The pan was set in a tube for grational measurement, where the height from the liquid levethe mouth of the pan was 4.80 mm. A long-chain 1-alka(C19OH) needed additional time to spread on the surfacsolution than the others. From dependence of the evaporrate on spreading time, at least 3 days standing was fonecessary for the alkanol to reach the lowest constant eoration rate or to reach the equilibrium spreading presin 100% relative humidity. Just 1 day standing for spreadwas enough for the rest of the 1-alkanols. This standingto allow the spreading monolayer to be in equilibrium wthe solid prior to starting the run. The run was started wout allowing thermal equilibrium with furnace temperatudue to the small thermal mass of the sample.

Equilibrium spreading pressure (ESP) was measurethe Wilhelmy plate method under atmospheric pressur298.2 K. The experimental error was±0.1 mN m−1.

3. Results and discussion

The evaporation rates of purified water over the tempture range 298.2–333.2 K were measured in order to obthe activation energy of the water evaporation [12]. Thdata were used as a reference scale to clarify the effean insoluble monolayer on the evaporation rate. The evration rate in units of mol s−1 cm−2 was calculated from thslope of weight with time after the initial drift. The activation energy of water evaporation from the surface cove

474 M. Rusdi, Y. Moroi / Journal of Colloid and Interface Science 272 (2004) 472–479

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d byaturece

t

. 2.

asovered

ora-ght

e sur-

sultssingin-ntalnolsr of

ratemallin-lost

oss

wa-rrtestas

isubil-ur-

Fig. 2. Weight decrease of water from the surface covered by C15OH withtime at different temperatures from 298.2 to 333.2 K.

by an insoluble monolayer can be calculated from the tperature dependence of the evaporation rate employinArrhenius equation (Appendix A),

(1)lnk =A− Ea

RT,

wherek is the evaporation rate per unit area (mol s−1 cm−2),Ea is the activation energy (J mol−1), T is absolute temperature,R is the gas constant, andA is an arbitrary constanThus,Ea can be obtained by analyzing the evaporation raccording to Eq. (1), where theEa value includes the contribution of the monolayer. This definition applies regardlof whether the Arrhenius plot is linear or not; if it is not, tactivation energy changes with temperature.

The evaporation rate of water from the surface coverea 1-alkanol monolayer was measured over the temperrange 298.2–333.2 K by flowing dry air through the furnatube at a flow rate of 46.4 ml min−1. The rates of weighdecrease of water from the surface covered by the C15OHmonolayer at different temperatures are illustrated in FigThe rate determinations for C13OH, C17OH, and C19OHwere similarly made. A quite linear decrease in weight wobserved against time up to 80% of the total weight, abwhich the bottom of the sample pan is incompletely coveby liquid and the platinum plate is exposed. The evaption rates were calculated from an initial slope of the wei

Fig. 3. Temperature dependence of evaporation rate of water from thface covered by 1-alkanols.

decrease and summarized in Table 1 and Fig. 3. The reindicate that the evaporation rates increase with increatemperatures for all 1-alkanols, while decreasing withcreasing alkyl chains of 1-alkanols over the experimetemperature range examined. The effect of the 1-alkaon the evaporation rate reduction increased in the ordeC13OH< C15OH< C17OH< C19OH. A very small reduc-tion effect on the evaporation rate was observed for C13OHat temperatures above 303.2 K. 1-Alkanols may evapofrom the monolayers, but in the present experiment the ssolid particles coexist with the monolayer at the air/waterterface. Therefore, they can easily supply the 1-alkanolsby evaporation, which can be verified by a quite linear lof weight.

The small difference in the evaporation rate betweenter covered by a C13OH monolayer and just bulk wateat temperatures above 303.2 K indicates that the sho1-alkanol which can certainly reduce water evaporationa spreading monolayer is C13OH. In other words, 13 isthe critical carbon number in alkyl chain length whichable to retard the water evaporation. The aqueous solity of C13OH above 303.2 K is quite low, because the sface tension of a C13OH-saturated solution (70.65 mN m−1)

Table 1Evaporation rate of water(k) from the surface covered by a 1-alkanol monolayer and the relative decrease in evaporation rate(1− φ)

T (K) k (×10−7mol s−1 cm−2) 1− φ

H2O C13OH C15OH C17OH C19OH C13OH C15OH C17OH C19OH

298.2 3.26 2.91 2.81 2.39 2.00 0.107 0.139 0.265 0.385303.2 4.25 4.02 3.88 3.38 2.78 0.0546 0.0856 0.203 0.345308.2 5.70 5.56 5.31 4.76 4.01 0.0242 0.0688 0.165 0.297313.2 7.71 7.53 7.27 6.68 5.68 0.0228 0.0563 0.133 0.263318.2 10.1 10.0 9.62 9.07 7.77 0.0151 0.0519 0.106 0.235323.2 13.2 12.9 12.5 11.9 10.4 0.0210 0.0476 0.0981 0.210328.2 17.0 16.8 16.3 15.7 13.9 0.0124 0.0418 0.0809 0.184333.2 21.8 21.3 20.9 20.2 18.0 0.0222 0.0402 0.0747 0.171


ered

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thetem-, theera-fromtionacti-hichogeniththe

d bythere-bic

Fig. 4. Weight decrease of water from purified water and the surface covby C13OH with time at temperature 333.2 K.

was almost the same as that of purified water at 303.271.20 mN m−1. This was also confirmed by a good lineaity for the weight loss vs time. The linearity means thatC13OH monolayer stills exist on the surface for more th1 h at 333.2 K (Fig. 4).

The water molecules require three steps to move fan inner water phase to a gaseous phase: (1) the moveup to just beneath an insoluble monolayer by self-diffusthrough the water phase, (2) the transport across an inuble monolayer, and (3) the movement from the top ofinsoluble monolayer into the gaseous phase. In this ctherefore, one additional process (step 2) is required forter molecules to evaporate into the gaseous phase (FigIn other words, the activation energy for the water evaption should be larger than that for just the liquid/air interfa

t

-

,

.

Fig. 6. Plots of lnk vsT−1 across a 1-alkanol monolayer.

For evaluation of the activation energy, the logarithm ofevaporation rates was plotted against the inverse of theperature (Fig. 6). The plots were not linear, and thereforeplots were analyzed by a quadratic curve fitting. Tempture dependence of the activation energy was calculatedthe slope at a given temperature. The values of activaenergy thus obtained are summarized in Table 2. Thevation energy decreases with increasing temperature, wis quite reasonable from the expected decrease in hydrbonds between water molecules for their evaporation wincreasing temperature. What is more important is thatactivation energy to cross an air/water interface coverean insoluble monolayer is larger than that just to crossuncovered interface, which means that extra energy isquired for water molecules to pass through a hydrophomolecular layer.

rt for watence of

Fig. 5. Schematic concentration profile of water molecules from bulk solution to the gaseous phase and schematic representation of transpormolecules from bulk solution to the gas phase, whererl , ri , rm, and rg are resistance of bulk water phase, resistance of interfacial region, resistamonolayer, and resistance of the gaseous phase. L, liquid phase; M, monolayer; G, gaseous phase.


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Table 2Activation energy (kJ mol−1) of water evaporation from just liquid wateand from the surface covered by a 1-alkanol monolayer

T (K) H2O C13OH C15OH C17OH C19OH

298.2 48.0 48.3 50.5 53.4 55.8303.2 47.4 47.7 49.8 52.5 54.9308.2 46.8 47.0 49.1 51.7 54.0313.2 46.2 46.5 48.4 50.9 53.1318.2 45.6 45.9 47.8 50.1 52.3323.2 45.1 45.3 47.1 49.3 51.5328.2 44.6 44.8 46.5 48.6 50.7333.2 44.1 44.3 45.9 47.9 50.0

The maximum evaporation rate(k) of water per unit areacan be estimated from the Hertz–Knudsen equation [13

(2)k =(

RT

2πM

)1/2

Ceq,

whereR is the gas constant,T is absolute temperature,M isa molecular mass, andCeq is concentration for equilibriumvapor pressure or saturation pressure. This equation prethe maximum evaporation rate at which molecules cancape from the liquid interface into a perfect vacuum (evaration coefficient (α) is unity). That is, this equation cannbe applied to estimate the evaporation rate of a real sysThe activation energy(Ea) of water evaporation from thsurface covered by 1-alkanol was found to be larger thanfrom just liquid water. The authors have derived an eqtion for the activation energy from kinetic theory of diffusio(Appendix B),

(3)Ea =�Hvap+ RT

2−R

{d lnβ(T )

d(1/T )

},

whereEa and�Hvap are the activation energy and the ethalpy change of evaporation, respectively, andβ is a para-meter depending on temperature. Based on Eq. (3), ifβ isassumed constant, theEa value is always greater than th�Hvap value byRT/2, 1.23 kJ mol−1 at 298.2 K. This wasverified by a separate study [12].

The rate of water evaporation is governed by the drivforce for the evaporation and by the total permeation retance through the transport pathway (Fig. 5), whose equais analogous to Ohm’s law for electrical conduction [15],

(4)J = �Cg

rg= �Ct∑

i ri,

whereJ is an evaporation flux at a stationary state,�Cg(= Ceq − Cv) is the difference between the water vapconcentrations driving the evaporation (Ceq is the concentration at a stationary evaporation rate just above the suof water bulk or above the monolayer andCv is the con-centration in the atmosphere at some distance abovesurface), and

∑ri is the total permeation or evaporati

resistance, the sum of all the resistances that arise fromtions of the pathway including the bulk water.�Ct is thetotal difference between water concentration in a bulk w(C0 = 55.39 mol dm−3 at 298.2 K) andCv (Fig. 5).

s

.

t

e

-

The difference in the vapor pressure between just abthe water surface(peq

s ) at a stationary flux and far above tsurface(p) becomes the driving force for the evaporatprocess. The present evaporation rates are rather slowtherefore,(P eq

s ) can be set equal to saturated equilibrivapor pressure [12]. When the surface covered by 1-alkasignificantly retards the water evaporation rate, this resula smaller difference betweenpeq

s andp at a stationary stateThe total resistance, which impedes the evaporation

of water, results from the bulk solution phase(rl), the in-terfacial region(ri), the monolayer(rm), and the gaseouphase(rg). A generalized and fairly typical arrangementshown in Fig. 5, where the concentration profile of wamolecules is also illustrated. The penetrating water mcules originate in a bulk solution, pass through the interfaregion and an insoluble monolayer, and then pass throthe gaseous phase as the final step. The important regiothe flux is the one where the water concentration startchange; that is, the region above the ordinate axis in FiThe insoluble 1-alkanol molecules on the surface becomsignificant resistance for water molecules to evaporate.magnitude of resistance of the gaseous phase dependsexperimental setup. In the present experiment, thereforemaximum flow rate of dry air (46.4 ml min−1) was used inorder to minimize the effect of drift of the flow rate on tevaporation rate. The effect of the flow rate on the evaption rate is very significant at a lower flow rate, while becoing smaller at a higher flow rate [12]. The evaporation rbecomes almost constant irrespective of the flow rate aabove flow rate. The vapor of liquid just above the surfforms a stagnant gaseous layer, which increases therg value.The thickness of the stagnant layer was almost constadifferent temperatures under the present experimentalditions, which was experimentally confirmed [12]. Usithe above flow rate, the stagnant gaseous layer could bchanged as much as possible for the present cell. This mminimization ofrg. For evaporation of just pure water, ttotal resistance is denoted by

∑rw, while for the evapora

tion from water surface covered by an insoluble monolaof resistancerm the total resistance becomes

∑rf (f refers

to a film):

(5)∑

rf =∑

rw + rm.

The performance for a monolayer is often reported asratio,φ, of the evaporation rate with an insoluble monolato the rate for just liquid water(φ = Jf/Jw) or as the relativedecrease in evaporation rate (effectiveness of evaporatioduction) defined by 1− φ, where the fluxesJw andJf referto an evaporation rate of just liquid water and that fromsurface covered by insoluble monolayer, respectively.

Provided that the experimental conditions for evaporaremain unchanged even when the 1-alkanol is spreadspecifically that the total driving force,�Ct, and the otheresistances,

∑rw, are unaltered, this results in the followin


ered.

yers

va-r all. Inhasetivaers.inles

her byall

omad ation

Thisthisrom2–boninly

ulkm-or

wasterted

lkyl

thelkyl

nesater-ratenter-hen

pore theouster-uitelesaused theole-ithi-

tarts,ffer-tu-ono-te ofcen-uldse-Thisond-rgehasen-selys inpassriorbe-as isase,les

lso

olssentree-atesderar-

heyand

t them-

ole-

Fig. 7. The activation energy of water evaporation from the surface covby 1-alkanol vs carbon number of 1-alkanols at different temperatures

expression [16]:

(6)1− φ = rm∑rw + rm

.

The values of relative decrease for the insoluble monolaare summarized in Table 1 for the present study.

From the values in Table 2 and Fig. 7, the actition energy decreased with increasing temperature fo1-alkanols, while it increased with increasing alkyl chainorder to escape from the liquid phase to the gaseous pwater molecules should overcome some barriers. The action energy becomes larger for longer 1-alkanol monolayIn addition, this energy is linearly related to alkyl chalength (Fig. 7), which clearly indicates that water molecumust cross an insoluble monolayer of 1-alkanols.

It is interesting to compare the interpretation of tpresent experimental data with that in a previous papeBarnes [8]. In Ref. [8], the author estimated that almostthe activation energy for water evaporation originated fra resistance generated by the long-chain 1-alkanol sprethe air/water interface. The author calculated the activaenergy of water evaporation to be ca. 45.5 kJ mol−1, whichis due to the presence of 1-alkanols at the interface.value is different from the present experimental data. Instudy, as shown in Table 2, the energy values varied f44.3 to 55.8 kJ mol−1 over the temperature range 298.333.2 K, depending on the type of 1-alkanol whose caratom number changed from 13 to 19. These values certainclude the activation energy of the evaporation from bwater itself (see Appendix B). Flowing dry air was not eployed in his study [17–20], which is another possibility fthe observed differences in activation energies.

Temperature dependence of the activation energyslightly larger for the longer 1-alkanol than for the shorone. In addition, the activation energy was linearly relato the alkyl chain length for lengths greater than C12. Thisfact suggests that the molecular interaction between a

,-

t

chains resists the transport of water molecules acrossmonolayer, which means a smaller barrier for a shorter achain.

The concentration profile of water molecules determithe evaporation rate because the evaporation is a mial flux. The concentration profile in the aqueous substphase beneath an insoluble monolayer at the air/water iface is the same as that in just the bulk water phase, wthe evaporation of water is not taking place or the vaphase is under the saturation pressure. This is becaus1-alkanols in the monolayer are insoluble in the aquebulk and because the concentration profile around the inface determines the evaporation rate of water. This is qdifferent from the concentration profile of water molecuin a soluble surfactant solution around the interface, becthe soluble surfactant molecules are concentrated arouninterface, and in addition, the concentrated surfactant mcules around the interface are in dynamic equilibrium wthose in the bulk substrate with a lifetime in the order of mcroseconds. When the evaporation of water molecules sthe concentration profile of water molecules becomes dient from that of no overall vaporization or under the sarated vapor pressure. If the presence of an insoluble mlayer at the interface does not retard the evaporation rawater, as is the case of a 1-tridecanol monolayer, the contration profile of water molecules at stationary flux shobe the same as that of just bulk water itself. As a conquence, the activation energy also remains the same.means that the energy required to break the hydrogen bing between water molecules for their evaporation is so lathat the presence of the monolayer of shorter moleculeslittle effect on both the evaporation rate and the activationergy. At the same time, shorter alkanol molecules are loopacked in a monolayer. However, as the alkanol moleculethe monolayer becomes longer, water molecules have tothrough a closely packed longer hydrophobic region pto their evaporation, and therefore, the evaporation ratecomes slower and the activation energy becomes larger,the case of 1-alkanols longer than 1-tridecanol. In this cof course, the concentration profile of the water molecubecomes different from that in just bulk water. This is athe case for the gas phase above the monolayer.

Equilibrium spreading pressures of long-chain 1-alkanare shown in Fig. 8, where the values obtained in the preexperiment are also given (triangle blank mark). Good agment between our data and the reference values indicthat the insoluble monolayer used in this study was unequilibrium spreading pressure. The molecular surfaceeas at ESPs were roughly 20 Å2/molec for longer alkanols(closely packed state) at 298.2 K [21–23]. Therefore, tcould reduce the evaporation rate significantly. HarkinsCopeland [24] measured the dependence ofπ–A curves oflong-chain 1-alkanols on temperature and showed thacurves shifted to a slightly larger area with increasing teperature. The shift amounted to about 0.03 Å2/molecT −1.Such a hole size should be sufficient among 1-alkanol m


] at

aturong

ionaationtem-n berain

redde-

rds,tion

rlye ac. Oneasin-dinge inost

eractian

ord-

Cosrofts.

lowtion

rateperndw-ionl to

tion

ined.nt isa-

ime,ndi-desulk

r-

ase,

ilib-a-

on

Fig. 8. Equilibrium spreading pressure (ESP) of 1-alkanols [21–23298.2 K, where the triangle marks are the present experimental data.

cules for the passage of a water molecule. As the temperrises, the 1-alkanols expand at the interface and room amthe alkanol molecules becomes larger so that less additexpansion is required for passage. Therefore, the activenergy of evaporation should decrease with increasingperature, as was observed. Nevertheless, the interactiotween the alkanol molecules still predominates to restwater molecules from evaporating.

4. Summary

The evaporation rate of water from the surface coveby an insoluble monolayer of four saturated 1-alkanolscreased with increasing alkyl chain length. In other wothe effect of the alkanols on the evaporation rate reducincreased in the order of C13OH < C15OH < C17OH <

C19OH. C13OH is the shortest 1-alkanol which can cleareduce water evaporation as a spreading monolayer. Thtivation energy decreased with increasing temperaturethe other hand, the activation energy increased with incring alkyl chain length, which indicates that this energycludes not only the energy to break the hydrogen bonbetween water molecules but also the energy to cross thsoluble monolayer. The activation energy increased almlinearly with alkyl chain length, when the length is longthan a dodecyl group. Temperature dependence of thevation energy was slightly larger for a longer alkanol thfor a shorter one.

Acknowledgments

This work was supported partly by a Grant-in-Aid fScientific Research No. 10554040 from the Ministry of Eucation, Science, and Culture, Japan, and partly by themetology Research Foundation. The authors thank to PTohru Inoue of Fukuoka University for ESP measuremen

e

l

-

-

-

-

-

-.

Appendix A

A rate constant is used for a chemical reaction to folthe time dependence of concentration. As for the reac

AkA−→B + · · · , the rate equation becomes

(A.1)dCA

dt= −kACA,

whereCA is the concentration of the species of A andkA isthe rate constant. In the present study, the evaporation(k in the text) corresponds to a flux of water moleculesunit area(J ), the left-hand side of the above equation, athe concentrationCA to a bulk concentration of water belothe surface, 55.39 mol dm−3 at 298.2 K. Therefore, the differentiation coefficient for the logarithm of the evaporatrate(J ) with respect to the inverse temperature is equathat of the logarithm of the rate constant (kA) with respectto the inverse temperature, which is equal to the activaenergy of the evaporation.

(A.2)Ea = −Rd lnJ

d(1/T )= −R

d lnkA

d(1/T ).

This is because the bulk concentration of water(−CA) canbe assumed constant over the temperature range examIn other words, the evaporation rate or the rate constadefinitely determined by the concentration profile of wter molecules above the bulk water, and at the same tthe concentration profile depends on the experimental cotions. This is the reason why the activation energy incluthe contribution from the insoluble monolayer above bwater.

Appendix B. Correlation between activation energy(Ea) and enthalpy change (�Hvap) for vaporization

Material flow rate(J ) at the gas/liquid interface is detemined by Fick’s first law,

(B.1)J = −Dl(∂cl

∂x

)x=0

= −Dg(∂cg

∂x

)x=0

,

whereJ is a rate of diffusion,Dl andDg are the diffusioncoefficients of material in a liquid phase and in a gas phrespectively, and∂c/∂x is the concentration gradient.

The relationship between temperature and the equrium vapor pressure(P eq) is defined by the Clausius–Clpeyron equation for an ideal gas,

(B.2)d(lnP eq)

d(1/T )= −�Hvap

R,

where�Hvap is the enthalpy change for vaporization.Based on the kinetic theory of diffusion, the self-diffusi

coefficientDg of gas molecules is given by

(B.3)Dg = 1λc̄,

3


lesof

enary

tedea

ion

di-at

hing

-ture-tion

97)

76

7.

o-–33.)

.s.

ter-

where λ is the molecular mean free path of molecu(1/(

√2πNd2)) and c̄ is the average molecular velocity

molecules(√

8RT/πM). Then,Dg is given by

(B.4)Dg = 2

3π3/2N

√RT

Md4 ,

whereN is the number of molecules per unit volume,d is amolecular diameter, andR, T , andM are the same as abovin the text. An average concentration gradient at a statiostate for gaseous phase is given by

(B.5)dc

dx= Cv −Ceq

l,

whereCeq is the concentration equivalent to the saturaequilibrium vapor pressure(P eq) for a slow evaporation ratas in the present case [12] andCv is the concentration atdistancel from the surface.

Combining Eqs. (B.1), (B.4), and (B.5), the evaporatrate or the flow rate of molecules then becomes

(B.6)k =Kα(T )√TP eq

l

with

(B.7)K = 2R1/2/(3π3/2M1/2),

(B.8)α(T )= 1

Nd2,

whereK is the constant irrespective of experimental contion, andl is the distance from the surface to the placeCv ∼= 0. If the logarithm of Eq. (B.6) is differentiated witrespect to inverse temperature, there results the followequation:

(B.9)d lnk

d(1/T )= d lnP eq

d(1/T )+ 1

2

d lnT

d(1/T )+ d ln(α(T )/l)

d(1/T ).

Substitution of Eqs. (1) and (B.2) to Eq. (B.9) leads to

(B.10)Ea =�Hvap+ RT

2−R

{d lnβ(T )

d(1/T )

}

with

(B.11)β(T )= 1

Nd2l,

whereβ(T ) is a function of temperature. Therefore, ifβ isassumed independent of temperature, theEa value is alwaysgreater than the�Hvap value byRT/2.

WhenP eq is different fromPeqs at a stationary evapo

ration rate, the experimental condition or the temperadependence ofβ(T ) plays an important role for the activation energy. In the present study, however, the evaporarate is so slow thatP eq can be assumed to be equal toP eq

s ,as the experimental results have substantiated [12].

References

[1] G. Hedestrand, J. Phys. Chem. 28 (1924) 1245.[2] N.K. Adam, J. Phys. Chem. 29 (1925) 610.[3] E.K. Rideal, J. Phys. Chem. 29 (1925) 1585.[4] F. Sebba, H.V.A. Briscoe, J. Chem. Soc. (1940) 106.[5] U. Navon, J.B. Fenn, Am. Inst. Chem. Eng. J. 17 (1971) 131.[6] U. Navon, J.B. Fenn, Am. Inst. Chem. Eng. J. 17 (1971) 137.[7] D.C. Walker, Rev. Sci. Inst. 34 (1964) 1006.[8] G.T. Barnes, Adv. Colloid Interface Sci. 25 (1986) 89.[9] K. Lunkenheimer, M. Zembala, J. Colloid Interface Sci. 188 (19

363.[10] P.C. Schulz, M.A. Morini, M.E.G. de Ferreira, Colloid Polym. Sci. 2

(1998) 232.[11] Y. Moroi, T. Yamabe, O. Shibata, Y. Abe, Langmuir 16 (2000) 969[12] M. Rusdi, Y. Moroi, Bull. Chem. Soc. Jpn. 76 (2003) 919.[13] H. Hertz, Ann. Phys. 17 (1882) 177.[14] M. Knudsen, Ann. Phys. 47 (1915) 697.[15] R.J. Archer, V.K. La Mer, J. Phys. Chem. 59 (1955) 200.[16] G.T. Barnes, J. Hydrol. 145 (1993) 165.[17] G.T. Barnes, V.K. La Mer, in: Retardation of Evaporation by Mon

layer, Transport Process, Academic Press, New York, 1962, pp. 9[18] V.K. La Mer, T.W. Healy, L.A.G. Alymore, J. Colloid Sci. 19 (1964

673.[19] G.T. Barnes, J. Colloid Interface Sci. 65 (1978) 566.[20] G.T. Barnes, D.S. Hunter, J. Colloid Interface Sci. 136 (1990) 198[21] A.V. Deo, S.B. Kulkarni, M.K. Gharpurey, A.B. Biswas, J. Phy

Chem. 66 (1962) 1361.[22] G.C. Nutting, W.D. Harkins, J. Am. Chem. Soc. 61 (1939) 1180.[23] G.L. Gaines, in: Insoluble Monolayers at Liquid–Gas Interfaces, In

science, New York, 1966, pp. 213–214.[24] W.D. Harkins, L.E. Copeland, J. Chem. Phys. 10 (1942) 272.

Study on water evaporation through 1-alkanol monolayers by the thermogravimetry method

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