THEORY OF MONOLAYER PERMEATION BY GASES
2793
An Approach to a Theory of Monolayer Permeation by Gases
by Martin Blank’ Department of Physwlogy, CO&ge of Physicians and Surgeons, Columbia University, Nnu Y o r k , Nnu York (Reeeiced February 88,1964)
The permeability of a monolayer to gases is compared to that of thicker layers. The current theory explaining monolayer resistance, i e . , the energy barrier theory, is also re-examined. ‘I’his study introduces a new interpretation of monolayer resistance to permeation in terms of known properties of monolayers. The new model suggests that free spaces in the monolayer are available for permeation from: (1) the natural free area in a lattice, (2) the equilibrium fluctuations in monolayer density a t a (gas molecule-monolayer) collision site, and (3) the work of expansion that the permeant molecule can perform against the monolayer forces. (This third factor turns out to be negligible.) From the entropy change associated with a n expansion of a monolayer it is possible to estimate the probability of a given expansion. The monolayer resistance can then be derived if it is assumed t h a t all local expansions which yield an area equal to or greater than the crosssectional area of the permeant result in passage through the monolayer. There is qualitative agreement between the derived resistance and the measured one, but the derived value is much too small. The model is refined and brought into closer agreement with obscrvations when two additional factors are considered: (1) gases that are present but a t equilibrium (e.g., air when water permeates) add to the measured resistance; and (2) the structure a t a vacant site in a monolayer causes a reflection of most molecules that are attempting to permeate because their angle of approach to the hole results in a collision with a monolayer molecule.
I. Introduction The effect of a monolayer on the transport of a gas across a gas-water interface is described in terms of a monolayer resistance, r (in sec./cm.). The total resistance to gas transport, p / U t , is set equal to the sum of the monolayer resistance and the bulk phase resistance, p / U .
- - -_p Ut
+ r
u
The resistance (or its inverse, the permeability) of the monolayer to the various gases has been estimated. The variety of experimental procedures (monolayers in different surface phases, composed of various polar and nonpolar groups, with different gases permeating and a t several temperatures) allow some general dcscriptions of the permeation process in monolayers. Table I coFpares the permeability of monolayers (about 30 A. thick) to much thicker membranes (composed of synthetic polymers, for exaniple, and microns or millimeters in thickness). Thesc properties have been discussed a t some length in a previous article.8 Several of the assertions are supported by few experiments because the necessary measurements are dif-
where U t equals transport rate of gas per unit time and area (cm. a/cm.2/sec.) when a monolayer is present, U equals transport rate in the absence of a monolayer, and p equals gas “prcssure” in cm.a/cm.a. Equation 1 is (1) Supported by a Research Career Development Award (GM-K3a steady-state equation and it has been applied by La 8158) and a Research Grant (GM-10101) from the U. S. Public Mer and co-workers2 to the steady-state transport of Health Service. water through a monolayer. More recently, Blank8 (2) Specific references will be cited at appropriate points in the text. has studied the transport of other gases (COz, 01, A bibliography can also be found in “Retardation of 1~;vaporationby Monolayers.” V. K. La Mer. Ed., Academic Press, New York, N. Y., N20) under conditions that are not steady state but 1962. which allow an approximation to the steady state. (3) M. Blank, J. P h y s . Chem., 66,1911 (1962).
MARTINBLANK
2794
ficult to make in a variety of systems. Xevertheless, the statements summarize repeatable observations and represent the best available information regarding the permeability properties of monolayers.
Table I : Permeability of Interfacial Layers to GaseR Thick rnem-
Behavior of system 1.
2.
3. 4.
5. 6.
brane
Permeant obeys Fick’s law for diffusion Yes a. Steady-state permeation rate proportional to pressure difference and area Yes b. Inversely proportional to barrier thickness Yes a. Temperature dependence of permeability constant, gives activation energy of 5-15 kcal./mole Yes b. The activat,ion energy varies with the NO barrier thic-kness Size and shape of permeant molecule influence rate of permeation Yes Composition arid physical state of barrier influence rate of permeation Yes Permeant obeys Henry’s law for solution Yes Several gases permeibte independently (excluding cases of chemical interaction) Yes
Monolayer XO
Yes
No Yes Yes Yes
Yes No S O
gas constant, and 7’ is absolute temperature. The form of eq. 2 can be derived from Boltzmann’s equation where E is the energy necessary to penetrate the barrier and the exponential gives the fraction of molecules possessing t,his energy. lii has been found t,o depend on t.he length of the hydrocarbon chain,, the surface pressure, the cross-sectional area of the pernieant, and some propert’ies intrinsic to the monolayer (surface phase, compressibility, free surface area, and polar group). The relatively large number of properties that influence the energy barrier have made it difficult to visualize a unified physical mechanism for monolayer resistance. The energy harrier formulation has also been unable to account for a number of observations. Briefly, some of the difficulties are: (a) The resistance is independent of the surface pressure, as,in the LC phase of fatty acids but not in other mon0layers.~~~6 (b) The energy barrier contribution per CH2 group, E m 2 ,is greater for the LC phase than for the S phase of fatty acids. This is unusual because the S phase is more close-packed t,han the LC phase and E C Hincreases ~ as the average spacings between molecules decrease.’ (e) For fatty acids about z/3 of E is attributed to the polar but oleic acid with the same polar group has no resistance18and molecules with different groups (OH, OC2H5) have comparable resistances.6 (d) Cholesterol nionolayers a t high as,are close-packed and have a low compressibility but, nevertheless, have innmeasureably low (e) The ability of two permeating species to interfere with each other is not considered by the theory. lo The complex dependence of the energy barrier on monolayer properties and the inability of the theory to account for some important findings have prompted the author to reconsider the process of monolayer permeation. This paper will suggest a new physical interpretation for the resistance due to a monolayer and attempt to arrive at an alternative to the energy barrier theory.
There are several significant differences between macroscopic arid monolayer processes. Some of the differences are reasonable: (a) (item 1) The inadequacy of Fick’s law is to be expected when the size of the pernieant approaches the thickness of the barrier. (b) (item 5 ) Since the monolayer has very little physical capacity for solute, we would not expect agreement with Henry’s law. Howwer, some of the differences imply that one is dealing with two qualitatively different phenomena. (c) (item 2b) The variation of the activation energy with the monolayer thickness implies a one-step, all-or-none p r o ~ e s s . ~(d) (item 6) The extra resistance ericountercd when several gases permeate at the same time is indicative of a different 11. The Monolayer Resistance rnec hanisni. A. Mechanism of Monolayer Permeation. For The permeability of a monolayer has been described a gas molecule to pass “through” a monolayer, the twoin terms of an extra energy barrier a t the i n t e r f a ~ e . ~ dimensional lattice must contain a free space at least This theory has been used extensively to account for the observations and it has proved very useful in giving (4) G . T. Barnes and V. K. La Mer in ref. 2, pp. 9-33. a quantitative description of the permeation process (5) (a) I. Langmuir and D. Langmuir, J . Phys. Chem., 31, 1719 (1927); (b) R. J. Archer and V. K. La Mer, i b i d . , 59, 200 (1955). in terms of monolayer and pernieant properties. The (6) H. L. Rosapo and V. K. La Mer, ibid., 6 0 , 348 (1966). resistance r = c exp({F)
(7) M ,Blank and V . K. La Mer in ref. 2, pp. 59-66. (8) F. Sehbn a n d H. V. A. Briscoe, .I. Chem. SOC.,106 (1940).
(9) >Bisnk ‘I. and P.J. W. Itoughton, Trans. Faraday SOC..56, 1832 (1960).
where c is a constant, R is the energy barrier, R is the The Jou rnnl of Phuaicnl Chemistrij
(10) M. Blank, .I. Phys. C h a . , 6 5 , 1698 (1961).
as largc as the pcrnicarit. ‘1 free spacr can arise by three dificrcnt iri(~chanisiiis. ( I ) RIoiiolaycr inolccules cannot pack together in a lattice without leaving frcc arca. ‘I’hc free a r w per collision site, -4,,dcpcnds upon the sizr arid shape of the inonolayer ~riolccules, the n u n h r r of ~ i i O l ~ ~per ~ l collision t5 site, and the statc of coinpression. (2) Since a colliding gas molecule will affrct few inonolayer nioleculcs, thcrc will be large fluctuations in monolayer dcrisity a t any collision site. The fluctuations givc rise to local cspansions t h a t incrrase the free arca. (3) The penetrating molecule has kinetic energy which can do work against the surface forccs in the nionolayer. Tf all the kinetic energy ( I 2?nv2)is utilized for expansion work ( n S A A ) , the largest frcc spacc a gas inolecule can forin is AA = mu2,2rs. Let us consider the effects of thcsc mechanisms in making holcs available for gas permeation and attempt to calculate a rnonolaycr "resistance ” We shall assuine that the perrncation process is all-or-none, i.e., either a gas nioleculc gcts through in one shot when it collides or c . 1 ~ ~it~ IS rcflected. The probability of pcrineat irig is, t herefore, cquivalcnt to the probability of the perincarit finding or forriiing a holc of sufficient niagnitut-fc. We can ronsider the insoluble monolayer at the gas-water intcrfacc and the gas phasc as equilibrium systcnis and t h e rnonolayer niolecules as a hexagonal close-packed latticc (condensed phasc). Equations valid for macroscopic systeins will be used when nccrssary sirice thc cvcnts discussed in trrms of several molecules are reprated many tirnes over the entire surface. Let us assuine t h a t n, the number of n-~oleculesthat arc influenccd by a collision, is equal to thc number of rnolecules hit plus t h e nearest neighbors. n is 7 if a gas molecule hits a inonolaycr nioleculc head-on, 10 if it hits hctiveen two inonolayer niolecules, or 12 if it hits at the spaces twtween three nionolayer molecules. The niagnitudc of n for an entire surface is obtained by averagiiig all valucs of n arid wcighting according to the fraction of nionolayer area t o which the value applies. Since this cannot be done unambiguously, we shall use an approximate value of n equal to 10. Although n for ariy collision is intrgral, the avcrage over the wholc surface is nonintrgral. Furthermore, ?z charigcs with comprrssion of a nionolaycr since the areas availablr for diffrrcnt kinds of collisions change. B. Local B.rpans?,ons in the Monolayer. The cntirc monolayer is a t constant arca, but a5 a rcsult of normal rnolecular motions tlicrc arc fluctuations in nionolayer density in small regions of the monolayer. Let us estimate the probability of fluctuations in monolayer derisity at a collision site by assuniing that n molecules
uridergo a change in the area (A) they occupy and that the rest of the moriolaycr acts as a resci.voir for this process. For this system, which is at constant ?’, p , V , and n, the variation in tho critropy clue to a local expansion is
(3) Sincc thc surface pressure ra = yo - y , where equals thc surface tension of pure water
yo
YO 1 AS= - - A A + - , f n , d A 7’ 1‘
The entropy change can be calculated on a pcr niolccule basis, AS,,,. For an LC or S monolayer a t constant tcinpcrature and composition, the arca pcr molecule, A , , and xwarc related. A~
=
-k’Am
+ k”
(4
where A , is in &2/rnolccule and k’ and k” are constants. Substituting for rs and integrating betwcen the cquilibrium arca per niolcculc, A,,, and A , , AA rn
+
YO
-- AAm
T
T
+ [(-k’A,
+ k”)AAm - 2- AA,!
For valucs of AAm < 10% A,, we can neglect the AAm2term. ‘The final cquation for the entropy changc per molecule, thereforc, is
Thc entropy changc a t a collision site is nkl times eq. 5 where kl (2.39 X lopz4)converts the AS into entropy units. We can set nAA, = AA, the area change a t a collision site. Thc probability, W , of observing a spontaneous dwense in entropy, AS is
w=w o W o is the probability of the equilihriuni statc (AkS = 0), and k is the Boltzmann constant. Substitut,ing for AS‘ duc to a local expansion at a collision site (7) The frequency distribution of fluctuations around the cquilibriuni area is not symmetric. However, it is Volumo 68,Number 10
October. 1864
reasonable to assume that the bulk of the distribution is due to the small fluctuations iminediately around Aeq and that these are symmctric. Thereforc, we can evaluate Wo by integrating eq. 7 over all values from AR = 0 to AA = a and setting thc intcgral cqual to '('2. The result is
Equation 7 gives the probability of an exparision, A n , a t a collision site in a monolayer. The probability of passing through a rnonolayer is equivalent to the probability that a permcating molccule xi11 encounter a hole equal to or larger than ao, its cross-sectional area. Since the hole can arise from a local cxpansion ( A A ) or from the free area in the lattice (ili), the eondition for permcation is a. 6 AA A f. The probability of permeating thc inonolaycr, W,,, if the availablc area is AA, is the probability of having AA 3 uo - At.
+
+
between v and v dv, or the probability that a molecule dv. The joint will have a vclocity betwecn v and v probability I , that a gas molecule will have a b'riven velocity (cq. 13) arid that it will gct through the monolayer if it has that velocity (eq. 12), is the product of thc two individual probabilities.
+
Substituting into cq. 14 and intcgratirig b c t w c n v = 0 Allti v = m ,
(15) Tho first factor in eq. 15 is cqual to unity sincc niTS >> 1016kly(by about five orders of rnagnitudc). 1:qiiation 1: rcduccs to cq. 10 arid I = TV,. I t appears that the permeating inolecule docs riot affect the formation of a hole. D. Derivation os the Resistance. I (or TY,) is the probability that a molecule will permeate whcn it collides with a nioriolaycr. The nuriibcr of niolcculcs that strike I cm.*of surfacc/sec. (1 6)
Since TVolcT/kly = the probability of getting through the monolayer is
n, cquals the nuiiibcr of molcculcs/crn.3 of gas, 111 . cquals thc tnolecular weight, R equals thc gas conC. Kinelic Energy 0)" the Permeating Molecule. Let us now take into account the ability of thc pcrmcating molcciile to expand the monoluyer. The inagnitudc of the maximum expansion is given as AA = 1016mv*/ 2rs, whcrc the factor 10l6 converts A i l into square angstroms. The condition for permcation is
arid eq. 10 should read lfly
1vp
= --
2l
cxp (
[
ao - Ar
___-
kT
-
109 7 2 v * __-
2rs
I)
(12)
T h e distribution of one component of velocity (normal to a siirfacc in one direction) is dN L V ._
( 2772))"' exp( 7' 2kT
m i 2 ) dv
ak
stant, and Q is sct cqiial to thc square root term. At equilibrium, the riuriibcr of niolccules cntcriiig or lcavirig a gas phase, the unidircctional flux, L" = N l a . cy, thc contlciisation cocfficicrit, is the fraction of niolcculcs that critcrs the. liquid phasc hfter striking the surfacc. When a monolayer is prcwtit, cach unidirectional flux, Ut' = NlcyI. (This ass itiics that the rnonolaycr has riot affected the condcrisation cocfiicicnt.) If there is a coriccntration difference (An,) across an interfacc, the net fluxes are I; = .(An,)& and I T t= a(An,')&l. An,', the concciitration difference across a monolayer, is srnallcr than Ana. Substituting for U and Ut into eq. 1
Substituting for I
(13)
dN,'N 'is the fraction of molecules with a velocity
If
2797
THEORY OF MONOLAYER PERMEATION BY GASES
then
effect than a CH2 group added to a longer chain. This would invalidate the linear extrapolation of resistance values to zero CH, groups and would explain the very high values found for for contribution of the polar group6bto the total energy barrier. Let us consider now the dependence of I3 y(ao At) in eq. 18. Substituting for y and multiplying through, we see that E, which depends on four tcrms
-
We can arrive directly a t eq. 18 by setting the monolayer resistance equal to a driving force ( p ) divided by a net flux (Ut). This would avoid using eq. 1, which is defined in macroscopic terms. However, inequality 17 emphasizes our assumption that the total resistance is due to the monolayer, i.e., the free surface has no effective resistance. Equation 18 is an expression for the monolayer resistance to gas permeation. The pre-exponential term depends on properties of the permeant arising from gas kinetic theory. The exponential (energy barrier) term depends on equilibrium monolayer properties and on the size of the permeant. Comparing eq. 10 and 18, we see that the energy barrier is equivalent to the probability of permeating a monolayer. The measured energy barrier, obtained from the temperature dependence of r, is largely determined by the exponential term but the pre-exponential term is also a function of temperature. 111. Discussion
A . Properties of the Derived Resistance. According to the proposed model and in contrast to the energy barrier theory, the kinetic energy of a penetrating gas moleculc does not contribute to monolayer permeation. A monolayer is permeable because, a t equilibrium, spaces from thc natural packing and from the density fluctuations are large enough for a permeant to pass through. Let us see how the monolayer resistance derived on the basis of this model can account for the properties of the observed resistance. Consider first the resistance or energy barrier increment per CH, group in the hydrocarbon chain, &E,. This can be attributed to two factors: (1) a smaller A f because of the greater attraction between hydrocarbon chains and ( 2 ) smaller fluctuations in monolayer density because of an increase in mass and volume of the kinetic unit. For these reasons, we can expect the different values observed6b for EcR, in the LC and S surface phases. I n the S state, the binding of fatty acid ions by calcium ions in the subphase decreases the density fluctuations and ECE,is lower than in the LC phase. (However, the S film still has a high resistance due to a large decrease in Af.) The value of ECR,should also vary with the chain length since the first CH, group has a proportionally greater
-
+
you0 - YOAf - 7ra@ 7raAf (19) (1) (2) (3) (4) can account for a number of additional observations. (a) The monolayer lattice (i.e., the number of molecules affected by a single collision and the free area per collision site) changes with surface pressure. For ar increase in 7rs, A f decreases and in eq. 19 (l), the largest of the four terms, is constant, ( 2 ) decreases, (3) increases, and (4) probably increases. The large constant term (1) tends to minimize the variation in E. The terms ( 2 ) and (4) tend to increase E , while (3) tends to decrease E. If (3) is much greater than (2) (4), E (or In r ) varies linearly with 7ra as in the S phase of fatty acids. If the variation of ( 3 ) is equal to (2) (4), In r is independent of 7r. as in the 1,C phase of fatty acid monolayers. (b) A comparison of monolayer permeation to two different gases (COz and HzO) indicates that the resistance depends on the crosssectional area, &, of the molecules.’O The agreement with eq. 19 is semiquantitative since the data do not warrant a more refined test. (c) The presence of a term that depends on A f explains why some nionolayers offer no detectable resistance even though they are‘ incompressible and a t high surface pressure (e.g., cholesterol). The free area in such monolayers is much greater than in saturated fatty alcohols.3 (d) I n mixed monolayers, if the two components pack together to minimize At, the resistance is high. Since the surface isotherm indicates the nature of the packing and the value of A f , it is possible to predict the resistance by comparing the partial areas of the components to the areas in one component monolayers. Recent work by Robbins and La Mer11J2on the incorporation of spreading solvents in monolayers supports this view. Further support comes from studies’s on mixed monolayers of fatty alcohols which pack like a single component monolayer and where the resistance varim linearly with the mole fraction. It appears that the proposed model can account for many qualitative observations on the resistance proper-
E
+
+
~
~~
(11) V.K.Lahlerand M.L. Robbins, J . Phys. Chem.,62,1291(1958). (12) M. L. Robbins and V. K. La M e r , J . ColloidSci.. 15, 123 (1960). (13) V. K. La Mer, L. A. G. Aylrnore, and T. W. Healy, J . Phys. Chem., 67,2793 (1983).
T701icme68, S u m h e r IO
October, l M 4
ties of iiionolsyers, iiiclutliiig soiiic piupcitics that arc not cxplaincd by thc ciiicrgy bauicr tliooiy. 'l'lic quaiititativc prcdictioiis of (XI. 18, h o \ \ c \ ~ ~Icad . t,o poor agrceiiicmt whcn \vtl substitlitc valucs arid put r into the forin of eq. 2. .lpproxiiiiatc valuos for \vatcr pcririeatirig a close-packed iiioiiolayctr (at 25 ") ai*(!: p = 0.03, a = IO-.-,, Q = 1.5 x I O ' ci;i. 'scc., y = 50 dyn(ts,lciir., uo = 7 and ..I i < 0.5 A.z 0 (estiniatcd for fatty alcohols), Usiiig thct oqiiatioii Ut = a(An',)Q and thc valucs givctii l)y .\rchc11, ant1 La (An'8) = 2 X l o 1 ?iiiolwdcs ciii.3. ('l'hc? tnaxiinuiii possible value is 8.4 x lo1', whcn there is iio backward flux.) Having t'hc values for ( A n ' , ) ? kl, and Avogadro's number, we ari,iv(t a t
-
(:qual. 'l'lic uiiidirctctioiial Nuxcs for €I& and S, arc, i~tspc~ct ivoly "S(an$Q)II,"oxp(-Kni,,o)
=
(Lr't)lI,O
(21)
alld (C"t)N,
=
I '2(GYL3Q)N,
cxp(--l(as,)
'.li has ~ N Y I I il(tgl(tct(?dailti K = lily 'ICY'. TO a first approxiiiiation, H'R is equal to K Ztiiiics thc product of thc siiiiiilt~ancous(and opposiiig) uriidircctiorial fluxes of 1 1 2 0 arid S,. (IC, is a coilstant related to thct frcquctiicy of hiiiaiy collisioiis.) Thcreforc? =
IZ'R
K2(L7't)l.I,0(
(22)
C-'t)S,
\l:hcii thcrt: is a iiot flux of watctr through thc iiioiiolayer,
we coriwt for the iiitcrfei~ciiiccby nitiogcn with an c~xprcssionsiiiiilar to cq. 2 2 .
The prc-exponential factor is iriucli lo\vci~than the measured5 value of c = 8 x 10 l l . Tho calculatcd E 0.5 kcal./rnolo is also iiiiich lower tliaii the iiimsured5 value of 15 kcal. ;iiiolc. 111 fact, the. calciilatcd value of 6, which is or1 the ordor of /?T, is iiot aii c~ioi'gy barrier of any significance. I-Zonc~vcr,tho dcrivat,ioti of eq. 18 has ncglectcd two aspccts of ~iioiiolaycri*csistancc that can greatly iiicrcasc thc inagnitutlc of 1'. They are: ( I ) the effect of a gas that is prctsc.iit hiit is riot diffusing bccausc it is at equilit)ri\iiii and ( 2 ) the greater selcctivity at' a vacaiit sit(: i i i a iiioiiolayctr due to thc aiigle at which thct pcriiiclating ~iiol(:culr is moving rclntive to the holc. 'I'hosc two factoi,s \vi11 iiow bo coilsidered. H . Resistance Due to a n Additional Gas at I.:qriilibriiitn, The niodcl for the procws of \vator wapoixtioii is a monolaycr at, a surface arid watc~i~ as h t l i tho liqiiitl subphase and t h e gas phaso. Ilowcvcr, thv i ~ a l system also coiit,airis a high coiiceiitixtioii of' air (which we can assuii~ois iiit rogcii), 'I'hc: S, iiiolwulw ai'(: iii equilibriuiii t)ct\vcori tho gas aiid licluitl pliasw a ~ i dso must pass through thc tiionolayor likc tlic: \vai,c:i. vapor. Since Nzreiiiains a t eqiiilihriuiii diiriiig a traiispoit of watcr, thc: uiiidirectioiial flux that is opposite. to the net flux of watcr is c:ffcativc~i i i causiiig gas cwllisiotis that rcductl thc iiicasimd i i o t flux:. I I I tliv c a w of nloriolayor pc:riii(:atiori by COz, w h e i ~thoi,c: is a n cqiii1it)riuiii c1istrit)ution of \vtLtvr, thc: ot)wrvcd iivt flux of COScan h cxplairicd i n tvriiis of ai1 additioiial wsistaticc diic to iiit,c:i~l'c:i~c~ricct by n a t w vapor. I o 1,ct us ostiiiiat,c: ljTI%, thv piwtmhility that a watvi. rnolocul(:, on iiioviiig thiuiigh a iiioriolayc>r. \vi11 c:ollitl(: with a gas iiiol(:culc~t ~ i i t ih: rcflcctcttf. ;it, cquilibriiini, rr,o alld s,Inoi(?cllics rlloVf: t,ilro~lgi1tile IliotlolaYcr in h t , h dirttctioiis and for tach gas thct two ratm arc
(C't)HZO
-
(cor.)
=
'2[.(An',)&]
oxp(-KKnIIzo) [l -
'l'lic iiitrodiictiori of
ai1
K2(L"t)sz]
x (23)
effect due to the prosciicc of
gasos c*haiigc?scertain aspects of our iritwprotation of
iiwnolayc.r pcriiioation mid of t,hc t~ccliriiqiwsused to rctsistaiicc. (a) If we us(: oq. 23 i,athcr thaii cq. 10 to dcrivc: t,hc irioiiolayor ivsistancc, thc niagiiitudc of I ' is sigiiificaiitly incroascd. (t)) 'I'lic last factor in ('(1. 2:{ iiidicutcs that the iict flux of natcr should vary directly with t'he concentration of intcrfcring gas (n, ol tx1. 21). In i,hc case of (XI2pcrincatioii, whcrc I-IzO is tho iiit(!rfcriiig gas, thc riionolayor rosistancc to (proportioiial to thct itiversc: of cq. 2 3 ) dcci*oascsi i i a n approxiiiiatdy lincar iiianiierlo with dccrcascts in watw CoiiCctiitratioii (vapor p i ~ m u r e ) . ( e ) 111 cietcrniinirig thc iiiagiiitridc of k: by varyiiig tlic: tcinpctraturc, the cliaixctci.istics of the iiitorfering gas inust bc considered. Iii CX), pcriiication, the water vapor prcssiire varies considcrahly with the teiiipcrature aiid greatly dfccts thcl i i i c m u i w l value of 15. l 4 In ivat'cr pci~mr!ation the tiitrogcw conccntratioii docs iiot vary significantly \vith tciiipcmtiirr. (d) If this ~iicchaiiisiii is valid, tho Sz R i i x c ~(assuiircd ctqual during watar waporation) iiiust t)c unttcltial bccaust: of thct asviiiiiiotric TI,O fluxes. C. Gcottiefric Considerations in Jfonoluyet, l'qrttzeution,. :\11 gas irioloculcs t'liat, collido with tho iiiorioAil il f \ v ( t r ~assuiiicd to l a y r at a sit0 w h e i ~uo pctriiic.at,cx. .lc:tiially t h c:oiid it ion for pcimiontioti is i i i o i ~stringoiit, I)ctcausc the vacant site is part of a .;t,rric:tiii.c: and t,hc! gas iiiolc~cul(tiiiiist approach t,hc vacaiit site siich that t,lioi,c will he iio collision aiid refl(wtion. I11 ordcr to cstiiiiat'c: this dfcct, let us assuiiict that tho vacwit sitjcs arc right cyliiidi-ical holcs of iiiwsiiix;
