Molecular orientation as a controlling process in adsorption dynamics

Colloids and Surfaces A: Physicochemical and Engineering Aspects 175 (2000) 51 – 60 www.elsevier.nl/locate/colsurfa

Molecular orientation as a controlling process in adsorption dynamics Francesca Ravera a,*, Libero Liggieri a, Reinhard Miller b b

a Istituto di Chimica Fisica Applicata dei Materiali-CNR, 6ia De Marini 6, I-16149 Geno6a, Italy Max Plank Institut fu¨r Kolloid und Grenzfla¨chenforschung, Max-Planck-Campus, D-14476 Golm, Germany

Abstract The formation of a surfactant adsorption layer at the liquid/fluid interface is basically described by a diffusion controlled adsorption. When adsorbed molecules undergo changes in their molecular interfacial orientation, the adsorption rate can also change simulating another adsorption mechanism. A generalised model is presented which takes into consideration both the transport of surfactant in the bulk phase by diffusion and rate equations describing the different steps of molecular arrangement at the interface: adsorption/desorption fluxes and the process of reorientation. The most general case results when the characteristic times of all three processes are of the same order of magnitude. Model calculations are used to estimate the effect of the three characteristic times: diffusion relaxation time, transfer rate constant, and reorientation rate constant. Moreover, the influence of the involved physical parameter on the dynamic surface pressure is analysed. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Adsorption kinetics models; Molecular surface reorientation; Diffusion controlled adsorption; Non-ionic surfactants; Dynamic surface tension

1. Introduction The adsorption dynamics of surfactants has been widely studied and diffusion has been accepted to be the principal process governing the time dependence of the surface pressure for most systems [1]. However, for some surfactants, the diffusion controlled adsorption model with classical isotherms is not adequate to describe the experimental data. To explain the adsorption dy* Corresponding author. Fax: +39-10-6475700. E-mail address: [email protected] (F. Ravera)

namics of these systems, different causes have been put forward, like the presence of adsorption barriers [2,3], or the interaction between the adsorbed molecules [4,5]. Moreover, it has been recently shown that the orientation process of the adsorbed molecules can play an important role in determining the equilibrium and dynamic behaviour of these systems. In fact, the surface isotherm derived from a model considering two states of the adsorbed molecules [6], with different molar surface area has been successfully applied to predict the equilibrium surface pressure of n-alkyl dimethyl phosphine oxides [7] and poly-

0927-7757/00/$ - see front matter © 2000 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 7 - 7 7 5 7 ( 0 0 ) 0 0 5 0 4 - 5

F. Ra6era et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 175 (2000) 51–60

52

ethoxylated [8 – 10] surfactants at the air water and water-alkane interfaces. This isotherm used in the framework of the diffusion controlled adsorption also provides a good description of the experimental data in several cases. A first attempt to develop a dynamic model in which the orientation of adsorbed molecules is taken explicitly into account as one of the controlling processes has been presented in ref. [8]. The aim of the present work is to give a general description of the adsorption dynamics in which all the principal processes involved are considered, i.e. the adsorption-desorption exchange between the surface and the bulk (kinetic process), the orientation of the adsorbed molecules and the diffusional transport in the bulk.

2. Theory In the framework of the thermodynamic model proposed by Fainerman et al. [6], it is supposed that, depending on the surface coverage, the molecules can adsorb in two different states characterised respectively by the molar surface areas v1 and v2 and by the parameters b1 and b2 which are related to the different surface activities. To ease the comparison between the surface activities of the two states, the relationship b2 = b1

  v2 v1

a

(1)

is also assumed. Thus, for a =0 the surface activity is independent from the molar surface area, and for a\ 0 the molecules adsorbed in the state with the larger surface area are higher surface active. The model provides the relationships between the bulk concentration c and the adsorptions corresponding to the two states, G1 and G2, c=

=

 

vSG1 v1 v1 a b2(1 − v1G1 −v2G2)vS v2 vSG2 v2 vS

b2(1−v1G1 −v2G2)

(2)

where vS is the average molar surface area, weighted by G1 and G2, i.e. vS =

v1G1 + v2G2 G1 + G2

(3)

the surface state equation P= − =−

RT ln(1− G1v1 − G2v2) vS RT ln(1− GvS) vS

(4)

where R is the gas constant and T is the absolute temperature, and the relationship expressing the principle of Braun-Le Chatelier [11],

  

G1 P(v1 − v2) v a = 1 exp − G2 v2 RT

n

(5)

As expressed by this latter equation, under orientation equilibrium condition, the partition between the two states of adsorbed molecules depends on the surface coverage. The state 1, with a larger surface area, corresponding to molecules adsorbed oriented along the surface, prevails at low surface pressure. On the contrary, the state 2, characterised by the smaller surface area, corresponding to molecules normally oriented with respect to the surface, prevails when the surface pressure, i.e. the surface coverage, is high. Moreover, these equilibrium relationships lead to the P-c isotherm PvS 1− exp − RT c= v1 a Pv1 Pv2 b2 exp − + exp − v2 RT RT (6)

  



  

n

It is important to notice that, as Eq. (2), Eq. (5) and Eq. (6), are valid only under orientation equilibrium, the state Eq. (4) expresses a relation between the surface pressure and the adsorptions which should also hold out of the adsorption equilibrium, and in particular out of the orientation equilibrium. For these reasons, the total adsorption, G= G1 + G2, does not define univocally the state of the interface, the surface pressure being dependent on G1 and G2 separately, unless the system is

F. Ra6era et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 175 (2000) 51–60

under local equilibrium conditions. In fact, in this latter case, by using Eq. (5) vS can be written as a function only of the surface pressure,

     

v1 a P(v1 −v2) v2 + v1 exp − RT v2 vS = v1 a P(v1 −v2) 1+ exp − RT v2

n

n

(7)

53

For can be expressed in terms of the rate coefficients k21 and k12, i.e. For = k21G2 − k12G1

(10)

Consequently, the evolution of the total adsorption is dG = Fa1 + Fa2 − Fd1 − Fd2 dt

Thus by considering Eq. (4) and Eq. (7) together, it is clear that there is a direct relationship between G and P.

2.1. Dynamics of adsorption with orientation of the molecules To give a complete description of the evolution of surface pressure during adsorption, the three dynamic processes which are involved have to be considered simultaneously, i.e. the adsorption-desorption exchange between surface and bulk, the change in the orientation of adsorbed molecules, and the diffusion process in the bulk. In the framework of the present model, adsorption is considered to proceed in the following way. The molecules, which are randomly oriented in the bulk, adsorb either in the state 1 or 2, with a probability x and 1− x, respectively. This induces a diffusion flux in the bulk. Meanwhile, the distribution of freshly adsorbed molecules between the two states is out of equilibrium, which is then attained by the re-orientation process. All these processes have their own characteristic time and can be separately considered when the other processes are in equilibrium or have a characteristic time much different. Therefore, the time evolution of the partial adsorptions G1 and G2 is described by dG1 =Fa1 − Fd1 +For dt

(8)

dG2 =Fa2 − Fd2 −For dt

(9)

where Fa and Fd are the adsorption and desorption fluxes, and For is the contribution of the orientation which is assumed to be a kinetic process involving only the adsorbed molecules.

(11)

As expected, the variation of the total adsorption G= G1 + G2 does not depend on the orientation process. The adsorption dynamics is completely described if the time dependence of two quantities among G, G1 and G2 is known. Concerning the orientation process an expression for the coefficients k12 and k21, in terms of the isotherm parameters, can be found by comparing the equilibrium relationship between G1 and G2 provided by the thermodynamic approach [6], with the equation obtained by imposing the orientation equilibrium, that is k21 =

 

G1 0 k G2 12

(12)

where the suffix ‘0’ refers to the orientation equilibrium. Thus, by using Eq. (5), Eq. (12) becomes k21 =

 

v1 − v2 v1 a (1−vSG) vS k12 v2

Therefore, the orientation term is

 

For = k12

v1 − v2 v1 a (1−vSG) vS G2 − G1 v2

(13)



(14)

Concerning the contributions from the bulk to the adsorption changes, the expression for the adsorption fluxes, Fa, can be found by assuming that among the molecules striking a unit surface in the unit time, whose number is proportional to the bulk concentration c, the amount of adsorbing molecules is proportional to the fraction of the free surface, 1− v1G1 − v2G2. Then, the adsorption fluxes are, Fa1 = xkac(1− v1G1 − v2G2)

(15)

Fa2 = (1− x)kac(1− v1G1 − v2G2)

(16)

A reasonable expression for the probability x can be found by assuming a simplified picture

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F. Ra6era et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 175 (2000) 51–60

where among the molecules adsorbing to the surface, those whose axis are inside a cone of spread 90° around the normal to the surface, adsorb in state 2, while all the other ones adsorb in state 1. Assumed that the orientation of the incoming molecules is randomly distributed, the probability to adsorb in the state 1 and 2 are respectively

 

vSG1 v1 v1 a b2(1−v1G1 − v2G2)vS v2 vSG2 + (1− x) v2 b2(1−v1G1 − v2G2)vS

cs = x

(20)

dG1 dG =x +For dt dt

(17)

To conclude, if G and G2 are chosen as variables of the problem, the dynamics of adsorption can be described by Eqs. (18)–(20) and Eq. (14) in which the orientation process is considered together with the kinetics transfer from the bulk to the surface. Moreover, the diffusion in the bulk has to be considered, using the Fick equation with boundary conditions given by the mass balance at the surface. An estimation of the characteristic time of this process is [21]

dG2 dG =(1−x) − For dt dt

(18)

tD =

2+ 2 2 − 2 and . 4 4 Moreover, x and 1 − x also define the fractions of the total adsorption flux coming from the bulk increasing respectively the adsorption G1 and G2. Thus Eq. (8) and Eq. (9) can be rewritten,

where dG/dt is given by Eq. (11) which, considering the expressions for the adsorption fluxes, reads dG = ka(1−vSG)(c− cs) dt

(19)

where the concentration cs has been introduced as variable instead of Fd1 and Fd2, for sake of simplicity. In fact, cs is a function of G1 and G2 which has the meaning of sublayer concentration at equilibrium of the kinetic transfer. The problem now reduces to find an expression for cs taking into account that at the limit of the orientation equilibrium it must tend to the equilibrium sublayer concentration given by Eq. (2). For this reason it is expected that this concentration is a combination of the two expressions of c appearing in Eq. (2). In fact, these functions are in general different and tend to be equal only at orientation equilibrium and, in that case, equate the sublayer concentration. Moreover, because the sublayer concentration is in general given by the sum of the concentrations of the molecules adsorbing in the states 1 and 2, and this also holds at kinetic equilibrium, the coefficients of the combination are expected to be linked to the probabilities x and 1 − x.The simplest expression meeting these requirements is

 

1 G0 D c0

2

(21)

which depends on the diffusion coefficient D and on the surface activity properties of the system which is here expressed by the ratio G 0/c 0. By using the adsorption kinetic equations, it is possible to give an expression for the characteristic times of the orientation process and of the kinetic transfer. The first one is found by assuming the other processes to be in equilibrium. Under these conditions, G is constant in Eq. (18), from which a solution for G2 is found with the characteristic time tor =



1

k12 1+

n

G 01 G 02

(22)

which depends both on the orientation rate constant k12 and on the partitioning between the two adsorption states. Applying similar considerations to the kinetic Eq. (19) allows an estimation of the characteristic time of the transfer kinetics tk =

1 kavSc0

(23)

The estimation of these characteristic times can be used to verify the possibility to have different processes governing the adsorption dynamics. In the following, particular cases will be treated in

F. Ra6era et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 175 (2000) 51–60

detail to describe the adsorption at a plane interface.

2.2. Diffusion controlled adsorption If both the kinetic exchange and the molecular orientation processes are at equilibrium with respect to the transport by diffusion, i.e. tD»tk and tD»tor, the adsorption increases only due to the diffusive flux to the surface. Because in this case the two sides of the kinetic equation vanish, it has to be replaced by

)

1 dG1 dG (c 1 dG2 = =D = dt (x x = 0 x dt 1 − x dt

(24)

where x is the co-ordinate normal to the surface located at x = 0. Thus, the evolution of the total adsorption G is given by the Ward and Tordai equation [13] G(t)=

'

D 2c0 t − p

&

t

0

c(0, t)

t −t

dt



(25)

where c0 is the initial concentration in the bulk and the expression of c(0, t) in this case is given by the orientation equilibrium condition, i.e, Eq. (2) or Eq. (6). A solution of this problem can then be found by solving Eq. (25) together with Eqs. (4), (6) and (7).

2.3. Diffusion-orientation controlled adsorption By assuming that the process of exchange of molecules between the interface and the sublayer (kinetic transfer) is at equilibrium in comparison to diffusion and orientation — i.e. tk«tor : tD — the time evolution of the total adsorption G(t) is controlled by these two latter processes. Like in the case of the diffusion controlled adsorption, G increases due to the diffusion flux to the surface, while the kinetic process is at equilibrium. This latter condition means dG/dt = 0 in Eq. (19), so that the sublayer concentration is given by Eq. (20).The process is then described by the Ward– Tordai Eq. (25), with c(0, t) =cs, together with Eq. (18).

55

2.4. Mixed adsorption kinetics To solve the general problem in which all the involved processes contribute to the adsorption dynamics, i.e. the three characteristic times are comparable, the approach proposed in [14–16] can be adopted.Instead of the Ward and Tordai equation, the following equation has to be solved [16] coming from the mass balance at the surface Eq. (19), as explained in Appendix A,

&

t

dG 1 dt= dt 1− vSG 0

' 

Da 2c0 t− p

&

t

0

c(0, t)

t− t

dt



(26)

where Da = D exp(− 2oa/RT)

(27)

is an apparent diffusion coefficient, which accounts a possible adsorption potential barrier oa [14]. c(0, t) is in this case the effective boundary concentration whose expression is again given by Eq. (20). In fact, this equation has been obtained considering the adsorption kinetic equation as a renormalised diffusive balance equation at the surface.To give a complete description of the process, the problem has to be complemented by Eq. (18). However, thought this case is the most general, it is difficult to meet, at least for the most common non-ionic surfactants. This can be shown by considering the expression of the characteristic time of the kinetic transfer [23], and by using the expression for the adsorption rate constant ka provided by molecular kinetics considerations [14], i.e. 6 ka = = 4

'

RT 2pm

where 6 is the mean velocity of the molecules in the solution and m is the molar mass. For most used surfactant it is found that tk is always much smaller than tD, if no potential barrier is present, which is in general true for a non-ionic surfactant. On the contrary, if a potential barrier exists, a factor exp(−oa/RT) has to be introduced in the expression for ka. For example, this barrier can be due to a transformation that a molecule has to undergo in order to adsorb or to the effect of an

56

F. Ra6era et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 175 (2000) 51–60

electrical double layer for ionic surfactant systems. The effect of this potential barrier is to increase the characteristic time, i.e. to make the kinetic transfer slower. For example, with oa of the order of 5 – 6 RT, and with typical values of the other quantities, the time of this process can become comparable to the diffusion time and a general approach could become necessary.

Fig. 1. Comparison of dimensionless G1 and G2 versus dimensionless time for diffusion controlled and diffusion-orientation controlled adsorption with different characteristic time ratios. The values of the isotherm parameters are those of C10E4: v1 =6.7 ×109 cm2 mol − 1, v2 = 2.6× 109 cm2 mol − 1, a =2.2, c0 = 6 ×10 − 8 mol cm − 3, P 0 = 21.6 mN m − 1.

Fig. 2. Dimensionless surface pressure versus dimensionless time, for diffusion controlled and diffusion-orientation controlled adsorption with different characteristic time ratios. The isotherm parameters are the same as in Fig. 1.

3. Results and discussion For a discussion of the results of the presented models, as mentioned above, it is reasonable to assume x= (2+ 2)/4. The influence of x will be also discussed later on. Moreover, it is useful to consider values of the parameters v1, v2, b2 and a consistent with the experimental data obtained in Refs. [9,10] for some CiEj surfactants at the water/ air and water/hexane interface. Fig. 1 shows the time evolution of dimensionless G1 and G2 for increasing values of the ratio tD/tor, for values of the adsorption isotherm and bulk concentration which are typical of CiEj surfactants. As shown at short time, i.e. at low surface coverage, most of the adsorbed molecules are in state 1 until a maximum value is reached for G1. Then, as the coverage increases, the number of molecules adsorbed in state 2 increases, while those adsorbed in state 1 decreases, and the ratio between G1 and G2 asymptotically approaches the equilibrium relationship given by Eq. (5). The corresponding time evolutions of the dimensionless surface pressures are plotted in Fig. 2. From these two figures, coherently with the above discussion on the characteristic times, it is clear that the diffusion controlled model is approached by the diffusion-orientation controlled model as the tD/tor ratio increases. The influence of the ratio between the molar areas corresponding to the two states is shown in Figs. 3 and 4. The behaviour of the surface pressure strongly changes only as the ratio v2/v1 is larger than 3. Besides, this values of the ratio is typical for CiEj’s. The effect of the ratio v2/v1 is more pronounced for the diffusion-orientation controlled than for diffusion controlled adsorption model. It is worth to note that v2/v1 =1 corresponds to the Langmuir model. The plots in Fig. 5 show the effect of a on the time evolution of the dimensionless surface pressure. For given v1 and v2, a expresses the ratio between the surface activity of the two states, i.e. b2/b1. The values of v2, v1, and a can be easily obtained by fitting the isotherm (6) to the P-c equilibrium values. On the contrary it is only possible to give a rough estimation of the probability x, by mod-

F. Ra6era et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 175 (2000) 51–60

Fig. 3. Influence of the ratio between the two surface areas, on the dimensionless surface pressure versus dimensionless time. k12 = 0.2 (corresponding to tD/tor : 1) and the other isotherm parameters are the same as in Fig. 1.

57

face can be fairly described by the diffusion controlled model. For CiEj surfactants, the possibility of different orientations for adsorbed molecules can be explained by considering the partial hydrophobic character of the ethylene groups of the hydrophilic part of the surfactant molecule [17], which allows the molecules to be oriented parallel to the surface, as long as the surface coverage is small. Coherently, the surface activity of C10E8 molecules adsorbing in the state with larger molar area, compared to the other state, is larger at the water-hexane than at water-air interface [9]. In fact, in this latter case, the value of a is about half of that for water-hexane. This means that the adsorption of C10E8 molecules in the state with larger area is relatively more energetically favoured at the water–hexane than water–air interface. As a consequence, the hypothesis is reasonable that in the dynamic of adsorption of CiEj’s at water–oil interfaces, the characteristic time for the orientation process could be larger than for water–air and comparable with the diffusion characteristic time, particularly for large bulk concentrations. Data of adsorption dynamics of CiEj’s at the water-hexane interface are reported in Ref. [9]. There are indications that these data cannot be simply interpreted by a diffusion controlled model. On the other hand, the interpretation of

Fig. 4. The same as in Fig. 3 but for diffusion controlled adsorption. v1 = v2 corresponds to Langmuir isotherm.

elling the early stages of the adsorption of the molecules just arrived from the bulk. However, as shown in Fig. 6 the influence of this parameter on the time evolution of the surface pressure is rather weak. In Refs. [8 – 10] the two state model has been used to describe the adsorption of some CiEj surfactants. The model successfully predicts the equilibrium properties of such surfactants both at liquid–liquid and liquid – air interfaces. Moreover, as also confirmed by the data for C10E5 shown in Fig. 7, the adsorption dynamics at liquid-air inter-

Fig. 5. Influence of a on the dimensionless surface pressure versus dimensionless time. The other isotherm parameters are the same as in Fig. 1.

58

F. Ra6era et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 175 (2000) 51–60

Fig. 6. Influence of x on the dimensionless surface pressure versus dimensionless time. The other isotherm parameters are the same as in Fig. 1.

Fig. 7. Dynamic surface pressure during the adsorption of C10E5 at fresh water/air interface. Short time (empty symbols) and long time (black symbols) experimental data are acquired by the Dynamic Maximum Bubble pressure and by the Drop Shape techniques, respectively. c0 = 6× 10 − 8 mol cm − 3, , , and c0 =1 ×10 − 7 mol/cm3, , . Theoretical curves are by diffusion controlled adsorption, using D = 2.10 − 6 cm2 s − 1, v1 =7.0 ×109 cm2 mol − 1, v2 = 2.6× 109 cm2 mol − 1, a =2.6, and b2 = 1.1 × 108 cm3 mol − 1.

these data according to the diffusion-orientation controlled model presented here is rather difficult due to the exchange of surfactant between the two liquids and the finiteness of the bulk phases, which have to be explicitly taken into consideration. However, some experimental and numerical works are currently under way, finalised to under-

stand the dynamic behaviour of these surfactants at water–oil interfaces. Another case in which the orientation process can become important is when the adsorption dynamics is investigated in a comparable timescale. This is, for example the case of the study of the response of adsorption to high frequency perturbations of the surface area or the dynamic of adsorption in conditions of very fast bubbling. Both these phenomena have a great technological relevance, for example to understand emulsion dynamics and multi-phase flows. Therefore, the utilisation of methods like the oscillating bubble or dynamic maximum bubble pressure tensiometry can be a way to characterise the orientation phenomena. So far, the coexistence of two states for adsorbed molecules has been attributed to the molecular orientation. The investigation of adsorbed CiEj layers at the water–air interface by the neutron reflection [18,19] shows that the conformation of the adsorbed molecules and the structure of the layer changes as the coverage increases. In fact, while the hydrocarbon chain are nearly oriented perpendicularly to the interface, at low coverage the orientation distribution of the oxyethylene chains presents a maximum in correspondence of the orientation along the surface, shifting towards the normal orientation by increasing the coverage. The picture that can be drown out from these experiments is a continuous distribution of states. Therefore, the two statemodel gives a simplified description of the physics of CiEj monolayers, where the molar area corresponding to the maximum of the distribution is the average molecular area vS. However, the upgrade of the model [8,20] to an arbitrary number of states with different molar areas, though more realistic, does not improves significantly the prediction of the surface pressure behaviour, which shows the effectiveness of the assumed simplified picture. This effectiveness suggests other considerations. The model relies on the coexistence of adsorption states characterised by a given area and a given surface activity, without specifying the physical reason underlying these properties. Therefore, it is possible to figure out that such a model can be

F. Ra6era et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 175 (2000) 51–60

used as the basis for developing new descriptions of the adsorption kinetics of molecules undergoing conformational changes at the interface. Conformational changes of the adsorbed molecules, have been already supposed to exist for proteins and, in fact, a model similar to the two-states it has been successfully applied to describe the adsorption equilibrium properties [21]. It is reasonable to suppose an important role of conformational changes also in the adsorption of polymers and of long-chain surfactants, like for example, long chain CiEj’s and Triton surfactants. Indeed, as the size of these surfactants increases, the classical models are generally not adequate in describing the adsorption properties [12].

Acknowledgements This work was partially funded by the Italian Space Agency (contract CNR-ASI ARS-98-43), by the DLR (50 WM 9822) and by the European Space Agency with the support given to the Facility for Adsorption and Surface Tension (FAST) studies project.

Appendix A. Diffusion based approach to mixed kinetics [16]



dG 6d c− f(G) = g(G) dt 4 d

59

n

(A2)

where d is a length small enough to consider a linear concentration profile and large enough to define a volume concentration. In practice d is of the order of few mean free paths of a molecule under Brownian motion. The length d, defined in this way, allows the diffusion coefficient D to be written as [23] D=

6d 4

(A3)

c is the concentration close to the surface (sublayer concentration) which can be considered to be the concentration at distance d from the surface, c(d, t). Thus, assuming the concentration at the surface, c(0, t)= f(G) in the macroscopic limit Eq. (A2) becomes

 

(c dG = Dg(G) (x dt

(A4)

x=0

By solving the diffusion problem by using Eq. (A4) as boundary condition and c(0, t)= f(G), G(t) is given by

&

G

0

du = g(u)

'

D 2c0 t− p

&

t

0

f(G)(t)

t−t

dt

n

(A5)

The kinetic equation can be written as [22] dG =kag(G)[c− f(G)] dt

(A1)

where g and f are functions which can be defined on the basis of the adsorption model. For example, for the Langmuir model we have G G −G

g(G)=1− G/G and f(G) =a

kac is the number of molecules striking the unit surface per unit time, which, in the framework of a perfect gas model for the molecules in solution 1 kac= 6c. 4 Therefore, the kinetic equation can be written

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. .