Stationary diffusion in the membrane systems with the ongoing reversible chemical reactions

Journal of Molecular Liquids 120 (2005) 71 – 74 www.elsevier.com/locate/molliq

Stationary diffusion in the membrane systems with the ongoing reversible chemical reactions K.V. Cherevko*, D.A. Gavryushenko, J.V. Kulyk, V.M. Sysoev Taras Shevchenko Kyiv National University, 2(1) Prosp. Glushkova, Kyiv 03127, Ukraine Available online 11 September 2004

Abstract The diffusion phenomena were analyzed using the phenomenological equations of the thermodynamics of irreversible processes. The diffusion coefficient was thought to be dependent on local concentrations and pressure, unlike it was done in the linear theories. The reversible chemical reactions were modeled as intermolecular interaction. The ideal and regular solutions and solutions, described by the Margules’s and Sketchard–Hammer’s equations, were investigated and analytical solutions were found. D 2004 Elsevier B.V. All rights reserved. Keywords: Stationary diffusion; Membrane sysltems; Reversible chemical reaction

1. Introduction Nowadays, there exist quite a few membrane systems in which diffusion processes play a very important role. There exists a theory that describes the diffusion in such systems that is based on the investigation of the chemical reactions of the substrate that takes part in the diffusion and the substances that are in the membrane itself [1]. This theory deals with the chemical reactions between those substances so many initial parameters are to be obtained to get the exact results (the reactions velocities, saturation function). This theory also does not take into consideration osmotic border conditions that exist in such systems and uses the particular case of the diffusion equation that has a constant diffusion coefficient. The diffusion coefficient is thought to be constant in the linear theories that exist for the description

of the diffusion phenomena, and the flow of the substrate is derived from Fick’s law, where diffusion coefficient is constant. In such a way, it is possible to obtain the standard diffusion equation [2]. However, in general, it does not correspond to the number of experiments. One of the experimental investigations that shows the dependence of the diffusion coefficient on the local concentrations and pressure as well as the nonlinear type of the flow was carried out by Bulavin et al. [3]. Therefore, the idea to find out the exact solution for the diffusion process using the physical properties of the membrane systems that gives the possibility to treat reversible chemical reactions as the intermolecular interaction and to take into consideration the dependence of the diffusion coefficient on the local concentrations and pressure, seems to be very important.

2. Model description The stationary diffusion of the substance of type one in the m-component solution which generates a membrane of the thickness l is investigated in this work. The borders of the membrane are permeable for the substance of the first type only. Thus, there is a flow of the first substance and all the other flows across the border are equal to zero. The problem that is * Corresponding author. E-mail address: [email protected] (K.V. Cherevko). 0167-7322/$ - see front matter D 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.molliq.2004.07.038

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described in the article is one-dimensional and the flow is perpendicular to the membrane. Stationary diffusion process is investigated in the absence of external fields and under constant temperature T. The fact that the process is stationary gives the possibility to eliminate the second spatial derivative in the diffusion equation in the correct way. In the same time, there still is a possibility to find the dependence of the substrate flow on the concentration difference. Since the membrane has the boundaries that are transparent for the substance of the first type only, the osmotic phenomena are to appear in such a system; hence, the change of the chemical potential with the change of the pressure P is taken into consideration. Those phenomena allow to explain zero flow of the substances in the membrane while the gradient of the concentration exists. Therefore, the chemical potential is thought to be dependent on the concentration and the pressure. The system that is built in the article is based on the fundamental phenomenological equations of the irreversible processes thermodynamics [4] that represent the linear relations of the flows and the thermodynamical forces that caused them: m 1   X Y Y lj Ji ¼  Lij d j T j¼1 where J i —flows, L ij —phenomenological coefficients and l j —chemical potential of the j component. As cross-effects are not taken into consideration, it is possible to change the phenomenological coefficients L ij for L i . After using the concentration n, we can obtain such a system of differential equations that describes the diffusion process:

f

Y J 1 ¼  LT1 d



Bl1 n1





Bl2 n1

dY j ð n2 Þ þ





Bl1 Bn2



Bl2 Bn2

0

L2 ¼  d T

0

¼ 

0

Y Y Y ¼ j ðn1 Þ þ j ðn2 Þ þ : : : þ j ðnm Þ

Lm d T



Blm n1

dY j ðn2 Þ þ



dY j ðn2 Þ þ





dY j ð n2 Þ þ : : : þ



Blm Bn2

dY j ð n2 Þ þ : : : þ







dY j ð n2 Þ þ : : : þ

Bl1 Bnm

Bl2 Bnm







Blm Bnm

dY j ð nm Þ þ



Bl1 Bp



dY jP



   Bl2 Y Y d j ðnm Þ Bp d j P



dY j ð nm Þ þ



Blm Bp



dY jP



2.1. Chemical potential definition It is necessary to know the explicit definition for the chemical potential in order to find the substrate flow from that system of equations. The ideal, regular Margules’s and Sketchard–Hammer’s solutions are investigated in the article. All the equations that give the definition of the chemical potential contain phenomenological constants. In the same time, it is possible to find them using perturbation theory results. That theory gives the equations for the activity coefficients c i for the non-valency molecules and far from the stability border. First-order approximation gives the regular solution model [5]: ln c1 ¼

n2 ð2U12  U11  U22 Þ; 2

ln c2 ¼

n1 ð2U12  U11  U22 Þ 2

where Uts ¼

Z

i  h drij gˆ 20 rij ; T ; hV ð P; T ; N Þi0 ebwts ðrij Þ  1 v1 0 ð P; T Þ;

hV ðP;T ;N Þi0

1 kb T g 20 is radial distribution function of the initial system, c—activity coefficient, r ij —radius vector between the particles i and j, N—number of particles. The molecular interaction potential w ts can describe the reversible chemical reactions. If we take 2a=(2U 12U 11U 22), then we get the regular solution model c i =(an2j /2). Second-order approximation gives the results for the Margules’s solution   1 lnc1 ¼ n22 U12  ðU11 þ U22 Þ  3U111 þ 2U112  U122 þ 2n32 ðU111  U222  U112 þ U122 Þ 2 b¼

K.V. Cherevko et al. / Journal of Molecular Liquids 120 (2005) 71–74

73

  1 lnc2 ¼ n21 U12  ðU11 þ U22 Þ  3U222 þ 2U122  U112 þ 2n31 ðU222  U111  U122 þ U112 Þ 2 a1 ¼ 2U12  U11  U22  6U111 þ 4U112  2U122 ;

lnc1 ¼

a1 2 a2 3 n þ n2 ; 2 2 3

lnc2 ¼

a2 ¼ 6ðU111 þ U122  U112  U222 Þ

a1 þ a2 2 a2 3 n1  n1 2 3

In this case, U ijk is a function that is derived similarly to the U ij in the first-order approximation. It is not shown here because the equation is quite long. 2.2. The link of the diffusion and phenomenological coefficients The detailed analysis of the system is carried out for the case of binary solution. In that case, it is possible to derive one equation from the system that defines the substrate flow. The equation that links the phenomenological coefficient with the diffusion coefficient D can be derived from the comparison of the obtained results with the generalized equation for the substance flow. 0 D¼



L B B Bl1  Bl1 d T @ Bn1 Bn2



 1 Blnc1   C BP T Bl2 Bl C   d þ  2 Blnc Bn2 Bn1 T ;P A 2 T ;P v02 þ RT BP T



v01 þ RT

Since the chemical potential is dependent on local concentrations and pressure, the diffusion coefficient is not constant according to the obtained equation. That result is in good agreement with the experimental data.

3. Results for the flow dependence on concentration in binary solutions Solutions for the different types of binary solutions were found within the above-described model. The analytical expressions for the flow dependence on the concentration were derived. The general equation for the chemical potential was used as starting point for all the solutions [6].

flow dependence is (Fig. 1) Jn x ¼  ðJln þ J1stn þ J2stn þ J3stn þ Jatln1 þ Jatln2 Þ þ Const   Jln ¼ 2lnðn1 Þ  2Alnð1  n1 Þ þ 2ðW þ E1 Þlnn1  l     N V02 n1 þ  þ ln 2  V01 þ V02  ðV01  V02 Þ

li ¼ li0 ð P; T Þ þ RT lnðc1 ni Þ Analytical results were obtained for flow dependence on concentration in ideal and regular binary solutions and in binary solutions where activity coefficients were described by Margules’s and Sketchard–Hammer’s equations. 3.1. Sketchard–Hammer’s solution  V01 lnðc1 Þ ¼ ð1  u1 Þ C þ 2u1 D C V02    V02 2 lnðc2 Þ ¼ ð1  u2 Þ D þ 2u2 C D V01 2





Fig. 1. Flow dependence for the Sketchard–Hamer’s binary solution.

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K.V. Cherevko et al. / Journal of Molecular Liquids 120 (2005) 71–74

½

J1stn ¼

"

OðV01  V02 Þ  N V02 þ 4 ðV01  V02 Þ ðV01  V02 Þ K

2

n1 þ

J2stn ¼ 

½

"



J3stn ¼ 

½

V02 V01  V02



S

2ðV01  V02 Þ4 !2 1 02 n1 þ V01VV 02 "

#

þ

LðV01  V02 Þ  2KV02 2ðV01  V02 Þ5



2 M ðV01  V02 Þ þ KV02  LðV01  V02 ÞV02

3ðV01  V02 Þ6 # !3 T ðV01  V02 Þ  V02 S 1 þ 02 n1 þ V01VþV 3ðV01  V02 Þ5 02

Jatln1 ¼

#



  3  P þ F1  2 V2 V02  lnY2 n1 þ ðZ2 þ lY2 Þn1  Y2 l 

2ðG1 þ QÞY2  ðF1 þ PÞðZ2 þ lY2 Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  ffi 3 V02 V02 Z2 þ lY2 2 Y2  Y2  l 2 2 3 Z2 þ lY2 6 7 Y2 n1 þ 6 7 2 7 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi s arctan6 6 7   3 4 V2 V02 Z2 þ lY2 2 5   Y2 l 2

Jatln2 ¼

Fig. 2. Flow dependence for the different types of binary solutions and for the model with constant diffusion coefficient.

the Marry’s theory. But the described approach allows not to investigate the ongoing chemical reactions but to model them through the intermolecular interaction. Moreover, it allows to get the equations with only first derivatives without making it less general; therefore, it is possible to use it for the wide range of different membrane systems. Moreover, in Marry’s theory, the second derivative in the general diffusion equation was neglected. Obtained results are in a good agreement with the Wittenberg experimental data for the oxygen diffusion in the mioglobin and hemoglobin solutions. It is possible to tell that taking into consideration the dependence of the diffusion coefficient on the local concentrations and pressure is essential when investigating real membrane systems. It is possible to use the described method for the systems with the higher number of components and for the different models of the solutions.

4. Conclusions and discussion The above-described model describes the nonlinear character of the substrate flow dependence on the boundary concentrations difference. Definite stabilizing and amplifying effects can be seen in comparison with the results for the constant diffusion coefficient. The substrate flow is almost the same for quite a wide range of the concentrations. This result is in good agreement with the real membrane systems. It is important to emphasize that the obtained results are completely different from those obtained with the constant diffusion coefficient (Fig. 2). The type of the obtained curves for the substrate general flow is the same as for the curves that can be obtained from

References [1] J.D. Murray, Lectures on nonlinear-differential-equations. Models in Biology, Clarendon, Oxford, 1977. [2] J.O. Hirschfelder, Ch.F. Curtiss, R.B. Bird, Molecular Theory of Gases and Liquids, Chapman & Hall, Lim., London, 1954. [3] L.A. Bulavin, P.G. Ivanitsky, V.T. Krotenko, G.N. Liaskovskaia, Russian Journal of Physical Chemistry 61 (Suppl. 12) (1987). [4] M.A. Zakharov, Russian Journal of Physical Chemistry 75 (Suppl. 12) (2001) 2189. [5] V.M. Sysoev, I.A. Fakhretdinov, S.G. Shpyrko, Russian Journal of Physical Chemistry 71 (Suppl. 12) (1997) 2142. [6] V.M. Glazov, L.M. Pavlova, Russian Journal of Physical Chemistry 75 (Suppl. 4) (2001) 644.