Nonequilibrium thermodynamics—A tool to describe heterogeneous ...

PAPER

www.rsc.org/pccp | Physical Chemistry Chemical Physics

Nonequilibrium thermodynamics—A tool to describe heterogeneous catalysis Dick Bedeaux,bc Signe Kjelstrup,bc Lianjie Zhua and Ger J. M. Kopera Received 13th July 2006, Accepted 10th October 2006 First published as an Advance Article on the web 23rd October 2006 DOI: 10.1039/b610041d In the study of multi-component mass transfer it is common to use the film model, in which all the resistance to mass transfer towards a catalytic surface is assumed to be localized in a diffusion layer in front of the surface. At the surface one furthermore assumes that the temperature and chemical potentials are continuous, while the coupling of a possible heat flux to the mass fluxes is assumed to be negligible. Both these assumptions are questionable. Using nonequilibrium thermodynamics we discuss how to integrate the coupling between heat and mass fluxes in the description of the film. Furthermore, following Gibbs, we introduce the surface as a separate thermodynamic system where the coupling between the vectorial heat flux and the scalar reaction rate is allowed and can be significant in heterogeneous catalysis. Non-equilibrium thermodynamic theory for surfaces allows one to find the proper rate equations. It allows for a consistent and complete description of mass and heat transfer through the film and subsequently from the film to the surface where the reaction takes place. Fast endo- or exothermic surface reactions in heterogeneous catalysis may give significant temperature gradients between a catalyst surface and the media, which will, when not accounted for, lead to an incorrect evaluation of the activity, stability and selectivity of a catalyst. Non-equilibrium thermodynamics is a useful tool for predicting the surface temperature as well as for analyzing the system. In this contribution we sketch how to systematically set up the complete description, in which the film and the surface ‘‘sum up’’ to one effective surface.

1. Introduction In the study of multi-component mass transfer to and from a catalytic surface it is convenient to use the film model, see for instance ref. 1 and references therein. In this model all the resistance to mass transfer is assumed to be localized in a film between the surface and the well-mixed flow region. The reactants diffuse through this film to the surface, where they react, and the products diffuse back. Though there may be heat exchange with the catalyst carrier, we will always consider running conditions such that the temperature of the catalyst pellets is uniform and equal to the temperature of the surface. In that case there is no heat exchange with the catalyst carrier. Two assumptions are commonly made when applying the film model. In the first place, one usually assumes that the temperature and chemical potentials of the surface are equal to those in the adjacent bulk region. Secondly, when describing the heat and mass fluxes through the film one neglects the coupling of these fluxes and as a consequence, the film then has neither a Soret nor a Dufour effect. Both assumptions are questionable. The reaction enthalpy can be substantial, which a

Delft ChemTech, Delft University of Technology, Julianalaan 136, 2628 BL Delft, The Netherlands Department of Chemistry, Norwegian University of Science and Technology, 7491 Trondheim, Norway c Process and Energy Department, Delft University of Technology, Leeghwaterstr.44, 2628 CV Delft, The Netherlands b

This journal is

c

the Owner Societies 2006

results in large heat flows. Coupling effects may therefore be significant. In a recent paper2 some of us documented the importance of the Soret effect for catalytic hydrogen oxidation, H2 + (1/2)O2 - H2O. A systematic method of combining heat and mass transfer is provided by nonequilibrium thermodynamics.3,4 Following nonequilibrium thermodynamics, one may easily incorporate the coupling between heat and mass fluxes in the description. How to do this is in fact explained by Taylor and Krishna.1 Similarly one may, following Gibbs,5 introduce the surface as a separate thermodynamic system.6–9 The nonequilibrium thermodynamic theory for surfaces allows one to find proper rate equations in these heterogeneous systems. It allows for a consistent and complete description of catalytic reactions at surfaces far from equilibrium. As we shall show, the surface features as an ‘‘additional film’’. As it is often sufficient to use a thin film approximation in the bulk phase, which is most easy to implement, we shall follow this procedure. It makes the similarity between the description of the film and of the surface all the more apparent. One of the aims of this contribution is to sketch how to integrate the description of the film with that of the surface. In the second section we discuss the thermodynamic description of the surface. In the third section we use nonequilibrium thermodynamics for the surface and derive expressions for the temperature and chemical potential differences between the surface and the film and for the Gibbs energy of the catalytic Phys. Chem. Chem. Phys., 2006, 8, 5421–5427 | 5421

The surface excess internal energy density is

reaction at the surface in terms of the fluxes. In the fourth section we discuss the description of the film. In the fifth section we combine the film and the surface to one effective film, for which explicit expressions are given for the temperature and chemical potential differences across this whole region. Expressions are also given for the resistances involved, in terms of the resistances of the films and the surface. Concluding remarks regarding the practical use of the description are made in the last section.

and Gibbs-Duhem’s equation becomes

2. Thermodynamic variables for a surface

2.2.

We choose the x-axis perpendicular to the planar surface. The thermodynamic properties of the surface are given by the values of the excess mass and energy densities, which were defined by Gibbs.5 Following Gibbs we define the location of the dividing surface, which separates the homogeneous phases, such that the excess concentration of a reference component is zero. In the present case the logical choice of the reference component is the carrier material. The surface as described by the excess mass and energy densities can be regarded as a 2-D thermodynamic system with properties that are given per unit of surface area. The dependence on the coordinates y and z remains.

We have so far considered systems that are in global equilibrium, i.e. the temperature and the chemical potentials are constant throughout the whole system. When the system is not in global equilibrium we need to introduce the assumption of local equilibrium. For a surface element, we say that there is local equilibrium when the local thermodynamic relations (4)–(6) are valid in each point along the surface and at each moment in time. Similarly, local equilibrium implies that all the usual thermodynamic relations are valid locally. The intensive thermodynamic variables for the surface, indicated by superscript s, are therefore given by the derivatives: s s du du s Ts ¼ and m ¼ ð7Þ j dss Gj dGj ss ;Gk

2.1.

Local thermodynamic identities for the surface

When excess surface densities are defined in this manner, the normal thermodynamic relations, like the first and the second law and derived relations apply for the densities.5 The Gibbs equation for the total excess internal energy, Us, of the equilibrium surface becomes: dU s ¼ TdSs þ gdO þ

n X

mi dNis

ð1Þ

i¼1

where Ss, O and Nsi are the total excess entropy, the surface area and the total excess density of component i. Furthermore T, g and mi are the temperature, the surface tension and the chemical potential of component i, respectively. In view of the extensive nature of Us, Ss and Nsi we can use Euler’s theorem and obtain U s ¼ TS s þ gO þ

n X

mi Nis

ð2Þ

i¼1

Gibbs–Duhem’s equation for the surface follows by differentiation of this equation and subtracting eqn (1): 0 ¼ Ss dT þ Odg þ

n X

Nis dmi

ð3Þ

i¼1

For the surface, we need the local variables given per unit of surface area. These are the excess internal energy density us = Us/O the adsorptions Gi = Nsi /O and the excess entropy density, ss = Ss/O. When we introduce these variables into eqn (1), and use eqn (2), we obtain the Gibbs equation for the surface: dus ¼ Tdss þ

n X

mi dGi

i¼1

5422 | Phys. Chem. Chem. Phys., 2006, 8, 5421–5427

ð4Þ

us ¼ Tss þ g þ

n X

mi Gi

ð5Þ

i¼1

0 ¼ ss dT þ dg þ

n X

Gi dmi

ð6Þ

i¼1

Definition of local equilibrium for the surface

The temperature and chemical potentials, defined in this manner, depend only on the surface excess variables, not on the value of bulk variables close to the surface. By introducing these definitions we therefore allow for the possibility that the surface has a different temperature and/or chemical potentials from the adjacent homogeneous systems. Molecular dynamics simulations support the validity of the assumption of local equilibrium for surfaces.10,11 The assumption of local equilibrium, as formulated above, does not imply that there is local chemical equilibrium.12 In that case the Gibbs energy of the reaction is also zero. The thermodynamic variables for the surface depend on the position along the surface and the time. As we shall not consider transport along the surface, we shall further restrict ourselves to cases where the variables are independent of y and z. Also we shall restrict ourselves to stationary states. This implies that all the excess densities have time independent values. This simplifies the description rather considerably. An essential and surprising aspect of the local equilibrium assumption for the surface and the adjacent homogeneous phases is the fact that the temperature and chemical potentials on both sides of the surface may differ not only from each other, but also from the values found for the surface, see Fig. 1.

3. The excess entropy production rate for the surface Consider a surface s between two phases i and o. We take the origin of the x-axis to coincide with the surface s. The phase i is located on the left of the surface, x o 0, and the phase o is located on the right of the surface, x 4 0. In our case i is the phase in which the diffusion takes place while o is the carrier This journal is

c


minus the total heat flux out of the surface to the right, Jo,i q . Both the molar and the heat fluxes in the above equations should be taken in a frame of reference in which the surface is at rest. These heat fluxes are related to the measurable heat fluxes by X 0 0 Jqi ¼ Jqi þ hij Jji and Jqo ¼ Jqo ð11Þ j

where hij is the partial enthalpy density of component j in the i-phase. The measurable heat fluxes are independent of the frame of ref. 12. 3.2. Fig. 1 Schematic diagram of the temperature variation in heterogeneous exothermic catalytic reactions, which can be modeled by nonequilibrium thermodynamics.

phase. The change of the entropy in a surface area element is a result of the flow of entropy in and out of the surface element, and of the entropy production rate inside: d s s ¼ Jsi;o Jso;i þ ss dt

ð8Þ

where Ji,o s is the asymptotic value of the entropy flux in the adjacent phase i left of the surface and into the surface, and Ji,o s is similarly the entropy flux in the phase o to the right of the surface and out of the surface. All fluxes are taken in a frame of reference in which the surface is at rest (the surface frame of reference). The first (or the only) roman superscript gives the phase, i, s or o in this case. The second superscript, o or i, indicates a value close to phase o or i. The combination i, o means therefore the value in phase i as close as possible to the o-phase at the interface. The excess entropy production rate is ss Z 0. We shall find explicit expressions for ss by combining: mass balances the first law of thermodynamics the local form of the Gibbs equation. In the derivation we follow ref. 6–8 We shall see that ss can be written as the product sum of thermodynamic fluxes and forces in the system. These are the conjugate fluxes and forces for the surface. 3.1.

Balance equations

The balance equation for the excess molar density of component j is: d Gj ¼ Jji;o þ n j rs dt

ð9Þ

c


n dss 1 dus 1 X dGj ms ¼ s s dt T dt T j¼1 j dt

ð12Þ

By introducing eqns (9) and (10) into eqn (12), and comparing the result to the entropy balance eqn (8), we find the excess entropy production rate in the surface frame of reference: 1 1 1 1 o;i ss ¼Jqi;o þ J q T s T i;o T o;i T s " !# ð13Þ n X msj mi;o D r Gs j i;o s þ Ji þ r T s T i;o Ts j¼1 where DrGs = surface. 3.3.

P s njmj is the reaction Gibbs energy for the

Stationary states

We will restrict the further analysis to stationary states. In that case the pellets carrying the catalyst have a uniform temperature equal to the temperature of the surface, To,i = Ts. This implies that 0

Jqo;i ¼ Jqo;i ¼ 0

ð14Þ

The excess entropy production simplifies to " !# X n msj mi;o 1 1 Dr Gs j i;o s i;o s þ r þ J s ¼Jq j T s T i;o T s T i;o Ts j¼1 X n h m i 1 Dr Gs j þ Jji;o Di;s þ rs s T T T j¼1 ð15Þ

ð10Þ

The change of the excess internal energy density of the surface is given by the total heat flux into the surface from the left, Ji,o q , This journal is

The time derivative of the entropy density is given by the Gibbs equation in its local form:

¼Jqi;o Di;s

is the molar flux into the surface and nj the where Ji,o j stochiometric coefficient of component j, while rs is the reaction rate per unit of surface area. The stochiometric coefficients are taken negative for the reactants and positive for the products. For simplicity we only consider one reaction. The first law of thermodynamics for the surface is: dus ¼ Jqi;o Jqo;i dt

The excess entropy production rate

In the last equality we introduced a short hand notation for the difference of a variable. For stationary states one furthermore finds that the molar fluxes, Jj, are constant throughout the diffusion layer while the total heat flux, Jq, is constant everywhere. Using the measurable heat flux J 0 iq, defined by eqn (11) and the thermodynamic identity @(mj/T)/@(1/T) = hj, one obtains after some algebra X n Di;s mj;T ðT s Þ 1 Dr Gs 0 s ss ¼ Jqi Di;s Jj þ r ð16Þ þ Ts Ts T j¼1 Phys. Chem. Chem. Phys., 2006, 8, 5421–5427 | 5423

The subscript T implies that the difference is calculated at a constant temperature, which in this case is given by Ts. 3.4.

Linear force-flux relations for the surface

The excess entropy production given in eqn (15) results in the linear relations n X 1 ¼rsee Jq þ rsek Jk þ rser rs Di;s T k¼1 Di;s

m j

T

¼rsje Jq þ

n X

s;e s rs;e jk Jk þ rjr r

ð17Þ

k¼1

n Di;s mj;T ðT s Þ s 0 i X s;q s ¼rjq Jq þ rs;q jk Jk þ rjr r s T k¼1

ð18Þ

Eji;o Jj with T i;o ¼ T s and rs ¼ 0

j¼1

Jq0i;o

¼

n X

ð20Þ Qi;o j Jj

with T

s;q s s s s rsee ¼rsqq ; rs;e rr ¼ rrr ; rre ¼ rer ¼ rrq ¼ rqr

i;o

s

s

¼ T and r ¼ 0

j¼1

E*i,o and Q*i,o are the total and measurable heats of transfer. j j Comparing this equation to eqns (11), (17), (18) and (19), it follows that rsej ¼ Qi;o þ hi;o j j rsee

Qi;o ¼ j

rsqj rsqq

ð21Þ

Using that the measurable heat flux is invariant when one changes the velocity of the frame of reference,12 it follows that n X

i;o xi;o ¼ hi;o and j Ej

j¼1

n Dr Gs s 0 i X s;q s ¼r J þ rs;q rq q rk Jk þ rrr r Ts k¼1

s;q s;e s;e o;i s s rsek ¼rske ¼ rsqk ho;i k rqq ; rrk ¼ rkr ¼ rrk hk rrq

n X

Eji;o ¼

where the resistance matrices satisfy the Onsager symmetry relations. The resistances in these matrices are related by

n X

i;o ci;o ¼0 j Qj

ð22Þ

j¼1

i,o i,o and ci,o where xi,o j cj /c j are the average mole fractions and molar concentrations and hi,o the enthalpy density in the i-phase. All these quantities of transfer are the same in the bulk when one uses a frame of reference in which the surface is at rest. One may therefore use the same values as those found in the next section, see e.g. eqn (38).

ð19Þ

s;q s;e o;i s o;i s o;i o;i s rs;e jk ¼rkj ¼ rjk hk rjq hj rqk þ hk hj rqq

This follows using eqn (11) and the thermodynamic identity @(mj/T)/@(1/T) = hj. Expressions for rsqq, rsqk = rskq, rsjk have been found for the liquid–vapor interface using kinetic theory.14–17 For the diagonal coefficient due to the reaction one can use a form typical for reaction rate theory. Molecular dynamics simulations for a one-component liquid–vapor interface seem to agree with kinetic theory for sufficiently short range interaction potentials.18 For longer range potentials19 and for binary mixtures20 molecular dynamics simulations indicate that the kinetic theory values are too small. For the off-diagonal coefficients due to the reaction nothing is known. Much work remains to be done to find reliable coefficients. On the basis of eqn (18) one of us attempted to calculate the resistance coefficients for a CO oxidation catalytic system and found that the coupling between the heat flow and the reaction rate in the 2-D reaction surface is significant and responsible for the surface temperature excess.25 By evaluating the reaction resistance coefficients rrr for the same reaction but using different catalysts, we are able to judge which catalyst is more efficient and therefore it is helpful for catalyst design. 3.5.

Jq ¼

n X Dr Gs s s;e s ¼rre Jq þ rs;e rk Jk þ rrr r s T k¼1

while the excess entropy production given in eqn (16) results in the linear relations n X 1 0 ¼rsqq Jqi þ Di;s rsqk Jk þ rsqr rs T k¼1

for the surface. For the total and measurable heat fluxes, when the temperature difference to the surface is zero and the reaction is in equilibrium, we can write

4. The film model In the usual studies of mass and heat transport to a surface, one uses the film model. The coupling of mass and heat flux is usually neglected, however. The derivation of the entropy production in the films has been discussed in many places, see for instance, ref. 1 and 21 and we just give the result in the form that is most convenient for our present purpose, see also:13 X n @ 1 @ mj si ¼ Jq Jj ð23Þ þ @x T @x T j¼1 The expression in the o-phase only contains the first term. It will not be needed to consider this phase in our further discussion of stationary states. For stationary states the total heat flux, Jq, and the molar fluxes, Jj, are constant. We already used this property in our analysis of the surface above. This makes it possible to integrate the entropy production across the film, which results in X n m 1 j sf ¼ Jq Df ð24Þ Jj Df þ T T j¼1

Quantities of transfer

It is common to introduce quantities of transfer in the description of transport in mixtures.12 This can be done also 5424 | Phys. Chem. Chem. Phys., 2006, 8, 5421–5427

where Df(. . .) gives the difference of the quantity between brackets across the film (the value on the right hand side This journal is

c


minus the value on the left hand side). The resulting linear laws are n X 1 ¼rfee Jq þ rfek Jk Df T k¼1 ð25Þ n m X j f;e f ¼rje Jq þ Df rjk Jk T k¼1 where the resistance matrix satisfies the Onsager symmetry relations. Using eqn (23) one may similarly give a continuous description, in which the gradients of 1/T and m/T are expressed in the energy and mass fluxes. It follows that the resistances in eqn (25) are the integrals of the resistances in the continuous description across the film. It is important to realize that the validity of eqn (25) is not restricted to the thin film case, which we will use below. In order to express the resistances in terms of coefficients familiar from the film theory, we replace the total energy flux by the measurable heat flux on the right-hand side of the film, J0fr q , by introducing eqn (11) into eqn (24) the result is the alternative expression: X n mj 1 1 0 þ ð26Þ þ hfj Df Jj Df sf ¼ Jqfr Df T T T j¼1 Here hfj is the average specific enthalpies in the film. In writing eqn (26) we assumed that the film thickness is small so that the variation of the specific enthalpies across the box is negligible. The film thicknesses are between 0.1 and 1.0 mm in a vapor film.1 In practice a choice is made that fits the data. The thin film approximation made above implies the assumption that this film thickness is small compared to the length scales which follow from the diffusion of mass, which are given by the diffusion coefficients divided by the center of mass velocity, and the length scale which follows from the diffusion of heat, which is given by the thermal conductivity divided by the heat capacity at constant pressure times the center of mass velocity. We will not go into the issue which film thickness to use any further. We only wanted to clarify what the assumption that a film is thin exactly means. Using the thin film approximation, one may write eqn (26) in the form ! X n Df mj;T T fl 1 0 sf ¼ Jqfr Df þ ð27Þ Jj T T fl j¼1 is the heat flux at the right end of the box where J0fr q and Dfmj,T(Tfl) is the difference of the chemical potential across the box at the constant temperature Tfl, the temperature at the left end of the box, which is equal to the temperature of the mixed fluid region. For cases when the film is too thick for this approximation we refer to Taylor and Krishna.1 The linear laws which follow from eqn (27) are n X 1 0 Df rfqk Jk ¼rfqq Jqfr þ T k¼1 ð28Þ n Df mj;T ðT fl Þ f 0 fr X f ¼rjq Jq þ rjk Jk T fl k¼1 This journal is

c


The relation between the resistances in eqns (25) and (28) is rfee ¼rfqq rfek ¼rfke ¼ rfqk hfk rfqq

ð29Þ

f;e f f f f f f f f rf;e jk ¼rkj ¼ rjk hk rjq hj rqk þ hk hj rqq

where the enthalpies are the average values in the film. 4.1.

Extended Maxwell–Stefan equations

In recent years the Maxwell–Stefan equations have become the favourite way to describe diffusion in a multi-component system. Traditionally the coupling of the mass fluxes to the heat flux, leading to the Soret and the Dufour effect, is neglected. Extensions of the Maxwell–Stefan equations to include these effects have been given. We will now give the relevant expressions, to clarify the relation between the resistances above and the Maxwell–Stefan diffusion coefficients. First we rewrite eqn (28) in the alternative form X n rf 1 1 0 fr qk Jq ¼ f Df Jk f rqq T r k¼1 qq ð30Þ " # X n rfjq rfqk Df mj;T ðT fl Þ rfjq 1 f Jk ¼ f Df rjk f þ T fl rqq rqq T k¼1 There is a heat flux due to the mass fluxes across the film, the Dufour effect. The total and measurable heats of transfer of the film are defined in the usual way, similar to the definitions given above. They are given (see eqn (21)) by rfej f ¼ Qf j þ hj rfee

Ejf ¼

Qf j ¼

and

rfqj rfqq

ð31Þ

and satisfy (see eqn (22)) n X

xfj Ejf ¼ hf and

j¼1

n X

cfj Qf j ¼0

ð32Þ

j¼1

where xfj cfj /cf and cfj are the average mole fractions and molar concentrations in the film, see ref. 12, pages 281–284. Furthermore we define lf

df

and Rfjk rfjk

rfqq T fl T fr

rfjq rfqk rfqq

ð33Þ

Here lf is (the harmonic average of) the heat conductivity of the film for given mass fluxes and df is the thickness of the film. Using these parameters eqn (30) becomes 0

Jqfr ¼

lf

Df T þ df

n X

Qf k Jk

k¼1

X n Df mj;T ðT Þ 1 f ¼ Q D Rfjk Jk þ f j T fl T k¼1 fl

ð34Þ

Now we use the thermodynamic property n X

cfj Df mj;T ðT fl Þ ¼ 0

ð35Þ

j¼1

Not all the thermodynamic forces are as a consequence of eqn (35) independent. This results, using also eqn (32b), Phys. Chem. Chem. Phys., 2006, 8, 5421–5427 | 5425

in the following property of the coefficients in eqn (34) n X j¼1

cfj Qf j ¼ 0 and

n X

cfj Rfjk ¼

j¼1

n X

cfj Rfkj ¼ 0

ð36Þ

j¼1

two sequenced surfaces. Below we shall discuss how to combine these two surfaces into ‘‘one effective surface’’. In order to calculate the total differences of 1/T and mj/T across both films and the surface we write

Here we also used the Onsager symmetry relations. The Maxwell–Stefan diffusion coefficients, Djk, are now defined by

1 1 1 D Df þ Di;s T T T m m m j j j Df þ Di;s D T T T

f

Rfjk

dR for j 6¼ k cf Djk

ð37Þ

The diagonal coefficients are found using eqn (36). The Maxwell–Stefan diffusion coefficients are symmetric, Djk = Dkj. The heats of transfer can be written in terms of the thermal diffusion coefficients, DTj, such that they satisfy eqn (36), as ! n X RT fl xfk DTj DTk f Qj ð38Þ f Djk rfj rk k¼1 Here rfj cfjMj is the average mass density in kg m3 and Mj the molar weight in kg mol1 of component j. Using eqns (36)–(38) we can write the linear laws (34) in the following form ! n X cf RT fl xfk xf‘ DTj DTk f cfj Df mj;T ðT fl Þ lf 0 fr Jq ¼ f Df T þ f v‘ Djk rfj rk df T fl d k;‘¼1 ! T n cf RT fl xf xf n cf Rxf xf X X DTk 1 j k Dj j k f D ¼ ðvk vfj Þ f Djk T rfj rfk df Djk k¼1 k¼1 ð39Þ where vfj Jj/cfj is the average velocity of the j-th component in the film. For the Maxwell–Stefan and the thermal diffusion coefficient one should also use average values. The Maxwell–Stefan diffusion coefficients are reasonably well known.1,21 The heats of transfer are not so well known. Approximate expressions have been given by Kempers22 and by Haase23 for binary mixtures. Given these coefficients and the heat conductivity one can first calculate the resistances rfqq, rfqk = rfjq, rfjk and then the resistances rfee, rfek = rfje, rf,e jk for the film. For an analysis of the influence of these coefficients we refer to Jenkinson and Pollard.24 In a recent paper2 we documented the importance of the Soret effect for catalytic hydrogen oxydation, H2 + (1/2)O2 - H2O. Equations similar to eqn (39), in which Df/df is replaced by a gradient and without the Latin superscripts, are given by Taylor and Krishna,1 page 268, in the context of the continuous description. In that case one may, analogously to the analysis given above, obtain expressions for the resistances ree, rek = rje, rejk. When the film is not thin, this is the appropriate procedure. As pointed out in the first remark in this section, the resistances rfee, rfek = rfje, rf,e jk should then be obtained by integrating ree, rek = rje, rejk across the cell.

5. Combining the films with the surface A very important feature of the description of the film in the fourth section is its great analogy with the description of the surface in the third section. The film and the surface are like 5426 | Phys. Chem. Chem. Phys., 2006, 8, 5421–5427

ð40Þ

Using eqn (17) for the surface and (25) for the film we find n X 1 1 1 ðrfek þ rsek Þ Jk þ rser rs ¼ s fl ¼ ðrfee þ rsee Þ Jq þ D T T T k¼1 ree Jq þ

n X

rek Jk þ rser rs D

k¼1

¼

m j

T

n X msj mflj s;e s;e s þ fl ¼ ðrfje þ rsje Þ Jq þ ðrf;e jk þ rjk Þ Jk þ rjr r s T T k¼1

rje Jq þ

n X

s rejk Jk þ rs;e jr r

k¼1

¼rsre Jq þ

n X

Dr Gs Ts

s;e s rs;e rk Jk þ rrr r

k¼1

ð41Þ These relations make it possible to replace the films and the surface by an ‘‘effective surface’’. One may, similar to the procedure above, eliminate the total heat flux Jq in favor of the measurable heat flux at either one end. We will not do this as it serves no real purpose.

6. Conclusions As shown, it is straightforward to integrate the contributions due to the surface into the usual film theory. It is therefore advantageous to use the full theory, which follows from and is compatible with nonequilibrium thermodynamics. This makes a systematic description of catalytic transitions on surfaces far from equilibrium possible, and relatively easy to integrate in already existing routines. The linear laws given in eqn (41) make it possible to obtain combinations of the various resistances from measurements in which fluxes, the reaction rates as well as the chemical potentials and temperatures of the surface are known. It is not straightforward to obtain these values for the surface. In the thesis of one of us25 a method is described to obtain the surface temperatures from the reaction rate using the Arrhenius equation.26 For both transport limited and kinetically controlled catalytic reactions, coupling effects are found to be significant25 and influence considerably the catalyst surface temperatures by using non-equilibrium thermodynamics. Therefore, non-equilibrium thermodynamics is a convenient tool to model a heterogeneous catalytic system including the coupling effects both in the diffusion boundary layer and the 2-D reaction surface. This journal is

c


References 1 R. Taylor and R. Krishna, Multicomponent Mass Transfer, Wiley, New York, 1993. 2 L. Zhu, G. J. M. Koper and D. Bedeaux, J. Phys. Chem. A, 2006, 110, 4080–4088. 3 D. Bedeaux and S. Kjelstrup, Chem. Eng. Sci., 2004, 59, 109–118. 4 D. Bedeaux and S. Kjelstrup, Int. J. Thermodyn., 2005, 8, 25–41. 5 J. W. Gibbs, The Scientific Papers of J. W. Gibbs, Dover, London, 1961. 6 D. Bedeaux, A. M. Albano and P. Mazur, Physica A, 1976, 82, 438–462. 7 D. Bedeaux, Adv. Chem. Phys., 1986, 64, 47–109. 8 A. M. Albano and D. Bedeaux, Physica A, 1987, 147, 407–435. 9 S. Kjelstrup and G. M. de Koeijer, Chem. Eng. Sci., 2003, 58, 1147–1161. 10 A. Røsjorde, D. W. Fossmo, S. Kjelstrup, D. Bedeaux, B. Hafskjold and J. Colloid, Interface Sci., 2000, 232, 178–185. 11 J. M. Simon, S. Kjelstrup, D. Bedeaux and B. Hafskjold, J. Phys. Chem. B, 2004, 108, 7186–7195. 12 S. R. de Groot and P. Mazur, Non-Equilibrium Thermodynamics, North-Holland Pub. Co., Amsterdam, 1962; reprinted by Dover, London, 1984. 13 S. Kjelstrup and D. Bedeaux, Elements of Irreversible Thermodynamics for Engineers, Int. Centre of Applied Thermodynamics,

This journal is

c


14 15 16 17 18 19 20 21 22 23 24 25 26

Instanbul, Turkey, 2001; reprinted by Tapir Academic Press, Trondheim, Norway, 2006. (a) Y. P. Pao, Phys. Fluids, 1971, 14, 306–312; (b) Y. P. Pao, Phys. Fluids, 1971, 1340–1346. J. W. Cipolla Jr, H. Lang and S. K. Loyalka, J. Chem. Phys., 1974, 61, 69–77. D. Bedeaux, L. F. J. Hermans and T. Ytrehus, Physica A, 1990, 169, 263–280. D. Bedeaux, J. A. M. Smit, L. F. J. Hermans and T. Ytrehus, Physica A, 1992, 182, 388–418. A. Røsjorde, S. Kjelstrup, D. Bedeaux and B. Hafskjold, J. Colloid Interface Sci., 2001, 240, 355–364. S. Kjelstrup, T. Tsuruta and D. Bedeaux, J. Colloid Interface Sci., 2002, 256, 451–461. M.-L. Olivier, J.-D. Rollier and S. Kjelstrup, Colloids Surf., A, 2002, 210, 199–222. G. D. C. Kuiken, Thermodynamics for irreversible processes, Wiley, Chichester, 1994. L. J. T. M. Kempers, J. Chem. Phys., 1989, 90, 6541. R. Haase, Thermodynamics of Irreversible Processes, Dover, London, 1990. J. P. Jenkinson and R. Pollard, J. Electrochem. Soc., 1984, 131, 2911. L. Zhu, Surface temperature excess in heterogeneous catalysis, PhD thesis, Delft University of Technology, 2005. L. Zhu and G. Frens, Indications for a Surface Temperature Excess in Heterogeneous Catalysis, J. Phys. Chem. B, 2006, 110, 18307.

Phys. Chem. Chem. Phys., 2006, 8, 5421–5427 | 5427