Ind. Eng. Chem. Res. 2006, 45, 5361-5367
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Onsager Reciprocal Relations for Stefan-Maxwell Diffusion Charles W. Monroe† and John Newman* EnVironmental Energy Technologies DiVision, Lawrence Berkeley National Laboratory, and Department of Chemical Engineering, UniVersity of California, Berkeley, California 94720-1462
The choice of diffusional driving forces and fluxes such that a sum of their products yields the entropy generation rate does not generally lead to a system with a symmetric transport matrix. One can apply Onsager’s fluctuation theory to a mass-transfer system with various driving forces, such as gradients of mole fraction, concentration, or chemical potential, and obtain a proper reciprocal relation among the transport properties so defined. Although Stefan-Maxwell coefficients are generally not symmetric, Dij * Dji, Onsager’s theory still supplies the reciprocal relation. This work employs these principles to derive the reciprocal relation among Stefan-Maxwell coefficients for isothermal, isobaric mass diffusion, with an illustration for ideal solutions. Introduction A reciprocal relation among transport coefficients is derived according to the following four physical principles employed by Onsager in his seminal works on the subject:1,2 1. The correlations between fluctuations decay to equilibrium according to the laws that govern macroscopic variations. 2. The principle of microscopic reversibility leads to a symmetry of the fluctuation correlations. 3. An instantaneous sampling of a microscopically fluctuating system obeys equilibrium statistics. 4. Diffusional fluxes are linearly related to diffusional driving forces. These physical principles are applied directly in deriving reciprocal relations by statistical mechanics. The fluctuation theory employed by Onsager is summarized correctly by Landau and Lifschitz.3 However, several other methods are commonly employed to derive reciprocal relations. The two most significant alternatives use irreversible thermodynamics or the kinetic theory of gases. In the theory of irreversible thermodynamics, it is stated that an appropriate choice of diffusional driving forces and fluxes can be ensured by examination of the entropy-generation equation.4,5 The entropy continuity equation can be manipulated into a form that expresses entropy generation as a sum of pairs of fluxes and gradients; the members of these pairs are identified as appropriate independent diffusional fluxes and driving forces. Flux laws are then written by expressing the driving forces as linear combinations of the fluxes (or Vice Versa). It is typically asserted that the Onsager reciprocal relation implies symmetry of the resulting transport matrix. However, Coleman and Truesdell call into question this line of reasoning involving the entropy generation.6 They show how a linear transport law can always be made symmetric by redefining the forces or fluxes in such a way that the sum of products of forces and fluxes still equals the entropy generation rate. Thus, transforming a nonsymmetric matrix derived with “improper” diffusion driving forces into a symmetric matrix is more of a mathematical than a physical exercise. Once the asymmetric matrix is known, a symmetric reconstruction is always possible. * Corresponding author. Tel.: (510) 642-4063. Fax: (510) 642-4778. E-mail:
[email protected]. † Present address: Room 443, Department of Chemistry, South Kensington Campus, Imperial College, London SW72AZ, U.K.
The derivation of reciprocal relations from the kinetic theory of gases employs the Chapman-Enskog perturbation of Boltzmann’s gas equation, which leads to diffusion equations for gas mixtures.7 This method gives correct results but is restricted to mixtures of monatomic ideal gases. The ChapmanEnskog result for binary dilute gas mixtures was generalized to n-component systems by Curtiss and Hirschfelder.8 The recent review article by Curtiss and Bird provides a summary of the subsequent discussion of the kinetic theory of diffusion and gives references that relate the Chapman-Enskog coefficients to the Stefan-Maxwell coefficients in three- and four-component gas mixtures.9 Chapman-Enskog theory suggests that the StefanMaxwell coefficients are symmetric in ideal-gas mixtures. The following development applies Onsager’s fluctuation theory to derive a reciprocal relation among the Stefan-Maxwell coefficients for isothermal, isobaric mass diffusion that is not restricted to ideal-gas mixtures. In ideal-gas mixtures, it is shown that Dij ) Dji, because in this case all the component partial molar volumes are equal. However, in condensed phases, the differences in component partial molar volumes and values of the activity coefficients lead to asymmetries. A general development is provided below to determine the reciprocal relations among Stefan-Maxwell coefficients in n-component solutions. The case of ternary ideal solutions is treated specifically to demonstrate the asymmetry of the Dij; the degree of asymmetry is illustrated for a ternary mixture of n-hexane, cyclohexane, and benzene. Transport Laws and Thermodynamic Relations Stefan-Maxwell Equations. Negative gradients of the chemical potential, -∇µi, are appropriate thermodynamic driving forces for mass diffusion, and the flux of component i can be expressed in terms of its velocity, b Vi. In a solution containing n components with mole fractions yi at constant temperature and pressure, there are n - 1 independent diffusional driving forces because the Gibbs-Duhem equation requires that ∑iyi∇µi ) 0. There are n - 1 independent flux densities because the diffusional fluxes can all be expressed in terms of n - 1 component Vi. This results in an (n - 1) × velocity differences, b Vj - b (n - 1) matrix of coefficients when driving forces are linearly related to fluxes in a homogeneous relation, that is, one for Vj - b Vi ) 0 for all i and j. which ∇µi ) 0 is equivalent to b To apply the principle that diffusional driving forces are linearly related to component fluxes, it is thus appropriate to adopt n - 1 extended Stefan-Maxwell laws,10,11
10.1021/ie051061e CCC: $33.50 © 2006 American Chemical Society Published on Web 06/24/2006
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ci∇µi ) RT
cicj
∑j c D T
(V bj - b V i) or ij
∇µi ) RT
yj
∑j D
(V bj - b V i) (1)
ij
where ci is the concentration of species i, R is the gas constant, and cT is the sum of ci. In this work, all sums will be stated explicitlysquantities such as ci∇µi should not be taken to imply summation over i. A reciprocal relation reduces the (n - 1)2 Stefan-Maxwell coefficients, Dij, that appear in eq 1 to 1/2n(n - 1), one for each pair of diffusing components. In general, Dij depends on composition. Note: One Stefan-Maxwell relation is not included in eq 1, that for i ) n. After the other transport properties have been determined, the Dnj can be determined from a kinematic relation, ∑kyk(1/Dki - 1/Dik) ) 0, which requires no statistical principles for its derivation. This equation provides n - 1 equations to get Dnj. The Stefan-Maxwell equations are consistent with the condition stated earlier for irreversible thermodynamics; that is, in isothermal, isobaric systems, the entropy generation per unit volume, gS, is given by
TgS ) -
∑i
(V bi - b V n)‚ci∇µi
(2)
The Stefan-Maxwell equations thus satisfy the condition that the sum of products of fluxes, (V bi - b Vn), and driving forces, ci∇µi, yields the entropy generation rate, and one would expect that Dij ) Dji. Property Basis in the Gibbs Function. Thermodynamic properties can be derived from the Gibbs function G,
G)
∑i ∫V cTyi[µ0i (T,p) + RT ln(yiγi)] dV
(3)
expressed as an integral over the volume of a slab, where V is the system volume, T is the temperature, p is the pressure, µ0i (T,p) is the chemical potential of pure species i, and γi is the activity coefficient for component i. Any parameters used in expressing the composition dependence of ln γi are intended to be functions of temperature and pressure only; γi must be thermodynamically consistent. The composition dependence of µi is
µi )
( ) ∂G ∂ni
) µ0i (T,p) + RT ln(yiγi)
(4)
T,p,nj*i
where ni is the amount of component i, in moles, and the component partial molar volumes, V h i, are
V hi )
( ) ∂V ∂ni
)
T,p,nj*i
( ) ∂µi ∂p
(5)
T,nj
In ideal-gas mixtures, γi ) 1, and µ0i is a function of temperature only. The partial molar volumes are all equal, V hi ) RT/p. In ideal solutions, the activity coefficients are equal to unity, but µ0i depends on temperature and pressure. The V h i are independent of composition, although they may not be equal h i may not be equal, to each other. In general cases, γi * 1, V and both properties may depend on composition. Macroscopic Transport Problem Governing Equations. The Stefan-Maxwell equations should be applied to describe the relaxation of small macroscopic
composition variations to a uniform equilibrated state. In an isothermal, isobaric closed system, the set of all instantaneous composition distributions constitutes an ensemble of states. When deducing the properties of an ensemble of isotropic solutions, it is sufficient to consider mole-fraction variations in a single direction. For an ensemble of one-dimensional slabs, the governing equations for the unknown mole fractions yi and species velocities b Vi are n material balances,
∂cTyi ) -∇·cTyib Vi ∂t
(6)
the n - 1 extended Stefan-Maxwell equations, eq 1, and one sum of the mole fractions,
∑i yi ) 1
(7)
In the ensemble of diffusing slabs, the component velocities obey n boundary conditions
b V i(0) ) 0
(8)
Also, the mole fractions are constrained by n mass-conservation relations
∫0L(t) cTyi dz ) c∞T y∞i L∞
(9)
where z denotes the position within the slab, L is the slab thickness, and a superscript ∞ indicates the value of a variable when the solution is in the homogeneous equilibrated state. The total solution concentration can be related to the component mole fractions and partial molar volumes. Taking the total differential of the system volume with respect to T, p, and ni, integration at fixed T and p, and division by the system volume V shows that
cT
∑i Vh iyi ) 1
(10)
With this relation and eq 4, the material balances, the StefanMaxwell equations, and the mole-fraction sum represent 2n Vi. 2n constraints are equations in the 2n unknowns yi and b supplied by eqs 8 and 9. Linearization. As the slabs approach a final equilibrium composition, a linearization of the problem is appropriate. The development by Chapman12,13 shows that, in the linear regime, L and Dij can be taken as constant and that higher-order perturbations of the governing system converge. The linearized governing equations take the form
∂y(1) y∞i ∂c(1) i T b(1) + ∞ ) -y∞i ∇·V i ∂t c ∂t
(11)
T
∇µ(1) i
) RT
∑j
y∞j
D∞ij
(V b(1) V (1) j -b i )
∑i y(1)i ) 0
(12)
(13)
where a superscript (1) denotes the linear correction to a variable, for instance, yi ≈ y∞i + y(1) Vi ≈ b V(1) i and b i . Because the linear correction to Dij does not appear in eq 12, the StefanMaxwell coefficients can be considered constant in the linear
Ind. Eng. Chem. Res., Vol. 45, No. 15, 2006 5363
regime. The boundary and mass-conservation conditions become
∫0L
b V (1) i (0) ) 0 and
∞
y(1) i dz ) 0
(14)
The linearization method shows that L(1) ) 0.12 From eq 4, corrections to the chemical-potential gradients are
∇µ(1) i )
RT (1) ∇yi + RT∇(ln γi)(1) y∞i
∑ j*n
( ) ∂ ln γi ∂yj
∞
y(1) j
( ) ∂ ln γi ∂yj
∞
(17) T,p,yk*j,n
where δij is the Kronecker delta, equal to 1 when i ) j and equal to zero otherwise, allows the linearized form of the chemical-potential gradients to be expressed as
∇µ(1) i )
RT
∑
y∞i j*n
∑QijRjm ) ∑j
L∞ j*n
(18)
(23)
∑i Rim ) 0
(24)
After elimination of βim by means of the material balances and elimination of Rnm by means of the mole-fraction sum, the Stefan-Maxwell equations become
QijRjm ) ∑ ∑ j*n j*n y∞i
( ) 1
D∞ij
-
1
dRjm
D∞in
dτ
-
(
dRim y∞n dτ D∞ in
+
∑ j*n
y∞j
)
D∞ij (25)
in which the terms coming from cT cancel and where τ ) (mπ/ L∞)2t. To simplify the notation further, the terms on the right of eq 25 can be grouped into a matrix R, the elements of which are given by
Rij )
Qij∇y(1) j
y∞i y∞j (βjm - βim) D∞ij
The mole-fraction condition 13 becomes
(16)
T,p,yk*j,n
A nonideal-correction matrix Q simplifies the notation of the chemical-potential gradients. The definition
Qij ) δij + y∞i
-
mπ
(15)
Activity coefficients can be expressed in the linearized form as
(ln γi)(1) )
and Stefan-Maxwell equations in the form
y∞i D∞ij
-
y∞i D∞in
if i * j, and Rii ) -
y∞i D∞in
-
∑ k*i
y∞k
D∞ik
(26)
Equation 25 then becomes
Near the equilibrium composition, eq 10 linearizes to
c(1) T c∞T
) -c∞T
∑i
∞ (V h ∞i y(1) h (1) i +V i yi )
(19)
c(1) T
This can be used to eliminate from the material-balance relations. Expansions similar to eq 16 also apply to V h (1) i . Fourier Space. A general solution to the macroscopic transport problem when composition variations are small can be obtained by translating the problem into Fourier space. To effect a solution, express the mole fractions as ∞
y(1) i )
( )
(20)
( )
(21)
∑ Rim(t) cos m)1
mπz L∞
and the component velocities by ∞
V(1) i
)
∑ βim(t) sin m)1
mπz L∞
The Fourier terms of different wavelengths do not interfere with each other in the linearized system. Substitution of series 20 and 21 into the linearized equations yields material balances in the form
dRim dt
-c∞T y∞i
[∑ j
V h ∞j
dRjm dt
+
∑j k*n ∑
y∞j
() ∂V hj
∞
∂yk
T,p,yl*k,n
]
dRkm
-
dt
mπy∞i L∞
)
βim (22)
∑ j*n
QijRjm )
∑ j*n
dRjm Rij
dτ
(27)
This is a system of n - 1 equations in terms of the n - 1 unknown Fourier coefficients Rim. Fluctuation Statistics Decay of Fluctuation Correlations. Equation 27 describes how macroscopic variations decay toward an equilibrium state. Onsager’s regression hypothesis states that fluctuations decay in the same way. One way to implement this is to treat the equations governing macroscopic variations as describing the decay of a microscopic fluctuation at time τ0 + τ, multiply by Rlm(τ0), and then take the ensemble average, interpreting the resulting combination as the correlation of fluctuations. (Fluctuations at differing wavenumbers are uncorrelated and do not need to be considered; this statement is justified more thoroughly in the initial-correlations section.) Use the shorthand notation
Cil(τ) ) 〈Rim(τ0 + τ)Rlm(τ0)〉
(28)
to define an (n - 1) × (n - 1) matrix of correlations, C. Here, the correlation is over all states in the ensemblesthat is, over all values of the fluctuating Fourier coefficients. As a function of the scaled time, τ, C is the same at every m, save for the partitioned normalization constant of the probability distribution. This constant cancels from both the reciprocal relations and the equations governing correlation decay. Thus, C is similar to Onsager’s correlations between the moments of fluctuating particles. With definition 28, the equations describing the decay of correlations are
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∑ j*n
QijCjl )
∑ j*n
dCjl Rij
dτ
or QC ) R
dC dτ
Here we use the convention for matrix multiplication from linear algebra, where XY ) Z means Zij ) ∑kXikYkj. Principle of Microscopic Reversibility. Fundamental or molecular processes can proceed equally well in the forward or backward direction. To apply this principle of microscopic reversibility as Onsager did means that
Cij ) Cji
Reciprocal Relation Application of the regression hypothesis to eq 27 shows that correlations decay according to eq 29. The principle of microscopic reversibility requires that C be symmetric. In typical derivations, reciprocal relations are extracted by formal manipulation of eq 29.3 Although direct, this method does not provide the time functionality of correlation decay. However, one may solve eq 29 to obtain the decays directly. We develop next four equivalent ways to obtain them; Laplace transforms allow particularly efficient implementation of the procedure. Solve for C with Exponentially Decaying Functions. Equation 29 is a differential equation with constant coefficients; its solution gives n - 1 decay constants and n - 1 sets of coefficients with the ability to satisfy the initial conditions for any disturbance. Equating Cij ) Cji yields reciprocal relations that are independent of time. We have carried this out in detail for ternary diffusion and also for binary diffusion with heat transfersan equivalent system. Use of Laplace Transforms for Ternary Diffusion. It is more economical to do the same thing with Laplace transforms, which incorporate the initial conditions directly. The Laplace transform of eq 29 is
or QC h ) R(sC h - C0)
(
)
R-1Q s
-1
C0 s
(33)
C h ) (Is - R-1Q)-1C0
(34)
In view of eq 30, the right side of eq 34 is equal to its transpose,
(Is - R-1Q)-1C0 ) (C0)T(Is - R-1Q)-T
(35)
To obtain this relation, the matrix identity (XY)T ) YTXT is employed. In addition, note that inversion and transposing are communicative; that is, (X-1)T ) (XT)-1 ) X-T. Equation 35 can be manipulated to obtain a relation that is independent of s as follows. Multiply both sides of the equation from the left by (Is - R-1Q) and from the right by (Is - R-1Q)T, yielding
sC0 - C0(R-1Q)T ) s(C0)T - (R-1Q)(C0)T
(36)
Because C0 is symmetric, terms containing the Laplace variable cancel by subtraction of sC0 from both sides. Using the symmetry of C0 and the matrix identity (XY)T ) YTXT, one obtains
R-1QC0 ) (R-1QC0)T
(37)
which constitutes the proper reciprocal relation. Because R-1QC0 is an (n - 1) × (n - 1) matrix, eq 37 results in 1/2(n - 1)(n - 2) relations among the Stefan-Maxwell coefficients, reducing the number of independent Dij from (n - 1)2 to 1/2n(n - 1). Implementation requires that the initial correlations be known. Initial Correlations
(31)
where an overbar denotes a transformed function and s is the Laplace variable. Initial correlations, C0, are given by entropy or Gibbs-function considerations of the fluctuations, as discussed in the next section. To treat a ternary solution, write this out for i ) 1, for i ) 2, for l ) 1, and for l ) 2, a total of four equations. C h 11 can be eliminated between two of these equations, and C h 22 can be eliminated between the other two. The resulting two equations contain C h 12 and C h 21, from which the time dependence of the cross-correlations can be determined. Instead, one can equate the correlations in Laplace space. The Laplace variable s can be eliminated along with the Laplace transform of the cross-correlations, yielding the reciprocal relation. This says, in essence, that the matrix R-1QC0 is symmetric. Inversion of Laplace Transforms for Small Time. Even for multicomponent systems, eq 31 is valid and can be solved for C h,
C h ) I-
-1 2 0 C0 R-1QC0 (R Q) C + + + ... s s2 s3
The first term gives the initial value for C(τ). The second was used by Onsager when he argued that, at short times, this term (whose Laplace inverse is proportional to τ) should be symmetric. Thus, Onsager stated that the matrix R-1QK-1 is symmetric, where K is related to C0 in a succeeding section. General Multicomponent Proof. The function C h can be isolated from eq 31, yielding
(30)
This symmetry of the correlations is the fundamental physical condition that ultimately results in a reciprocal relation among the phenomenological coefficients.
QijC h jl ) ∑Rij(sC h jl - C0jl) ∑ j*n j*n
C h)
(29)
(32)
where I is the identity matrix. For short times, this can be expanded for large s:
In a thermally isolated system, the probablity of finding a state with a particular entropy is proportional to exp(S/k), where k is Boltzmann’s constant. The entropy is a maximum at the final equilibrium state, and any finite value of Rim will decrease the entropy. For a system at constant temperature and pressure, statistics should be performed with the Gibbs function; the probability of finding a state with a particular Gibbs energy is proportional to exp(-G/kT). This distinction does not make a difference for ideal solutions of nonelectrolytes where Qij ) δij. The exponent can be expanded in a Taylor series about the equilibrium state, and this will take a quadratic form:
G kT
)
G∞ kT
∞
+
∑ ∑ ∑ KijRimRjm + higher order terms i*n j*n m)1
(38)
To obtain the quadratic coefficients Kij, one needs to evaluate eq 3 with the Fourier expansions of the mole fractions and cT. (Performing the integral in eq 3 decouples Fourier components at different wavenumbers in the quadratic terms of G/kT. Thus, the correlations 〈Rim(τ0)Rjl(τ0)〉 are all zero for all m * l. This justifies the neglect of cross-correlations between different wavenumbers.) Constant terms are incorporated into G∞, and linear terms integrate to zero when the Fourier components are
Ind. Eng. Chem. Res., Vol. 45, No. 15, 2006 5365
integrated from 0 to L∞. Because the mole numbers are specified in each slab, the terms with µ0i (T,p) in eq 3 can be integrated directly; these terms enter into G∞, and Kij has no contribution from µ0i (T,p). Terms higher than the quadratic coefficients are generally ignored, based on an argument that, when they become significant, the integrand of the correlation is sufficiently small to be neglected. This paper does not pursue terms beyond the quadratic; they were also neglected by Onsager. Only ratios of terms (either Kij or C0ij) are required, as the example with a ternary, ideal solution shows. To get the magnitude of a correlation of fluctuations at a given time τ ) 0, one needs to evaluate the integral (for a ternary)
C0ij ) A
∫RimRjm exp(-K11R1m2 - 2K12R1mR2m K22R2m2) dR1m dR2m (39)
where A is a normalization factor so that the integral of the probablility distribution of all states will equal unity. Also, K11 > 0 and K122 < K11K22 for a stable system. The result for a ternary solution can be expressed concisely as
C011 K22
)-
C012 K12
C022
)
K11
)
Aπ/2 (K11K22 - K122)1.5
is not zero unless K12 is zero, even for an idealNote that gas mixture. In essence, C0 is proportional to K-1. To finish the reciprocal relation in general is formidable because of the complexity of the Gibbs function. Numerical evaluation of the needed terms is straightforward. Ideal solutions are treated here in more detail. Such solutions are almost tractable, and they constitute the simplest case that demonstrates that the conventional statement of the Onsager reciprocal relation (D12 ) D21) needs to be revised with physical-property variations more complicated than ideal gases. Thermodynamics of Ideal Solutions. Setting γi ) 1 in eq 3 defines an ideal solution. The molar volume, enthalpy, and entropy of the pure fluids are contained in µ0i . For example,
V hi )
( ) ( ) ∂G ∂ni∂p
)
T
∂µ0i ∂p
+ RT ln yi
∑j yj ln yj
(43)
( )
yk*i,n
)
) 2cT3(V hi - V h n)(V hj - V h n) (46) yk*n
To obtain the contribution to G of the ideal terms, these two expansions, one for Y and one for cT, need to be multiplied, and the quadratic terms identified to go into Kij. The constant terms contribute to G∞, and the linear terms integrate to zero when the Fourier terms are introduced and the integration is over the volume, that is, from z ) 0 to z ) L. This is equivalent to noting that
Kij )
(
)
2 V∞ ∂ (G/V) 4kT ∂yi∂yj
∞
(47)
T,p,yk*i,j,n
For an ideal solution, Qij ) δij, and the reciprocal relation 37 reduces to
R21C011 + R22C021 ) R12C022 + R11C012
(48)
The kinematic relationship (in the text following eq 1) permits a symmetric definition of a parameter φ to express the deviation from the conventional reciprocal relation (Dij ) Dji),
φ)
(
) (
) (
)
1 1 1 1 1 1 1 1 1 ) ) ) y3 D12 D21 y1 D23 D32 y2 D31 D13 φ1/D12 + φ2/D13 + φ3/D23 y3(y2C011 - y1C012)
(49)
where
φ1 ) y2(C011 + C012) - y1(C022 + C012)
(50)
φ2 ) y1(C022 + C012) + y3C012
(51)
φ3 ) -(y2 + y3)C012 - y2C011
(52)
) ln(yi/yn)
1 1 + and yi yn
The expression is still cumbersome because the initial values of the correlations must first be evaluated from the Gibbs function of the fluctuations. For an ideal-gas mixture, cT is constant, and the derivatives of Y can be used directly in the reciprocal relation, which now reduces to
D12 ) D21 (44)
yk*i,n
and second derivatives are
∂ 2Y ∂yi2
( )
(42)
Appropriate first derivatives are
( )
yk*i,n
∂2cT ∂yj∂yi
(41)
The first term has already been discussed and contributes to G∞. Let us define an ideal term as
∂Y ∂yi
) -cT2(V hi - V h n) and
and
µ0i (T,p)
Y)
∂cT ∂yi
T
independent of composition. For the ideal solution,
µi )
( )
(40)
C012
2
Note that the last derivative is not zero. In eq 3, Y is multiplied by cT. For an ideal ternary solution, eq 10 gives
( ) ∂2Y ∂yj∂yi
yk*n
)
1 yn
(45)
(53)
as shown by Curtiss and Hirschfelder8 by means of the Chapman-Enskog method. This result would also be obtained for the ideal solution if the variation of cT with composition (due to different partial molar volumes) is ignored. Equation 53 remains valid for multicomponent ideal gases, for which Dij ) Dji. However, when partial molar volumes are not equal, as in an ideal solution, then volume transport and mass transport
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where NA is Avogadro’s number and ηi is the viscosity of pure component i. Molar volumes are used in this equation to compute the Stokes radius. Two terms are contained in the brackets. The diffusional resistance is taken to be the mean of the resistances at the extremes of infinite dilution. The reciprocal relation provides the values of the remaining diffusion coefficients (D21, D31, and D32), which turn out not to be constant and not equal to the values expected from the conventional reciprocal relation. On Figure 1, φ is multiplied by D12 to form a dimensionless quantity. There is one contour where φ is zero, but generally it is not. The quantity plotted becomes somewhat larger than 0.5 in part of the diagram. It should be understood that quite large values could be obtained for arbitrary values of the parameters. On the other hand, φ is uniformly zero if either all the partial molar volumes are equal (as for an ideal gas) or all the diffusion coefficients are equal. Discussion and Summary Figure 1. Asymmetry of the Stefan-Maxwell coefficients in an ideal liquid mixture of n-hexane (1), cyclohexane (2), and benzene (3), expressed as a ternary contour plot of φD12 as a function of the component mole fractions. The zero contour corresponds to the conventional reciprocal relation, Dij ) Dji. Table 1. Properties of the n-Hexane/Cyclohexane/Benzene Systema index
component
V h i (mL/mol)
ηi (mPa‚s)
1 2 3
n-hexane cyclohexane benzene
131.598 108.744 89.399
0.2985 0.8980 0.6028
Stefan-Maxwell coefficients, Dij (× 109 m2/s) D12 ) 1.2 D13 ) 1.7 D23 ) 1.1 a Component properties are taken from the tables edited by Rossini,15 and the binary diffusion coefficients are estimated by eq 54.
are not identical. This leads to reciprocal relations more complicated than symmetry of the Stefan-Maxwell coefficients. Numerical Results Because of the complexity of the Gibbs function, it is convenient to explore the system numerically; how does the value of φ vary over the composition diagram, and does it depart significantly from zero? Equation 49 shows that the usual relation holds for the binaries, even though φ itself is not necessarily zero there. Equation 49 makes one ask whether C012 can be positive and whether φ could possibly go to infinity on a curve across the diagram. Could there also be a line where φ is zero, even though it is not zero across the whole diagram? Figure 1 shows a ternary diagram constructed for a liquid mixture of n-hexane, cyclohexane, and benzene. This system is nearly ideal; here it is taken to be exactly ideal with the property values given in Table 1. The three diffusion coefficients D12, D13, and D23 were estimated for this system and taken to be constant over the entire diagram. This is what one would have done to make estimates for an ideal system. The Stefan-Maxwell coefficients were estimated according to the following form of the StokesEinstein equation,14
[( ) ( )]
hj 3π ηi V 1 ) Dij kT 2 NA
1/3
+
ηj V hi 2 NA
1/3
(54)
The principles of Onsager, based in statistical physics, can be applied to diffusion laws expressed in terms of forces and fluxes that are defined by the convenience and preference of the user or of the community, and proper reciprocal relations can be obtained so that there are 1/2n(n - 1) independent transport coefficients. One could use, for example, gradients of chemical potential, mole fraction, concentration, or mass fraction for the driving forces. Similarly, one could use flux densities chosen variously from species velocity differences or molar or mass flux densities relative to a velocity such as the molaraverage, mass-average, volume-average, or solvent velocity. Even when forces and fluxes are chosen so that the sum of their products is the rate of generation of entropy, as in the extended Stefan-Maxwell equations used here, the resulting transport matrix will not be symmetric, in general. The correction can be significant, as shown by an example. Although a proper reciprocal relation still exists and there is the same number of independent transport coefficients, the situation is more complicated than if Dij were equal to Dji. All transport properties may be involved simultaneously. To do a complete evaluation of transport properties, the same number of experiments still needs to be performed. These generally need to be deconvoluted to find Dij anyway. Now it may be more desirable to use a computer program to handle the reciprocal relations. There are only a few cases where complete sets of transport properties are measured. The Onsager reciprocal relation is important because it is a general result of statistical mechanics, not restricted to any particular intermolecular-force laws. Thus, it is comparable to other general results, such as Liouville’s theorem16 and the more recent theorem by Jarzynski17 showing that the equilibrium free energy difference between states can be determined by measuring the work done in nonequilibrium experiments. This work still needs to be reconciled with that of Miller,18 who examined extensive and diverse experimental data and concluded that the Onsager reciprocal relations are satisfied within experimental error. Also in need of reconciliation is the work of Wheeler and Newman,19 who extended the GreenKubo analysis to show that Dij ) Dji in general for the extended Stefan-Maxwell equations. Equation 37 has a compelling derivation and is in harmony with Onsager’s original work, which used driving forces proportional to ∂S/∂Ri. These are not forces commonly used in mass transfer. It is a little disconcerting that one would obtain Dij ) Dji for ideal solutions by ignoring the composition
Ind. Eng. Chem. Res., Vol. 45, No. 15, 2006 5367
dependence of cT. The derivation of the proportionality of C0 to K-1 depends on an expansion of the Gibbs function in Rim with neglect of cubic and higher terms. Equation 37 is also valid for nonideal solutions. Acknowledgment This work was supported by the Shell Foundation and the Assistant Secretary for Energy Efficiency and Renewable Energy, Office of FreedomCAR and Vehicle Technologies of the U.S. Department of Energy, under Contract DE-AC0376SF0098. C.W.M.’s work during the final stages of manuscript preparation was also supported by the Leverhulme trust, Grant F/07058/P. Nomenclature A ) normalization factor for the probability distribution, dimensionless C ) (n - 1) × (n - 1) matrix of correlations, defined by eq 28, dimensionless C h ) Laplace transform of C with respect to τ, dimensionless ci ) molar concentration of component i, mol/m3 cT ) sum of component concentrations, mol/m3 Dij ) Stefan-Maxwell coefficient characterizing diffusional interactions between components i and j, m2/s G ) Gibbs function, given by eq 3, J gS ) entropy generation per unit volume, W/K‚m3 I ) identity matrix, dimensionless K ) (n - 1) × (n - 1) matrix of quadratic terms from the Taylor expansion of G, defined by eq 38, dimensionless k ) Boltzmann constant, 1.381 × 10-23 J/K L ) length of one-dimensional slab, m m ) harmonic number of Fourier expansion, dimensionless NA ) Avogadro’s number, 6.022 × 1023 mol-1 n ) number of components, dimensionless ni ) amount of component i, mol p ) system pressure, Pa Q ) (n - 1) × (n - 1) nonideal-correction matrix, defined by eq 17, dimensionless R ) gas constant, 8.3143 J/mol‚K R ) (n - 1) × (n - 1) diffusional-resistance matrix, defined by eq 26, s/m2 s ) Laplace variable, m2/s T ) system temperature, K t ) time, s V ) system volume, m3 V h i ) partial molar volume of component i, m3/mol b Vi ) velocity of component i, m/s Y ) term accounting for ideal contributions to G, defined by eq 43, dimensionless yi ) mole fraction of component i, dimensionless z ) position within the one-dimensional slab, m Greek Rim ) Fourier coefficient for the mole fraction of component i at harmonic m, dimensionless βim ) Fourier coefficient for the velocity of component i at harmonic m, m/s
γi ) activity coefficient of component i, dimensionless δij ) Kronecker delta, equal to 1 if i ) j and 0 if i*j η ) viscosity of pure component i, Pa‚s µi ) chemical potential of component i, J/mol τ ) scaled time, τ ) (mπ/L∞)2t, s/m2 φ ) parameter to characterize the reciprocal relation in ternary solutions, defined by eq 49, s/m2 Superscripts and Subscripts 0 ) value of a variable at τ ) 0 (1) ) linear perturbation of a function near equilibrium ∞ ) value of a variable at equilibrium T (superscript) ) transpose of a matrix Literature Cited (1) Onsager, L. Reciprocal Relations in Irreversible Processes I. Phys. ReV. 1930, 37, 405-426. (2) Onsager, L. Reciprocal Relations in Irreversible Processes II. Phys. ReV. 1931, 38, 2265-2279. (3) Landau, L. D.; Lifschitz, E. M. Statistical Physics, Part 1, 3rd ed.; Pergamon Press: Oxford, U.K., 1980; p 365. (4) de Groot, S. R. Thermodynamics of IrreVersible Processes; Interscience Publishers: New York, 1951; p 7. (5) Callen, H. B. Thermodynamics and an Introduction to Thermostatistics, 2nd ed.; John Wiley and Sons: New York, 1985; p 309. (6) Coleman, B. D.; Truesdell, C. A. On the Reciprocal Relations of Onsager. J. Chem. Phys. 1960, 33, 28-31. (7) Chapman, S.; Cowling, T. G. The Mathematical Theory of Nonuniform Gases, 2nd ed.; Cambridge University Press: Cambridge, U.K., 1964; Chapter 9. (8) Curtiss, C. F.; Hirschfelder, J. O. Transport Properties of Multicomponent Gas Mixtures. J. Chem. Phys. 1949, 17, 550-555. (9) Curtiss, C. F.; Bird, R. B. Multicomponent Diffusion. Ind. Eng. Chem. Res. 1999, 38, 2515-2522. (10) Onsager, L. Theories and Problems of Liquid Diffusion. Ann. N.Y. Acad. Sci. 1945, 46, 241-265. (11) Lightfoot, E. N.; Cussler, E. L.; Rettig, R. L. Applicability of the Stefan-Maxwell Equations to Multicomponent Diffusion in Liquids. AIChE J. 1962, 8, 708-710. (12) Chapman, T. W. The Transport Properties of Concentrated Electrolytic Solutions. Ph.D. Thesis, University of California, Berkeley, CA, 1967; p 99. (13) Newman, J.; Chapman, T. W. Restricted Diffusion in Binary Solutions. AIChE J. 1973, 19, 343-348. (14) Diffusion Coefficient. In International Encyclopedia of Heat and Mass Transfer; Hewitt, G. F., Shires, G. L., Polezhaev, Y. V., Eds.; CRC Press: Boca Raton, FL, 1997; p 306. (15) Rossini, F. D., Ed. Selected Values of Physical and Thermodynamic Properties of Hydrocarbons and Related Compounds: American Petroleum Institute Research Project 44; Carnegie Press: Pittsburgh, PA, 1953; pp 162 and 226. (16) Tolman, R. C. The Principles of Statistical Mechanics; Oxford University Press: Oxford, U.K., 1962; p 48. (17) Jarzynski, C. Nonequilibrium Equality for Free Energy Differences. Phys. ReV. Lett. 1997, 78, 2690-2693. (18) Miller, D. G. Thermodynamics of Irreversible Processes: The Experimental Verification of the Onsager Reciprocal Relations. Chem. ReV. 1960, 60, 15-37. (19) Wheeler, D. R.; Newman, J. Molecular Simulations of Multicomponent Diffusion. 1. Equilibrium Method. J. Phys. Chem. B 2004, 108, 18353-18361.
ReceiVed for reView September 21, 2005 ReVised manuscript receiVed May 3, 2006 Accepted May 15, 2006 IE051061E
