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On the formulation of the diffusion coefficient in isothermal binary systems
Chemical Engineeting Science, 1975, Vol. 30, pp. 151-154.
Pergamon Press.
Printed in Great Britain
ON THE FORMULATION OF THE DIFFUSION COEFFICIENT IN ISOTHERMAL BINARY SYSTEMS J. C. R. TURNER Department of Chemical Engineering, Pembroke Street, Cambridge, England (Received 10May 1974;accepted 29 July 1974) Abstract-The diffusion of a component in a binary isothermal mixture can be regarded as arising either from its concentration gradient or from the gradient of its chemical potential. The ‘kinetic’and the ‘thermodynamic’approach can both be related to the simple, measurable, mutual diffusion coefficient II,,,. When considering the concentration dependence of 0, the ‘thermodynamic’ approach leads naturally to the presence of the term (d In a,/d In ni), while the ‘kinetic’ approach does not. In systems showing marked non-ideality, of which two examples are discussed, allowance for this term greatly simplifies the concentration dependence of D,. This lends support to the proposition that the gradient of chemical potential should be regarded as the ‘natural force for diffusion’. In ideal systems it does not matter which approach is used,but where non-ideality is marked, the ‘thermodynamic’approach would appear to have advantage.
There are two different approaches to the formulation of the mutual diffusion of the two components of an isothermal binary mixture. The first, which one might call the ‘kinetic’ approach, considers the random movement of molecules as a result of thermal motion. It is assumed that molecules on the left of a reference plane have the same chance of moving through the plane to the right as molecules on the right have of moving to the left. There will thus be a nett flux of a given component to the right if there are more molecules of that type on the left. It follows that a natural formulation for the diffusive flux of component 1, J,, is J, =
A proper treatment requires care in the choice of reference frames, for which see, e.g. Tyrrell[l], or de Groot[2]. It results that Eq. (1) can be more precisely written dc, J,” = -Dmdt
-D,Qg,
in which we need for present purposes consider only a unidimensional system. The ‘thermodynamic’ approach, on the other hand, says that a molecule will move, on average, with a velocity u which is related to the nett force which it experiences. For isdthermal diffusion in a binary system the appropriate ‘force’ is the gradient of chemical potential. This is not, by itself, a very convincing picture to some, though it is re-inforced by considerations of irreversible thermodynamics. For example, the right rate of entropy production per unit volume is obtained if this force is multiplied by the resultant flux. This velocity, v,, of molecules of type 1, is associated with a flux J, = clul and so one arrives at the formulation
where K, is a mobility.
where J1” is the flux relative to a ‘volume fixed’ frame of reference and D, is the mutual diffusion coefficient. D,,, can be a function of composition, but not of composition gradient, and at any given composition it has the same value for component 1 as for component 2. If the partial molal volumes of the components are constant over the range of concentration of interest this frame of reference is the same as the ‘apparatus-fixed’, or Fickian frame of reference. Thus D,,, is experimentally determinable. For a discussion, see Agar[3]. The usual formulation of irreversible thermodynamics relates the flux of a component to the gradient of its chemical potential, whereas a physically more acceptable relationship would be between the v&city of a component and the gradient of its chemical potential. As usually formulated, however, the result is to connect the D, of Eq. (3) to a phenomenological coefficient Lii via an expression which is symmetrical in i and j. This must be so if there is to be one D,,, only for a two-component system. The expression does, however, have a term in, e.g. clcz in the denominator. The coefficient, e.g. L2, must therefore be composition dependent if the flux of a component is to vanish as its concentration -+ 0. To relate the chemical potential gradient to a composition gradient it is necessary to introduce a factor such as (d In a4 d In n,), or (d In al/d In cl). The relationship between D, and K, is therefore not simple and straightforward.
151
J. C. R.
152
TURNER
possible to produce a zero chemical potential gradient in the presence of a non-zero concentration gradient, (or the other way around), that might be used to decide the matter. This could be done if a third component J, = -K,RTc, y were added, but this greatly complicates the formulation of diffusion, and means that one value of 0, will no =_KRTc dlncl dlna, dlnn, longer describe the system. It is possible to make I ‘Jr’--- dlnn, dlnc, (d In al/d In nJ tend to zero, if one chooses to work at the temperature and composition at which the twoOr component mixture shows incipient separation into two J =_KRTdlnal dlnnl dc, I I -.-.(4) phases (the critical solution, or consolute, temperature dlnn, dlncl dz and composition). This poses experimental difficulties and we shall treat If we compare Eq. (4) with Eq. (3) we sqe that we can here two examples where (d In al/d In nJ is low, but not write zero. The first relates to the system water + triethylamine, for which Dudley and Tyrrell[4] report values of the D =KRTdlnal dlnn, m I -.(5) mutual diffusion coefficient, 0,. The vapour pressure dlnn, dlnc, data of Counsel1 et a1.[5] were used to calculate (d In al/d In n,), which we shall henceforth call B. provided J, can be equated to J,“. This last proviso At 15°C the experimental diffusion coefficient, D,,,, falls requires not only the restrictions discussed after Eq. (3) rapidly from a value of about 6 X lo-” m2sec-’ in very above, but also requires that dilute solutions of triethylamine in water to a trough value of about 3.5 x lo-” m’sec-’ which then slowly rises to K dln al .a= K2 dlnaz dInn2 about 1 x lo-” m’sec? in 70 wt% triethylamine. D, then (6) ‘d dlnc, -‘dlnnz dlnc2 increases much more rapidly to a value of 2 x 10F9for dilute solutions of water in triethylamine. This extreme By application of the Gibbs-Duhem relationship we can variation of D,,, with composition is matched by an reduce Eq. (6) to extreme variation of B. If one divides D,,, by B one obtains RTK,,,, and as seen in Fig. 1 this quantity has a less dlnnz dlnn, K,d=Kz(7) pronounced composition dependency than 4. These d In cz results refer to lS”C, and similar conclusions apply to their 5°C results. Both sets of results, particularly those at If the mixture is dilute in either component, or if they Y’C, are subject to some scatter, especially at very low have equal partial molar volumes, then d In ni /d In ci = 1 and We can express Eq. (2) in terms of the activity aI and the mole fraction nl as follows
K,=Kz=K,,,,
were
(8)
and this result is analogous to that which shows that the same value of D,,, applies to each component of a two-component mixture. If we may henceforth disregard the factor d In n,/d In cl in Eq. (5) we have to equate 0, of Eq. (3), derived from a ‘kinetic’ viewpoint, with K,,,RT(d In al/d In n,), derived from a ‘thermodynamic’ viewpoint as expressed in Eq. (2). Put more succinctly, does one get a simpler relationship if the driving force for isothermal diffusion is the concentration gradient or if the driving force is the chemical potential gradient? The preamble before this question is required to place the question precisely in the environment of diffusion studies. If it turned out that D, was a constant, independent of composition, whereas K,,, was not (or the other way around) one would lean towards that formulation which gave the constant value. But it does not so turn out. If it
Weight
Fig. I. Triethylaminelwater at
fraction 15°C; 4
Triethylamine
and 0, /B vs composition.
153
On the formulation of the diffusion coefficient in isothermalbinary systems
and at very high values of the weight fraction of triethylamine, where both D,,, and B are changing very rapidly. The second example is that of thermal diffusion in the system cyclohexanelmethanol. The thermal diffusion coefficient, V, was measured in a flow-apparatus by Story and Turner[6]. The quantity cr is the ratio of a thermal diffusion coefficient to the mutual diffusion coefficientD/D,,,-and relates to the steady state separation obtainable with a given temperature difference. The rate of attainment of this steady state is controlled by D,,,,and Story[7] has described how thermal diffusion measurements can be used to measure D,,,, as well as (T. The system cyclohexanelmethanol shows an immiscibility gap at 25”,and we carried out measurements close to the mole fraction of 0.13 cyclohexane which is the lower end of this gap. As the gap is approached B becomes small, and Fig. 2 shows that D, decreases rapidly. This is reinforced by the behaviour of u = D'ID,,,, which increases rapidly as the gap is approached. This rapid decrease of D,,, in parallel with a rapid decrease in B would suggest that the ‘thermodynamic’ approach is more profitable than the kinetic approach. One can proceed somewhat further with this example if one assumes that cyclohexane and methanol form a regular solution. This may well be incorrect, but probably not sufficiently so to invalidate the general conclusions to be drawn from the following treatment. The thermodynamic factor B for a regular solution can readily be
“0 ;; b
shown to be
B_ddlnn,
1-!$r,(l
- nr)
in which T, is the (upper) critical solution temperature for a strictly regular solution showing partial miscibility. For such a solution the miscibility limits should be symmetrical about nl = 0.5 at any temperature T below T,, and knowledge of these limits and T enables T, to be estimated. For cyclohexane/methanol at 25°C the limits of miscibility are at n, = O-132and n, = 0900. These are not quite symmetrical, but a value of T, / T = 1.3 f 0.05 can be estimated from them. Thus, on the assumed model, we can say B = 1- 52n,(l-
n,).
If this expression for B is used with the measured diffusion coefficient D,,, of Fig. 2, one can calculate that D,,,/B = D,,,IB = RTK,,,. The result is 2.2x 1O-9 m*/sec within about 1 per cent for the whole range of n, up to the immiscibility gap-for which the measured values of D cover the range 1.9 x 1O-9-O+3X 10-9. Too much stress should not be laid on the constancy of D,,,/B, as this results from an assumed, if reasonable, model of the liquid non-ideality, and there is also no reason why D,,,/B should not itself be compositiondependent. But if one accepts that the driving force for isothermal diffusion is the gradient of chemical potential one then comes to consider the concentration dependence of D,,,, after the thermodynamic factor B has been allowed for, and this is a much milder concentrationdependence. The mobility of a molecule could well depend on its environment, and Dudley and Tyrrell[4] discuss this ‘residual’ concentration dependence. This is difficult enough to explain for strongly non-ideal solutions, and would seem an impossible task if-as the ‘kinetic’ approach would indicate-the thermodynamic factor B should not be allowed for. For dilute solutions of electrolytes in water, the thermodynamic factor by itself accounts for nearly all of the concentration dependence of the observed diffusion coefficient, and these solutions present particularly strong non-idealities. The examples discussed above add weight to the argument that it leads to a greater understanding of diffusion phenomena if the gradient of chemical potential, rather than the concentration gradient, is regarded as the ‘driving force’ for diffusion. NOTATION
activity of component i B thermodynamic factor, (d In ai /d In s 1 ci concentration of component i, kmol me3 ai
Fig. 2. Soret coefficients and diffusion coefficients for cyclohexane (I)-methanol (2).
(10)
154
J. C. R. TURNER
IA’ kinetic diffusion coefficient, see Eq. (1) m2set-’ D, mutual diffusion coefficient, as measured, see Eq. (3), m* set-’ D’ thermal diffusion coefficient, m2sect’ K-’ Ji flux of component i, kmol m-’ set-’ .A” volume-fixed flux of component i, kmol m-* see-’ Ki mobility constant, see Eq. (2) Lii phenomenological coefficient ni mol fraction of component i R gas constant, J kmol-’ K-’ T temperature, K z co-ordinate distance, m /.L~chemical potential of component i, J kmol-’
CT Soret coefficient, thermal diffusion coefficient, K-’ REFERENCES
[I] Tyrrell H. J. V., LX&on
and Heat Flow in Liquids,
Butterworths,London, 1961. [2] de Groot S. R., Thermodynamics of Irreversible Processes, North-Holland, Amsterdam, 1951. [3] Agar J. N., Trans. Faraday Sot. l%O 56 776. [41 Dudley G. J. and Tyrrell H. J. V., J. Chem. Sot. Faraday Tram: 1, 197369 22bo. I51 . _ Counsel1J. F.. Everett D. H. and Munn R. J.. Pure mol. .& Chem. 19612 335. [6] Story M. J. and Turner J. C. R., Trans. Faraday Sot. 1969 65 1523. [7] Story M. J., Ind. Engng Chem. Fund. 1%9 8 777.
Pergamon Press.
Printed in Great Britain
ON THE FORMULATION OF THE DIFFUSION COEFFICIENT IN ISOTHERMAL BINARY SYSTEMS J. C. R. TURNER Department of Chemical Engineering, Pembroke Street, Cambridge, England (Received 10May 1974;accepted 29 July 1974) Abstract-The diffusion of a component in a binary isothermal mixture can be regarded as arising either from its concentration gradient or from the gradient of its chemical potential. The ‘kinetic’and the ‘thermodynamic’approach can both be related to the simple, measurable, mutual diffusion coefficient II,,,. When considering the concentration dependence of 0, the ‘thermodynamic’ approach leads naturally to the presence of the term (d In a,/d In ni), while the ‘kinetic’ approach does not. In systems showing marked non-ideality, of which two examples are discussed, allowance for this term greatly simplifies the concentration dependence of D,. This lends support to the proposition that the gradient of chemical potential should be regarded as the ‘natural force for diffusion’. In ideal systems it does not matter which approach is used,but where non-ideality is marked, the ‘thermodynamic’approach would appear to have advantage.
There are two different approaches to the formulation of the mutual diffusion of the two components of an isothermal binary mixture. The first, which one might call the ‘kinetic’ approach, considers the random movement of molecules as a result of thermal motion. It is assumed that molecules on the left of a reference plane have the same chance of moving through the plane to the right as molecules on the right have of moving to the left. There will thus be a nett flux of a given component to the right if there are more molecules of that type on the left. It follows that a natural formulation for the diffusive flux of component 1, J,, is J, =
A proper treatment requires care in the choice of reference frames, for which see, e.g. Tyrrell[l], or de Groot[2]. It results that Eq. (1) can be more precisely written dc, J,” = -Dmdt
-D,Qg,
in which we need for present purposes consider only a unidimensional system. The ‘thermodynamic’ approach, on the other hand, says that a molecule will move, on average, with a velocity u which is related to the nett force which it experiences. For isdthermal diffusion in a binary system the appropriate ‘force’ is the gradient of chemical potential. This is not, by itself, a very convincing picture to some, though it is re-inforced by considerations of irreversible thermodynamics. For example, the right rate of entropy production per unit volume is obtained if this force is multiplied by the resultant flux. This velocity, v,, of molecules of type 1, is associated with a flux J, = clul and so one arrives at the formulation
where K, is a mobility.
where J1” is the flux relative to a ‘volume fixed’ frame of reference and D, is the mutual diffusion coefficient. D,,, can be a function of composition, but not of composition gradient, and at any given composition it has the same value for component 1 as for component 2. If the partial molal volumes of the components are constant over the range of concentration of interest this frame of reference is the same as the ‘apparatus-fixed’, or Fickian frame of reference. Thus D,,, is experimentally determinable. For a discussion, see Agar[3]. The usual formulation of irreversible thermodynamics relates the flux of a component to the gradient of its chemical potential, whereas a physically more acceptable relationship would be between the v&city of a component and the gradient of its chemical potential. As usually formulated, however, the result is to connect the D, of Eq. (3) to a phenomenological coefficient Lii via an expression which is symmetrical in i and j. This must be so if there is to be one D,,, only for a two-component system. The expression does, however, have a term in, e.g. clcz in the denominator. The coefficient, e.g. L2, must therefore be composition dependent if the flux of a component is to vanish as its concentration -+ 0. To relate the chemical potential gradient to a composition gradient it is necessary to introduce a factor such as (d In a4 d In n,), or (d In al/d In cl). The relationship between D, and K, is therefore not simple and straightforward.
151
J. C. R.
152
TURNER
possible to produce a zero chemical potential gradient in the presence of a non-zero concentration gradient, (or the other way around), that might be used to decide the matter. This could be done if a third component J, = -K,RTc, y were added, but this greatly complicates the formulation of diffusion, and means that one value of 0, will no =_KRTc dlncl dlna, dlnn, longer describe the system. It is possible to make I ‘Jr’--- dlnn, dlnc, (d In al/d In nJ tend to zero, if one chooses to work at the temperature and composition at which the twoOr component mixture shows incipient separation into two J =_KRTdlnal dlnnl dc, I I -.-.(4) phases (the critical solution, or consolute, temperature dlnn, dlncl dz and composition). This poses experimental difficulties and we shall treat If we compare Eq. (4) with Eq. (3) we sqe that we can here two examples where (d In al/d In nJ is low, but not write zero. The first relates to the system water + triethylamine, for which Dudley and Tyrrell[4] report values of the D =KRTdlnal dlnn, m I -.(5) mutual diffusion coefficient, 0,. The vapour pressure dlnn, dlnc, data of Counsel1 et a1.[5] were used to calculate (d In al/d In n,), which we shall henceforth call B. provided J, can be equated to J,“. This last proviso At 15°C the experimental diffusion coefficient, D,,,, falls requires not only the restrictions discussed after Eq. (3) rapidly from a value of about 6 X lo-” m2sec-’ in very above, but also requires that dilute solutions of triethylamine in water to a trough value of about 3.5 x lo-” m’sec-’ which then slowly rises to K dln al .a= K2 dlnaz dInn2 about 1 x lo-” m’sec? in 70 wt% triethylamine. D, then (6) ‘d dlnc, -‘dlnnz dlnc2 increases much more rapidly to a value of 2 x 10F9for dilute solutions of water in triethylamine. This extreme By application of the Gibbs-Duhem relationship we can variation of D,,, with composition is matched by an reduce Eq. (6) to extreme variation of B. If one divides D,,, by B one obtains RTK,,,, and as seen in Fig. 1 this quantity has a less dlnnz dlnn, K,d=Kz(7) pronounced composition dependency than 4. These d In cz results refer to lS”C, and similar conclusions apply to their 5°C results. Both sets of results, particularly those at If the mixture is dilute in either component, or if they Y’C, are subject to some scatter, especially at very low have equal partial molar volumes, then d In ni /d In ci = 1 and We can express Eq. (2) in terms of the activity aI and the mole fraction nl as follows
K,=Kz=K,,,,
were
(8)
and this result is analogous to that which shows that the same value of D,,, applies to each component of a two-component mixture. If we may henceforth disregard the factor d In n,/d In cl in Eq. (5) we have to equate 0, of Eq. (3), derived from a ‘kinetic’ viewpoint, with K,,,RT(d In al/d In n,), derived from a ‘thermodynamic’ viewpoint as expressed in Eq. (2). Put more succinctly, does one get a simpler relationship if the driving force for isothermal diffusion is the concentration gradient or if the driving force is the chemical potential gradient? The preamble before this question is required to place the question precisely in the environment of diffusion studies. If it turned out that D, was a constant, independent of composition, whereas K,,, was not (or the other way around) one would lean towards that formulation which gave the constant value. But it does not so turn out. If it
Weight
Fig. I. Triethylaminelwater at
fraction 15°C; 4
Triethylamine
and 0, /B vs composition.
153
On the formulation of the diffusion coefficient in isothermalbinary systems
and at very high values of the weight fraction of triethylamine, where both D,,, and B are changing very rapidly. The second example is that of thermal diffusion in the system cyclohexanelmethanol. The thermal diffusion coefficient, V, was measured in a flow-apparatus by Story and Turner[6]. The quantity cr is the ratio of a thermal diffusion coefficient to the mutual diffusion coefficientD/D,,,-and relates to the steady state separation obtainable with a given temperature difference. The rate of attainment of this steady state is controlled by D,,,,and Story[7] has described how thermal diffusion measurements can be used to measure D,,,, as well as (T. The system cyclohexanelmethanol shows an immiscibility gap at 25”,and we carried out measurements close to the mole fraction of 0.13 cyclohexane which is the lower end of this gap. As the gap is approached B becomes small, and Fig. 2 shows that D, decreases rapidly. This is reinforced by the behaviour of u = D'ID,,,, which increases rapidly as the gap is approached. This rapid decrease of D,,, in parallel with a rapid decrease in B would suggest that the ‘thermodynamic’ approach is more profitable than the kinetic approach. One can proceed somewhat further with this example if one assumes that cyclohexane and methanol form a regular solution. This may well be incorrect, but probably not sufficiently so to invalidate the general conclusions to be drawn from the following treatment. The thermodynamic factor B for a regular solution can readily be
“0 ;; b
shown to be
B_ddlnn,
1-!$r,(l
- nr)
in which T, is the (upper) critical solution temperature for a strictly regular solution showing partial miscibility. For such a solution the miscibility limits should be symmetrical about nl = 0.5 at any temperature T below T,, and knowledge of these limits and T enables T, to be estimated. For cyclohexane/methanol at 25°C the limits of miscibility are at n, = O-132and n, = 0900. These are not quite symmetrical, but a value of T, / T = 1.3 f 0.05 can be estimated from them. Thus, on the assumed model, we can say B = 1- 52n,(l-
n,).
If this expression for B is used with the measured diffusion coefficient D,,, of Fig. 2, one can calculate that D,,,/B = D,,,IB = RTK,,,. The result is 2.2x 1O-9 m*/sec within about 1 per cent for the whole range of n, up to the immiscibility gap-for which the measured values of D cover the range 1.9 x 1O-9-O+3X 10-9. Too much stress should not be laid on the constancy of D,,,/B, as this results from an assumed, if reasonable, model of the liquid non-ideality, and there is also no reason why D,,,/B should not itself be compositiondependent. But if one accepts that the driving force for isothermal diffusion is the gradient of chemical potential one then comes to consider the concentration dependence of D,,,, after the thermodynamic factor B has been allowed for, and this is a much milder concentrationdependence. The mobility of a molecule could well depend on its environment, and Dudley and Tyrrell[4] discuss this ‘residual’ concentration dependence. This is difficult enough to explain for strongly non-ideal solutions, and would seem an impossible task if-as the ‘kinetic’ approach would indicate-the thermodynamic factor B should not be allowed for. For dilute solutions of electrolytes in water, the thermodynamic factor by itself accounts for nearly all of the concentration dependence of the observed diffusion coefficient, and these solutions present particularly strong non-idealities. The examples discussed above add weight to the argument that it leads to a greater understanding of diffusion phenomena if the gradient of chemical potential, rather than the concentration gradient, is regarded as the ‘driving force’ for diffusion. NOTATION
activity of component i B thermodynamic factor, (d In ai /d In s 1 ci concentration of component i, kmol me3 ai
Fig. 2. Soret coefficients and diffusion coefficients for cyclohexane (I)-methanol (2).
(10)
154
J. C. R. TURNER
IA’ kinetic diffusion coefficient, see Eq. (1) m2set-’ D, mutual diffusion coefficient, as measured, see Eq. (3), m* set-’ D’ thermal diffusion coefficient, m2sect’ K-’ Ji flux of component i, kmol m-’ set-’ .A” volume-fixed flux of component i, kmol m-* see-’ Ki mobility constant, see Eq. (2) Lii phenomenological coefficient ni mol fraction of component i R gas constant, J kmol-’ K-’ T temperature, K z co-ordinate distance, m /.L~chemical potential of component i, J kmol-’
CT Soret coefficient, thermal diffusion coefficient, K-’ REFERENCES
[I] Tyrrell H. J. V., LX&on
and Heat Flow in Liquids,
Butterworths,London, 1961. [2] de Groot S. R., Thermodynamics of Irreversible Processes, North-Holland, Amsterdam, 1951. [3] Agar J. N., Trans. Faraday Sot. l%O 56 776. [41 Dudley G. J. and Tyrrell H. J. V., J. Chem. Sot. Faraday Tram: 1, 197369 22bo. I51 . _ Counsel1J. F.. Everett D. H. and Munn R. J.. Pure mol. .& Chem. 19612 335. [6] Story M. J. and Turner J. C. R., Trans. Faraday Sot. 1969 65 1523. [7] Story M. J., Ind. Engng Chem. Fund. 1%9 8 777.