May, 1963
AXALYSIS O F
LIQUIDDIFFUSIVITY hIEASUREXIEKTS
pected from IV because of the large size of the inolecule. One additional comparison which is extremely interesting involves the methyl radical yields from 2,4dimethylpentane and 2,3,4-trimethylpentaiie. One would a t first think that the yield from the latter molecule would be the greater. It is seen however that the fractional increase expected to be produced by the aciditional methyl group (12.52111.52 = 1.09) is more than balanced by the decrease due to the increased molecular size (36/49 = 0.74). ,4 net decrease is thertfore expected and is observed. As iiidicatcd iii Table 111, the methyl radical yields predicted by IV also seem to apply reasonably ncll to the methyl substituted cyclopentanes and cyclohexanes. KOmethyl radicals are expected from cyclopentane and cyclohexane tlxmselves. Small yields are in fact observed. It w m shown by collecting separate fractions that these yields are not attributable to contaminating vinyl iodide. I n the case of cyclopentane one experiment was carried out with an A I ” sample and gave a methyl iodide yield quite similar to that mitli the Ressarch Grade material. It would therefore seem that these inetliyl radicals probably do not result from
1011
impurities but rather that a certain amount of rearrangement may be possible. The methyl radical yield froin methyl cyclopentane is higher than predicted to about the same extent as the yield from cyclopentane alone. The theoretical foundation for equation IV is not obvious a t the moment. It would seem that such a foundation awaits the detailed application of the more general treatments of energy flow within the molecule before bond dissociation occurs. Apropos of this the (n - 1)-2 dependence of the yield on molecular size indicates very strongly that energy must be exchanged between various degrees of freedom. I n the absence of such energy flow one would expect that the yields would be more nearly proportional to the frequency of occurrence of the methyl groups. A related manifestation would seem t o be the large influence of adjacent methyl groups on the probability of rupture of a given C-CH3 bond. Acknowledgment.-The authors wish to express their appreciation to Mr. Gerald Buzzard for his aid with many of the methyl iodide determiiiations reported here.
ATC’ALYSIS OF LIQCID DIFFUSIVITY MEASUREMENTS TO ACCOUNT F0R T’OLTJME CHANGES ON RIIXIS G-THE DIAPHRAGM CELL BY DO~YALD R. OLANDER Depnrtnienb o j IYuclear Enganeerang and Inorganic Materaals Research Dzuzsion, Lccwrence Raclzatzon Laboratory, Gnzversaty of Calzfornaa, Berkeley, Cala.fornza Received September 11, 1966 The method1 of computing differential diffusion coefficients in liquids from diaphragm cell data has been reexamined to allow for volume changes upon mixing. The diffusion coefficient iypon which the new calculational method is based is of greater theoretical interest than the common Fick diffusivity. The difference between the two is usually small in diaphragm cell n-ork, but deviations as large as 6% ha7.e been found for the ethanolwater system.
Most experimental liquid phase diff usioii work is based upon a diffusilon coefficient defined by Fick’s law (in the sense originally used by Fick)
M
-DF,~~VC (1) However, a more sztisfactory definition of the niutual diffusion coefficient for kinary systems is n
E
= - p,DYTB + W(n
+ +
- C,DV.?: x(N N = - DBC C(NV
N
+ n,)
+ N,)
+ NsVs)
(24 (2b)
(2c) I n eq. 2 , the subscript s denotes the component chosen as solvent, and no subscript denotes the solute. ’cli and li: are the solute mass and mole fractions, respectively; D is the mutual diffusion coefficient; n and N the mass and molar fluxes of solute relative to a stationary observer; pt and Ct the total mass and molar density, respectively; P and Vqdenote the partial molal volumes of the solute and solvent, respectively. The quantities in the parentheses of eq. 2 are: (a) the total density times tbe mass average velocity, (b) the total concentration times mole average velocity, and (c) the volume average velocity. Each of eq. 2 is an equivalent definition of the same diffusion
coefficient.lt2 The derivation of eq. 2c from 2b is shown in the appendix. The quantity defined by eq. 2 has numerous advantages over the Fick diffusion coefficient: (1) The diffusivity is defined in the same wav for liquids and gases; eq. 2b is the Stephan-Maxwell equation for a binary gas. ( 2 ) The Fick diffusion coefficient depends upon the component in terms of which concentrations are measured (Le., D A B # D R A ) . The coefficient defined by eq. 2 characterizes the system, and is independent of the method of measurement. (3) Eq. 2 defines the diffusion coefficient to which all theoretical interpretation of transport phenomena in liquids refer ( e . g . , it is the Dv of Hartley and Crank3), (4) The convective term in eq. 2 is essential if a diffusion equation is to be coupled with the hydrodynamic equations, as in the analysis of mass transfer in laminar flow systems.4 ( 5 ) Unless the last t e r n s of eq. 2 are included in the definition of D , experimentally measured diff usivities (1) R. B. Bird, W. E. Stewart and E. N. Lightfoot, “Transport Phenomena,” John Wiley and Sons, S e w York, S . Y . , 1960, pp. 502, 518. (2) I. Prigogine, Bull. Acad. R o y . Sci. Belo., [SI, 34, 530 (1548). (3) G. S.Hartley and J. Crank, Trans. Faraday hoc., 46, 801 (1549). ( 4 ) D. R. Olander, Intern. J . Heat Mass Transfer, 6 , 765 (1962).
DONALD R.OL.4iYDER
1012
mill differ according to the geometry and convective flow conditions of the measuring device. Gross fluid motion need not be the result of “forced” or “natural” convection; some sort of average motion is inevitably present as a result of the diffusive movement of the two components of the mixture. The Fick diffusion coefficient and that defined by eq. 2 are identical if the volume average velocity in the experimental device is zero a t all points of the diffusion path. This condition is precisely fulfilled for diaphragm and capillary cells only if there is no volume change on mixing (Le., if the partial molal volumes of each component are constant in the range of concentration over which the experiment is conducted). The following discussion will develop the means by which volume change effects can be incorporated into the treatment of the data obtained from a typical diffusion cell. For simplicity, the diaphragm cell will be investigated, although a similar analysis should be applied to all experimental diffusivity devices. Descriptions and analyses of diaphragm cells for systems with no volume change on mixing are given in detail elseurhere.6s6 Two compartments (usually of equal volume) are separated by a porous glass diaphragm, through which transfer occurs solely by molecular diffusion. To minimize natural convection effects, the diaphragm is mounted in a horizontal position, with the denser solution beneath the disk and the less dense one above it. The lower compartment is completely filled with liquid, while the upper one contains a small free volume open to the atmosphere to accominodate the small volume changes which may occur.7 An analysis of volume change effects upon diaphragm cell measurement has recently been presented by Dullien and Shemilt However, their derivat’ioiicontains an error; even if it were rigorous, the problem of extracting differential diffusivities from cell measuremeiitmsis so complex a task that the authors did not even apply it to their own experiment’aldata. lo ( 5 ) R. H. Stokes, J . Am. Chem. Soc., 72, 763, 2243 (1950). (6) .4. R . Gordon, Ann. N. Y . Acad. Sei., 46, 285 (194B). ( 7 ) I n principle i t is possible t o avoid volume changes b y sealing off t h e top compartment. However, this will lead to a large pressure buildup in the cell which (if i t does not rupture the apparatus) will affect the diffusion coefficient. ( 8 ) F. A. L. Dullien and L. W. Shemilt, Trans. Paraday S o c . , (18, 214 (1962). (9) F. A . L. Dullienand L. W. Shemilt, Can. J . Chem. Eng., 39,242 (1961). (10) Equation 21 of ref. 8 can he written (in the nomenclature of t h e reference) as
where the symbol X is
dy
=
’/2
f
bAB(WA’rB.4
f K)
WBi)-l(l
K and p are defined by eq. 22 and 23 of the same reference. eq. 21 from the initial t o final conditions yields
Jo@Spd$ =
-
(1nR
Interrating
+I)
where I and R are defined b y eq. 2 5 and 26. Eq. 23 reads
For both of the precedinp equations t o be valid, time average
p
must be the X-weighted
T‘ol. 67
Despite the considerable mathematical difficulties which the problem in its completely rigorous form poses, the generally small perturbation of the common Fick coefficients caused by volume change effects permits an approximate correction factor to be calculated. Since the perturbation from Fick’s Law is caused in part by non-equi-volume flow conditions, eq. 2c has been chosen as the most convenient starting point. Diffusion through the diaphragm is assumed to be oiie dimensional and quasi-steady state. From eq. (2c), the flux is
where y is measured perpendicular to the plane of the Now rewrite eq. 21 as
Integration from the initial t o the final conditions yields
E q . 30 reads
BO =
$ x1- [f
PA’’
0
4pa
dV” __ V“
+ dln
For both of the preceding equations t o be valid, average
Ip.41
p must be the simple time
Thus, Dullien and Shemilt have erroneously employed t w o values of p, neither of which is equal to their eq. 27. The simple average definition reduces t o eq. 27 if p is assumed t o be a linear function of 0. Equations 24 a n d 30 are simply two different definitions of p, and since they are not equal, the combination of eq. 29 (which is based upon the first 6) and eq. 30 (which is based upon the second) cannot be performed t o yield eq. 31. Equation 33, therefore, is not a rigorous formulation of the problem; the error incurred by its use will depend upon the difference between the simple time average and the X-weighted time average of fl as derived above. I n addition, since appears in eq. 29, the difference between the X-weighted 6 and t h e 6 of eq. 27 will influence the result. E v e n if eq. 33 were rigorous, calculation of differential diffusivities from ~ Vo“and the i t presents a formidable problem. If values of PAO’, p ~ f i ” Vo‘, density-composition curves are specified,over-all and one component material balances provide V”, PA”. p’, p ” , OA‘, and WA” as functions of P A ’ . -4 value of TBA is then computed from eq. 8 (with p and PA primed). If a D.44~composition variation is assumed, F can be computed as a function of p.4‘ from eq. 34 (by numerical integration, unless the density-composition curve is amenable to analytic approximation. .I series of F ( P A ’ ) values must be computed for PA“’ 5 PA‘ 5 PAf’, which then permits the two integrals in the denominator of eq. 33 to be integrated numerically. The parameter I is then evaluated by numerical integration of eq. 25. This yields a single value of DABfor a single run and for a n assumed D ~ ~ - c o m p o s i t i o n v a r i a t i o n . These calculations must be repeated for as many assumed DAB-composition curves (each time for all experiments involved) as are required for all of the computed b . 4 ~from eq. 33 to amee with those obtained from the d a t a and b y eq. 29 (there is a n obvious misplacement of I and R here). However, the previous discussion has indicated t h a t the 6 which should appear in eq. 29 (the X-weighted average) is not t h a t used b y Dullien and Shemilt. I n any case, the authors never performed these calculations for their own d a t a on the ethanol-water system. Instead, they computed the effect of volume changes for a single hypothetical run, found i t to be small, and interpreted their own experimental data by the conventional constant volume methods. The reason t h a t the rolume change effect was small in the example treated b y Dullien and Shemilt (in addition to possible errors associated with a n incorrect treatment of 4) is t h a t the concentration range considered (0-60 g. EtOH/100 ml.) was not in the region of the most drastic volume change effects, According to Fix. 1 of this paper the region from -65-78.5 g . / l O O ml. (-14-17 moles/l.) is the most serious in this regard. They explained the 7 7 , discrepancy between their value of the limiting diffusivity a t 100% ethanol and t h a t of Hammond and Stokes12 as due to the increased number of d a t a points and a minor improvement in the conventional graphical method of Stokes. It will be shown in the paper t h a t a discrepancy of this order is due t o volume cha,nae effect8 in ihe work of Hsmmend and Stokes,
ANALYSIS OF LIQUID DIFFUSIVITY MEASUREMENTS
May, 1963
diaphragm, with y = 0 a t the lower face. A material balance 011 the lower or fixed volume compartment shows that the molar flux ratio in eq. 3 equals - ( VO/ V80),where the subscript 0 denotes the lower or fixed volume compartment. Since the right-hand term in the denominator of eq. 3 is zero for ideal solutions and is small even when volume changes on mixing are substantial, the following approximation will be made : the denominator will be written as 1/(1 - x) = 1 x; the solvent molal volume in the correction term will be taken as the average value in the concentrat'ion range between the upper and lower compartments (Vsavg); the ratio of the solute and solvent partial molal volumes will be assumed linear in solute concentration over the same interval
1013
+
v/vs= L + rC,
cl' I c I ci
(4)
where the superscripi; i denotes initial conditions, and the subscript 1 denotes the variable volume (top) compartment. L and r are constants determined from the best linear fit of the partial molal volume ratio-concentration curve. Only a numerical value of r is required for this analysis. With these simplifications and the material balance relation for the bottom compartment, eq. 3 can be integrated to yield
Here t is the time and ,6 the cell constant, which for equal volumes of both compartments is twice the effective pore area divided by the product of the cell volume and the effective pore length. Further treatment of eq. 5 requires a material balance relation between the concentrations in each compartment. If the initial volumes in each compartment are equal to uo, and the pore volume denoted by up} material balances on both the solute and solvent yield
where v1 is the variable volume of the solution in the top compartment. The solvent concentration in the above expression can be eliminated by the relation CV CsVs= 1. The reciprocal partial molar volume of the solvent will be approximated by a linear relation in solute concentration l/Vs = l / V s b &C (7)
+
+
The constants for the region CO' 2 C 2 C1' are obtained from the partial molal volume data. The desired relation between Co and C1 can be written as
c1 = (c:
+ c1' - C,)(1 + F )
(9)
Two special cases of eq. 9 are applicable to most diaphragm cell work (1) Pure solvent is initially placed in the upper compartment, of C? = 0. (2) The upper compartment is pre-loaded with solute to reduce the total concentration driving force. Since the factor F (which represents the fractional volume change of the upper compartment) is zero initially} and a t its maximum at the end of the run, replacement of Co in the denominator of the first bracketed term in eq. 9 by the final concentration in the lower compartment, Cof, is a justifiable simplification. Furthermore, since volume changes are small, CO' C1' - Cof rn Clf. The last term in the second bracket is multiplied by (Coi - Co), which is usually kept small compared to the driving force in high concentration experiments. I n addition, the two components of this term are of opposite sign and tend to cancel. Neglecting the last term in the second bracket mill only introduce a small error into what is already a small correction factor. In both of these cases, eq. 9 reduces to
+
F
2A(C; - Co)(Co - Ci')
(10)
where in the first case (Cli = 0)
and in the second case (C1' # 0)
Utilizing the smallness of F compared to unity and approximating Co in eq. 10 by its equivalent for the constant volume case, the left-hand side of eq. 5 can be written in terms of the driving force e = Co - C1. Gordon0 and Stokes6have shown for r = 0, the righthand side of the eq. 5 can be considered constant a t CO = Corn and C1 = Clm, where Cm = ' / z ( C ' Cf). Since the partial molal volume changes introduce a small correction to the integrand} it mill be assumed that the same approximation is valid for the entire integral of eq. 5 . Expanding the differential diffusion coefficient is a power series in C
+
D
(8)
where, for small volume changedl
=
=
Do
+
j=1
ajCj
(13)
the integration of ( 5 ) yields
(11) Equation 9 IS correct to the extent that
d
r
x
0
I
+ 1/2x - 1/8x2,where x =
2 T v 4 $ o ~ -(4C,' - c+ o~ Z(C0' - C,) [I + '/4?.VSD(CO~ + 11-7- T VBD(C11)2]2 CON - [I - TV8b(C11)21 (COI + C0)l Since r and (00' satinfartorye
-
Co) are usually bath small, this approximation is quite
where
D
is the measured integral coefficient
DOSALD R. OLAXDER
1014
CEtOH
’
Fig. 1.-Partial q~olelvolume ratio for the ethanol-water system a t 25°-Frorn “ThermQd3’namics,” 2nd Ed., by K. S.Pitzer, L. Brewer, G. N. Lewis, axid M. Randall, hfcCraw-Hill Book Co., Xew York, K. Y., 1961, p. 209.
? .. II I
Whether or not the precision of the experimental measurements warrants full scale computation of the usually small difference between the Fick diffusion coefficient and that of eq. 2 can be readily determined from eq. 14. The bracketed terms in eq. 14 can be evaluated for the experiment which involves the largest driving force in the region of largest r. If these terms differ from unity by the same order as the experimental precision, then the correction for volume changes will just barely be significant. It may prove inconvenient to fit the data with a polynomial of a reasonable number of terms, and the graphical method may be easier to employ if r is assumed to be zero. In this case, eq. 14 can be used to provide a correction to the Fick coefficients. The experimental data are fitted to eq. 14 for r = 0 and for the true values.’3 This yields two sets of coefficients, Do*and aj* for r = 0, and Doand aj for r from partial molal volume data. The correctioii to the graphically determined Fick diffusion coefficient is then
,
I
I !
O5I\Z
T’ol. 67
1
I
144 146 148
1
I
’
150 152 154 156 158 160 I 6 2 16/4
Fig. 2.--?vlethod of obtaining the constants describing the diffusivity of the ethanol-water system-with and without consideration of volume changes.
-
1
D = - In Pt
(2)
Application to the Ethanol-Water System.-The system ethanol-water exhibits rather large volume changes on mixing, as indicated by the partial molal volume ratio-concentration curve shown in Fig. I . Since Hammoiid and Stokes12have employed the diaphragm cell technique to measure the diffusivity of this system without accounting for volume changes, the methods derived above will ke used to correct their values. These authors attempted to minimize the effects of volume changes by dividing the concentration range into three groups, 0-6.5 moles EtOH/l., 6.5-13, and 13 to 17 (pure ethanol). It is obvious from Fig. 1 that the last of these groups was most severely subjected to volume change effects; therefore, the analysis was applied to these data (reported in Table I of ref. 12). TABLE 1 DIFFERENTIAL DIFFUSION COEFFICIENT IK THE ETHAKOL-WATER SYSTEM G. EtOH/100 ml. soh.
6j = Corn [>+I
- tj+2
(18)
a
65 70 75 78 51“ Pure ethanol.
Diffusion coefficient, cm.Z/sec. X IO5 With volume changes
No volume changesb
0.561 .766 .981 1.132 From Table 11, ref. 12.
0 561 0 797 1 034 1.200
Equation 14 is the basic relation from which the constants DOand q can be obtained from the data. The AG term accounts for volume changes in the upper compartmeat, and the terms containing r reflect the definition of the diffusion coefficient by eq. 2 . If r = 0, eq. 14 reduces to the relation which has been derived previously by Hammond and Stokes12 and its solution is equivalent to the usual graphical techniques.&I6 Non-integer values of j are permitted, and may be quite useful for solutions of electrolytes. If Do is known (as a Sernst limiting value), eq. 14 is divided by Do and used as with noli-electrolytes.
Complete accounting for volume change effects requires the initial and final concentrations in each compartment, the integral diffusion coefficient, and r , and vsb for each run. Hammond and Stokes, however, have reported only the mean concentrations (Corn and Clm) and the integral coefficients. They stated that the initial ethanol-rich solutions in their group 3 ranged from 70 to 78 g. EtOH/100 ml. solution. On this basis, the data can he divided into three subgroups, according to estimated initial ethanol coiitents of the top compartment: (a) two runs with CIi = 15.19 (70 g./lOO ml.); (b) two runs with CI‘ =
(12) B. R. Hammond and R. H. Stokes, Trans. Faraday SOC.,49, 890 (1963).
(13) Since the ratio D/BF,Dk will usually be close to unity, only a n approximate fit is required to yield a satisfactory correction factor.
rsavg,
AKALYSIS OF LIQUIDDIFFUSIVITY MEASUREMESTS
May, 1963
16.28 (75 g./lOO ml.); (c) five runs with Cli = 17.04 (pure ethanol). With Con‘, Clmand the assumed values of Cli, all of the required concentrations can be coinputed by use of eq. t3, 10, and 12 and the definitions of Corn and Clm. Values of r , VBavg,and v , b are then computed for each of the subgroups from the partial molal volume data and Figure 1. They are14 (a) 13.4
5 C 5 15.2, r = 0.11, VSay,= 0.0164, Tsb = 0.0252
(b) 13.4
5 C 5 16 3, r
= 0.16,
rSavg = 0.0161, Vssb = 0.0314
(c)
15.0
5 C 5 17.0, r = 0.39,
~za,,
= 0.0149,
T,b = -0.0294 Since the differential diffusivities were found to be linear in ethanol concentration, the calculations were performed using only the first term of the series of eq. 14. The results are shown in Fig. 2; the lower line essentially reproduces the calculations of Hammond and Stokes, and the constants are Do* = - 2.230 X loF6and ul* = 0.197 X The upper line shows the effect of including volume changes; its constants are Do = - 2.503 X lW5and al = 0.217 X lo+. The effect on the differential diffusion coefficients caused by this change in the constants is shown in Table I. The maximum change occurs for pure ethanol, where a 6% error in the diffusion coefficient is introduced if volume change effects are neglected. This deviation is substantially greater than the reported experimental precision of 0.57,. These results are subject to the errors involved in approximating the partial molal volume ratio curve by straight lines and in the estimate of the initial concentrations associated with the data of Hammond and Stokes. The major source of the 67, error a t 10070 ethanol is due to Hammond and Stokes’ choice of ethanol as the solute (the component whose concentration appears in Fick’s law) for all experiments. This accentuates the discrepancies resulting from volume changes in the high ethanol content runs. If the computation had been based on wat,er as the solute, the deviiations from Fick’s law would have been much less prominent. The volume change effect is practically negligible in the low ethanol experiments. Considerable cancellation of the effects due to volume changes of the top component (the AG term of eq. 14) and the use of a diffusion coefficient defined by eq. 2 (the rVsavg&term on the left of eq. 14) rendered the net effect of volume changes less pronounced than it would have been if only one of the two effects had been considered. In recent experimental work on the CuSO4-H20 system, l6 by contrast, the r values were considerably larger than in the ethanol-water case. However, the
-
-
(14) r , lisa%g, and l i s b are in liters per mole. (15) A. Ernanuel and D. R. Olandei, J . Chem. Ene. Data, 8 , 31 (1963).
1015
solute (CuSO4) was always a t a molar concentration an order of magnitude smaller than the ethanol concentrations in the experiments of Hammond and Stokes. As a result, the deviations from Fick’s law were found by the method described in the paper to be far smaller than the experimental precision. The small value of r for the system n-butyl alcohol-water rendered volume change effects negligible as well. Conclusion For most binary liquids, the effects of volume changes on mixing are too small to alter appreciably the diffusion coefficients measured in diaphragm cells. However, for systems such as ethanol-water, HC104water, HzS04-water, and HK03-water, volume changes may be of sufficient magnitude to warrant treatment of diaphragm cell data by the method outlined here. Although the treatment here bas been limited to the diaphragm cell, interpretation of data from all types of diffusivity measuring devices should be based upon eq. 2 rather than Fick’s law. Appendix Equation 2b can be written as
C(U - V*)
=
- CtDVx
(A-1)
+
where V * is the mole average velocity, V* xu x,u,. u, and u are the velocities of the solute and solvent, respectively, defined by their relation to the molar fluxes in the lab system, N = Cu and N, = Csu,. The solute velocity relhtive to the mole average velocity can be written as 11 -
v* = C z-
Ct
(u - u,)
(A-2)
Similarly, the solute velocity relative to the volume average velocity is
u-
v+ = C,VS(U - us)
(A-3)
where V + is the volume average velocity, V + = CVU C,VSUS. For a binary system a t constant pressure and temperature VC VsC* = 1 (A-4)
+
+ VdC + VzdCz = 0 (A-5) Using eq. A-4 and A-5, and the relation x = C/Ct, IC,= C,/Ct, Ct = C + C,, the following relation be-
tween mole fraction and molar concentration gradients can be derived
vc = Ct2VsVx
(A-6) Substituting eq. A-2, A-3, and A-6 into A-1, there results
C(U
-
V+)
=
This is equivalent to eq. 2c.
- DOC
(A-7)
