R. C. Srlvastava, M. 0. Abraham, and A.
906
K. Jain
Electroosmosis of Liquid Mixtures. Studies on Aqueous Methanol R. C. Srlvastava;
M. George Abraham, and Abhay K. Jain
Chemlstry Depafiment, Birla Institute of Technology & Science, Pilanl 333031, Rajasthan, Indla (Recelvsd June 1 I , 1876)
Electroosmotic transport of aqueous methanol through a pyrex sintered glass membrane (porosity G 4) has been studied and the data have been analyzed in light of nonequilibrium thermodynamics. The validity of the linear laws and Onsager's relations has been demonstrated for all compositions of the mixtures. Concentration dependence of the phenomenological coefficients has also been studied. The efficiencies of electrokinetic energy conversion for both electroosmosisand streaming potential have been calculated and the results thus obtained have been shown to be in accordance with thermodynamic theories.
Introduction Nonequilibrium thermodynamic studies on electroosmosis of some liquid mixtures through a pyrex sintered glass membrane have been conducted recently's2 and the data thus obtained have been used to test the validity of linear phenomenological relations and Onsager's reciprocal relations. In our own recent studies on acetonemethanol mixtures3 and acetone-water mixtures4 an attempt has been made to study the concentration dependence of phenomenological coefficients in light of Spiegler's model, and to study the efficiency of electrokinetic energy conversion in addition to verifying the Onsager formalism. In the present communication we report our studies on the electroosmosis of methanol-water mixtures through a pyrex sintered glass membrane (porosity G-4 and thickness 0.450 cm) with a view to studying the concentration dependence of phenomenological coefficients and electrokinetic energy conversion. The efficiencies of electrokinetic energy conversion for both the modes of conversions, viz., electroosmosis and streaming potential, have been calculated for various compositions of the mixtures. The results thus obtained support the nonequilibrium thermodynamic theory of electrokinetic energy conversion. Experimental Section Triple distilled water and methanol purified by the method described in literature5 were used in the present studies. The electroosmotic cell used in the present studies and the procedure followed for the measurement of ordinary permeability, electroosmotic velocity, and streaming potentials have already been described in earlier publicat i o n ~ . All ~ ~measurements ~ ~ ~ were made at constant temperatures by placing the electroosmotic cell in a thermostat maintained at 40 f 0.1 "C. Results and Discussion The linear phenomenological relations for the simultaneous transport of matter and electricity, as obtained from the nonequilibrium thermodynamic treatment of electroosmotic effects, are written as7,'
Ll2
(3)
= L2l
among them holds on account of Onsager's theorem. From eq 1and 2 the following equalities can be deduced: ( J " ) A G = ~ = LllAP (J,)AP=o
= LI2A4
The Journal of Physical Chemistry, Vol. 81, No. 9 , 1977
(for electroosmotic
velocity) (5) (A$/AP)r=, = -LZ1/Lz2 (for streaming potential) (6) ( I ) A G ==~ LZlAP (for streaming current) (7) which all predict straight line plots passing through the origin. The values of various phenomenological coefficients L11, L12, LZ1, and LZ2for various compositions of mixtures were calculated from the slopes of such straight line plots and are given in Table I. The validity of Onsager's reciprocal relation (eq 3) for d the compositions of mixtures is obvious from the values of the cross coefficients given in Table I. (a) Concentration Dependence of Phenomenological Coefficients. The values of the phenomenological coefficients Lll, LlZ,and L22have been plotted against the composition of mixtures in Figures 1 and 2, from which it is apparent that the concentration dependence of the coefficients Lll and Llz is represented by the linear relationships
Lii = ( L ~ I ) M X+ M (Lii)wXw (8) and L21 Or = (L12)MXM + (L12)WxW (9) respectively. In eq 8 and 9 X represents the mass fraction and the subscripts M and W represent methanol and water, respectively. Following the procedure outlined in our earlier papers3l4it can be shown that eq 8 and 9 for the concentration dependence of the phenomenological coefficients are consistent with the Spiegler frictional model.g The concentration dependence of the coefficient LZ2,however, is nonlinear (Figure 2) and was found to be represented by the equation L~~= ( L ~ -~0.75 ) ~x 10-5xM - 2.1
x 10-5~~2
where J, represents the volume flux, Z the flow of electricity, and AP and A4 are the pressure difference and electrical potential difference, respectively. The coefficients Lik are called the phenomenological coefficients and the following equality
(for ordinary permeability) (4)
(10) It can be seen from Figures 1 and 2 that the coefficient Lll decreases as we go from pure methanol to pure water while coefficients Llz and LZ2increase. These trends can be rationalized in terms of structural modification that are likely to occur in aqueous methanol because of hydrogen bonding.
907
Electroosmosis of Liquid Mixtures
TABLE I: Values of the Phenomenological Coefficients, Figure of Merit, and (
) at~Various ~ Compositions of
Q
Methanol-Water Mixtures ~~
Mass fractions lo6 L , , , of cmJ MeOH dyn-ls-l 0.0 1.410 0.1 1.834 0.3 2.550 3.307 0.5 4.059 0.7 4.732 0.9 1.0 5.081
105L,,, ohm-' 5.50 5.40 5.26 4.54 4.00 3.12 2.50
~
104 L , , , cm3 A J" 10.52 9.75 9.09 8.33 7.66 7.02 6.34
104 L , , , cm3 A J-' 11.03 9.80 8.81 8.19 7.78 7.39 6.29
105
105
(qe)max
(q,lmw
(from
lo4 pio 14.20 9.50 6.10 4.60 3.61 3.30 3.17
lo4 Poi 16.75 9.70 5.70 4.4 3.73 3.60 3.12
(from
Pi,)
Poi)
35.00
41.60 24.25 14.25 11.00 9.32 9.00 7.80
23.70 15.20 12.50 9.02 8.25 7.91
105
105
(Qelmax
(qslmax
(from the (from the plot of plot of qeVS.
AP)
35.60 23.90 15.50 13.00 9.10 8.50 7.88
1)sVS.
A@)
40.00 24.00 14.00 11.50 9.40 8.80 7.60 1
- 70
-60
f
-50 W
-40
2 3 0:
- 30
e
v
X
-20
- 10
MASS FRACTION OF METHANOL-
Flgure 1. Variation of methanol.
2
L 11 and L 12 coefficients with mass fraction of
, t A P cms OF LIQUID
---+
Flgure 3. Dependence of qe on output force AP.
'2
and
77 = - - = JAP
'50 ul
0 x N N
-I
0
0
0.2
0.4
0.6
0.8
M A S S FRACTION OF METHANOL
1.0
---+
Figure 2. Variation of L P Pwith mass fraction of methanol.
(b) Electrokinetic Energy Conversion. Osterle and co-workers10-12and Kedem and Caplan13have discussed the efficiency of electrokinetic energy conversion on the basis of nonequilibrium thermodynamics. According to their definition the efficiency of energy conversion, for both the modes of conversion, viz., electroosmosisand streaming potential, can be written as
JAP IAdJ
77e=--=-
JAP (AdJ)2/R
(AdJ)'lR JAP
where the subscripts e and s represent the phenomenon of electroosmosis and streaming potential, respectively, R being the electrical resistance of the system. From the discussion of the efficiency of electrokinetic energy conversion defined by eq 11and 12, the following conclusions have d e d u ~ e d . ' ~(i) ' ~The ~ maximum values for the energy conversion efficiency, (q)-, for a fixed value of input force, always occurs when the output force equals half its steady state value. (ii) The values of (&, are independent of = (q,),,, which in fact is a the input forces. (iii) consequence of the Onsager theorem. The values of qe and qs for various composition of the mixtures, corresponding to two fixed values of input forces, were calculated using the method described earlier14from the transport data on ordinary permeability, electroosmotic velocity, streaming potential, and streaming current a t several values of output forces ranging between zero and their steady state values. The results have been shown in Figures 3 and 4 for three typical compositions of the mixtures as a sample. Figures 3 and 4 clearly demonstrate the validity of all three conclusions listed above. The values of (q),,,= for all compositions of mixtures obtained from the curves of the type shown in Figures 3 and 4 are given in Table I. The values of & ,(,, for various comThe Journal of Physlcal Chemistry, Vol. 87, No. 9 , 1977
R. C. Srivastava, M. G. Abraham, and A. K. Jaln
908
O--O
AP-5 FOR X ~ ‘ 0 . 0
M(
AP-IO FOR XM*O*O
L----.,
AP.5
Acknowledgment. Financial support from the Council of Scientific and Industrial Research, New Delhi, in the form of a Junior Research Fellowship to MGA and a Post-doctoral Fellowship to AKJ is thankfully acknowledged.
FOR X ~ ‘ l . 0
15.0
450
Y
A + x io4
VOLT
--+
Figure 4. Dependence of qs on output force A@
positions, were also computed from the values of the phenomenological coefficients (Table I) using the relationship”
References and Notes
where
the subscripts o and i represent the output and input quantities. The values of thus computed match (Table I) the values obtained from the experimental curves of the type shown in Figures 3 and 4. The quantities fl in eq 13 have been termed as figures of merit” and it follows as a consequence of Onsager’s relation that Pi0 = Poi
(15)
The values of ,f3 for both modes of conversion were calculated from the values of the phenomenological coefficients and are given in Table I. These data (Table I) confirm the validity of the eq 15.
The Journal of Physical Chemistry, Vol. 81, No. 9, 1977
Appendix. List of Symbols volume flux, cm3 s-l I flow of electricity, A Ai’ pressure difference, dyn cm-2 A+ electrical potential difference, V L11 phenomenological coefficients representing ordinary permeability, cm6 dyn-’ s-l L12 phenomenological coefficient representing electroosmotic velocity, cm3 A J-l Lzl phenomenological coefficient representing streaming current, cm3 A J-l LZZ phenomenological coefficient representing electrical conductivity, ohms-’ X mass fraction of the species denoted by the subscripts 9 efficiency of energy conversion for the phenomenon denoted by the subscripts R electrical resistance, ohms B figure of merit Subscripts input and output quantities, respectively i, o W, M water and methanol, respectively To convert the phenomenological Coefficientsinto MKS the conversions 1 dyn = 1 N X 1 N m = 1 J = 1A V s may be used.
J,
(1) R. L. Balokhra and T. C. Singhal, J. Phys. Chem., 78, 2302 (1974). (2) R. L. Balokhra and T. C. Singhal, Electroanal. Chem. Interfacial Electrochem., 57, 19 (1974). (3) R. C. Srivastava and M. G. Abraham, J. Colloid Interface Sci., 57 58 (1976). (4)R. C. Stivastava and M. G. Abraham, J. Chem. Soc., Faraday Trans. 7, 72, 2631 (1976). (5) A. I. Vogel, “A Text Book of Practical Organic Chemlstry”, 3rd ed, Longmans, Green, and Co., London, 1956,pp 163-179. (6) R. C. Srivastava and P. K. Avasthi, KolloidZ. Z. Polym., 250, 253 (1972). (7) S.R. DeGroot, “Thermodynamics of Irreversible Processes”, North Holland Publishing Co., Amsterdam, 1966. (8)A. Katchalsky and P. F. Curran, “Non-equilibrium Thermodynamics in Biophysics”, Harvard University Press, Cambridge, Mass., 1965. (9) K. S. Spiegler, Trans. Faraday Soc., 54, 1408 (1958). (IO) J. F. Osterle, Appi. Sci. Res., 12, 425 (1964). (11) F. A. Morrlson and J. F. Osterie, J. Chem. Phys., 43, 21 1 1 (1965). (12) R. J. Gross and J. F. Osterie, J. Chem. Phys., 49, 228 (1969). (13) 0.Kedem and S. R. Capian, Trans. Faraday Soc., 61,1897 (1965). (14) R. C. Srivastava and A. K. Jain, J . Hydroi., 25, 339 (1975). (15) R. C. Srivastava and A. K. Jain, J. Polym. Sci., Polym. Phys., 12, 1603 (1975).
