1017
CROSS-PHENOMENOLOGICAL COEFFICIENTS
Cross-Phenomenological Coefficients. 111.
Studies on Electroosmosis
by R. P. Rastogi and K. M. Jha Department of Chemistry, University of Gorakhpur, Gorakhpur, India
(Received July 27, 1066)
The electroosmotic permeability of conductivity water through sintered Pyrex glass has been measured for varying magnitudes of potential difference, A+, at 35”. Measurements have also been made for electroosmotic pressure differences per unit potential difference which have been found to be constant up to 60 v. Measurements on electroosmosis show that the cross-phenomenological coefficient is also constant up to 60-70 v. Experiments for the direct test of phenomenological equations have been performed. The temperature dependence of cross-phenomenological coefficients has also been studied. The results appear to be consistent with the classical theory of electroosmosis.
Introduction Linear phenomenological laws are used as axioms in thermodynamics of irreversible However, direct experimental tests of linear phenomenological laws when two or more forces are involved have not been attempted so far. These laws are valid within a certain range and it is of considerable interest to know this range. Further, very little information is available regarding the cross-phenomenological coeffi~ients.58~The objects of this paper are (i) to ascertain the magnitude of cross-phenomenological coefficients in electroosmosis of water through sintered glass, (ii) to study its dependence on temperature and potential difference, and (iii) to test the validity of linear phenomenological equation in this case, and to assess its range of validity. The phenomenon of electroosmosis has been chosen for detailed study because various electrokinetic effects which are related to phenomenological coefficients can be studied experimentally. However, experimental difficulties are known to be considerable.’ It may be remarked that the experimental studies recorded in the paper are the first of their kind. Phenomenological Equations for Electrokinetic Phenomena The expression for entropy production, u at temperature T (OK), for electrokinetic phenomena can be written in the form’ T u = JAP $- l A +
The fluxes are the volume flow, J , and the current I .
The forces are, respectively, the pressure difference A P and the potential difference, A+. The linear phenomenological equations are consequently written as
L2l
1 = -AP
T
+ -A+ T L22
where Lm (i, k = 1, 2) are called the phenomenological coefficients. When i # k , the coefficients are called cross-phenomenological coefficients. When A+ = 0 Lll
J = -AP T
(3)
From Poisseuille’s law, flow through n capillaries is given by dV dt
&APn
J=-=------
8171
(4)
(1) K. G. Denbigh, “The Thermodynamics of Steady State,” Methuen and Co. Ltd., London, 1958. (2) 5. R. de Groot and P. Mazur, “Non-equilibrium Thermodynamics,” North-Holland Publishing Co.,Amsterdam, 1961. (3) I. Prigogine, “Introduction to Thermodynamics of Irreversible Process,” Interscience Publishers, Inc., New York, N. Y., 1961. (4) D.G.Miller, C h m . Rev.,60, 15 (1960). ( 5 ) R. P. Rastogi, R. L. Blokhra, and R. K. Agarwal, Trans. Faraday
Soc., 60, 1386 (1964). (6) R. P. Rastogi, R. C. Srivastava, and K. Singh, ibid., 61, 854 (1965). (7) H.R. Kruyt, “Colloid Science,” Vol. 1, Elsevier Publishing Co., Amsterdam, 1952.
Volume 70, Number 4
April 1966
1018
R. P. RASTOGI AND K. M. JHA
where V is the volume of the liquid flowing through the capillaries, .r is the radius of the capillary, q is the viscosity of the liquid, and 1 is the length of the capillary. Comparing eq 3 and 4 it follows that
(5) Similarly, when A P = 0
I
L22 T
= -A$
but from Ohm's law I = A$/R, where R is the resistance. On comparing eq 6 with Ohm's law, we find that 4522
=
T
jj
(7)
Ohm's law has a very large domain of validity; i.e., LZ2is independent of A$ up to a very high value of A$. The extent to which L22/T depends upon temperature is largely governed by the temperature dependence of R which is generally given by
R
=
Ro
+ CYT+ PT2 + y T 3 + . . .
where Ro,a,0,and y are constants. As long as the flow does not become turbulent and the Reynolds number is not greater than 1O00, Poisseuille's law remains valid and hence L I T remains constant up to a fairly high value of A P . Thus, whereas we find that we have reasonably reliable information regarding the limits up to which L11 and LZZare ineependent of the corresponding forces, we do not have any information about LZIor LIZ. From eq 1 it follows that when A P = 0
where ( J )4p -0 is called the electroosmotic velocity. The quantity L12/T may be called the electroosmotic permeability. It is clear that measurements of electroosmotic velocity for various values of A$ a t a constant temperature can directly yield the value of LIZ. In the same fashion, when A$ = 0
AP
(1)A+ = 0 = L Z l 7
(9)
where ( I ) 4 + = 0 is called the streaming current. Measurements of streaming current for various values of AP would directly yield the value of L z ~ . From eq 1it follows that when J = 0 (lo)
The Journal of PhU8icrrl. C h m i s t r y
where (AP)Jpo is the electroosmotic pressure. Equation 1 can be written as (J)tOtSl
=
(J)A+-O
+
(J)AP=O
(11)
where ( J ) ~ t , is l the resultant flow. The three quantities involved in eq 11 can be directly and simultaneously measured and hence the equation can be conveniently tested. Experimental details for the measurement of L12, (J)totsl, ( J ) A , = o , ( J ) A P , o and electroosmotic pressure are given below.
Experimental Section Membrane. Sintered Pyrex glass was used as membrane. The Gd variety was chosen to get maximum value of electroosmotic pressure, since according to the classical theory of electrokinetic phenomena it is inversely proportional to the square of the radius of the capillary. It may be noted that it is difficult to get a single capillary with a radius equal to that of the pores of sintered Pyrex. Since the electroosmotic velocity depends directly on the fourth power of the radius, it would tend to decrease when pore size is reduced. Fortunately, this factor is compensated for the case of sintered Pyrex since it acts as a bundle of capillaries and the amount of liquid flowing is multiplied by the number of such capillaries. Treatment of the Membrane. The disk was cleaned with boiling hot nitric acid followed by steaming from either side and washing with filtered boiling hot, deionized water several times. The washing with the deionized water was continued until the disk was practically free from acid. This was ensured by noting the conductivity of the washings. It was then dried in an electrically controlled air oven a t a temperature of 180" for a period of 3-4 hr. Before starting the experiments, boiling hot deionized water which was previously filtered was sucked three or four times from either side of the disk to ensure the wetting of the maximum number of pores. Prior to starting the experiment, the conductivity of the washing was checked once again. The apparatus was then filled with freshly degassed and filtered deionized water and kept in an air thermostat for 45 min to attain the temperature of the bath. The fluctuation in the thermostat was of the order of j=0.05". Material. Fresh distilled water was twice distilled over potassium permanganate. The specific conductivity of the deionized water was of the order of (1.2-3) X ohm-' cm-l. The specific conductivity was measured by an ac conductivity bridge based on Jones' design.*
1019
CROSS-PHENOMENOLOGICAL COEFFICIENTS
M
L
respectively. The tubes L and M each of 1.138-cm diameter were kept parallel. When the liquid in the limbs L and M was the same, a constant dc voltage was supplied. The rise of water level in L or M a t different time intervals was noted with a vernier telescope. Time was measured with the help of a stop watch reading up to 0.1 sec. The measurements were made for voltages varying from 10 to 120 v. If X is the rise in the water column in one limb, the fall in the other tube would be to the same extent. The volume rate of flow would be given by AdXldt, where A is the cross-sectional area of the tube. Now according to eq 1
In the beginning of the experiment, AP = 0 and electroosmotic flow predominates which is instantaneous; hence
Figure 1. Apparatus for the measurement of electroosmotic pressure difference and electroosmotic permeability.
Source of Emf. An electronically variable voltage supply waa used as a source of emf. The strength of the current drawn was in the range of 2-5 ma. This greatly facilitated the reduction of polarization. In order to maintain the consistency of the voltage during the run, a voltage stabilizer was connected in series. Fluctuations in the voltage were of the order of 2 4 % . Electrode. Silver-silver chloride electrodes did not work well. Hence, spiral-type platinum electrodes were used which were kept pressed against the two sides of the sintered disk, covering the maximum area of the disk. Measurement of Electroosmotic Permeability and Electroosmotic Pressure Diference. Apparatus. The apparatus used for the measurement of electroosmosis is shown in Figure 1. The central part consisted of a Pyrex tube 30 cm long and 2.5 cm in diameter with a sintered glass disk, A, fixed in the middle. The tube had two female joints, B and C, at the ends. Platinum wires, F and G, were fixed to two standard male joints as shown in Figure 1 through tubes P and N. The electrical contact was made with mercury. The length of the wires was adjusted in such a way that when the standard joints were kept in position, the spiral ends of these just pressed the disk. The central tube, BAC, had two side tubes, H and K. The latter was joined at Z. The tubes H and K had standard joints D and E,
Expressed differently, in the initial stages the rise of the water column would be entirely due to electroosmosis, and the rate of rise would be constant at constant temperature when voltage is kept constant. Thus when X is plotted against time, the limiting slope is LIZA$/TA from which the magnitude of L12 can easily be determined. The method is quicker and more convenient than the conventional methods. One typical plot of X against t is given in Figure 2. Another way of getting the same information from the same set of data is as follows. A P would be equal to 2Xpg where p is the density of the fluid and g is the acceleration due to gravity so that
Equation 14 can be integrated, but the integrated form is of little value from the point of view of the analysis of the data. However, since for a particular value of A 4 the second factor would be constant at constant temperature even if LIZdepends upon A4, one can plot dX/dt against 2X, and a straight line should be obtained (dX/dt can be determined graphically either directly or by finding a suitable analytical equation for the curve). The slope of this straight line would give the value of ordinary permeability and the intercept would be LIZ. It is difficult, however, to estimate the value of dX/dt with precision. Hence, ac( 8 ) G . Jones and R. C. Joseph, J. Am. Chem. SOC.,50, 1049 (1928).
Volume 70, Number 4
April 1066
1020
R. P. RASTOGI AND K. M. JHA
tion is negligible. In other words, (J),,,,, is equal to (JL-0.
Test of Linear Phenomenological Equations. Experiments were performed in order to test the range of validity of eq 1. The experimental setup was quite similar to that described above. It was modified to have simultaneous values of (J)total,( J ) A p , o and (J),+-o. For convenience the flow rates in these experiments were measured by noting the time taken by the water-air meniscus to move through a definite distance in the exit capillary, the cross-sectional area of which was 2.78 X cm2. nu&
(nrmwrs)
Figure 2. Plot of AP as a function of time in a typical experiment.
curate values of ordinary and electroosmotic permeability cannot be obtained, but they are useful as rough correlations. I n order to study the temperature dependence of L12, measurements of electroosmotic velocity were made at different temperatures for A$ = 70 v. The results are recorded in Table 111. The electroosmotic pressure difference was measured with a similar apparatus. The steady-state value of (AP)J-o was measured with a vernier telescope for various values of A$. The steady state was usually attained in 2-3 hr. Measurement of the Ordinary Permeability. I n order to have information about the magnitude of Lll and the domain of validity of Poisseuille’s law in the region under study, careful studies of ordinary permeability were undertaken. The measurements for the ordinary permeability of deionized water through the disk were carried out in two different ways using essentially the same apparatus as was used for measuring electroosmotic permeability in order to assess the influence of capillary active forces on fluid flow in the present case. In one case, the flow was measured in a capillary by noting the movement of the water-air meniscus, while in the other the flow was measured by noting the mass of water that came out of the capillary in a certain time interval. We will call the former the volume-flow method and the latter the mass-flow method. The amount collected in a definite interval of time was weighed and the amount flowing per unit time was thus calculated. The experiments were repeated using a capillary of 4.323 X 10-*-cm radius instead of the Pyrex disk. Blank experiments were performed by inducing the streaming potential to zero. These showed that the effect of the streaming potential on ordinary permeaThe Journal of Physical Chemistry
Reproducibility and Sources of Errors The following factors are likely to affect the reproducibility of the results for ordinary and electroosmotic permeability. Incomplete Wetting of the Disk. The rate of flow of liquid depends upon the number of capillaries available for the transmission of the liquids; hence, if all the capillaries are not wet, poor reproducibility would be expected. Prior suction of boiling hot deionized water from both the sides of the disk several times ensured wetting of maximum number of the capillaries. This treatment also minimized the number of air bubbles in the capillsries. Blocking of the Capillaries. If periodic blocking and opening of the capillaries in the sintered glass disk were to occur, reproducibility would be badly affected. Blocking is mainly due to suspended impurities, and opening of the capillaries is due to the force of the flowing liquid. In order to avoid this, carefully filtered deionized water was used. Care was also taken so that no suspended impurities were introduced during the run. Cleanliness of the Disk. Since ordinary flow is affected by the presence of grease in the pores, its use was avoided. Further, since inorganic impurities affect the 1 potential, which in turn affects electroosmotic flow, only deionized water was used. Action of Watey on Glass Membrane. This is a very serious source of error and affects the reproducibility tremendously. Even with the use of carefully purified water after filtration, the permeability of the sintered glass disk was found to vary with time. The permeability could be reasonably reproducible only when the disk was subjected to the specific treatment described earlier. The reason for this behavior is not definitely known.9 The following reasons have been ascribed: (a) chemical action, and (b) swelling. The evidence, although very meager, is perhaps in favor of the for(9) H.George, “Properties of Glass,’’ Reinhold Publishing Corp., New York, N. Y., 1954.
CROSS-PHENOMENOLOGICAL COEFFICIENTS
1021
mer. Consequently, experiments had to be devised which could be finished in a short period of time. Polarization of Electrodes. Due to evolution of gases a t the electrodes, there is the likelihood of bubbles sticking to the sides of the capillaries, thus causing significant errors in precisely knowing the actual height of water column in the tube. This was, however, minimized by using freshly boiled conductivity water for the filling of the apparatus, and currents of very small magnitude (of the order of 2-5 ma). The conductivity of the water used was checked before and after the experiments. The per cent change in the conductivity of deionized water, on the average, for a run of shorter duration (up to 2 min) in which we were only interested in the measurement of the electroosmotic velocity, is given below. The increase in conductivity for the measurements up to 80 v was of the order of 570, whereas for 90 v and beyond, the increase was more than 20%. It is thus apparent that in the measurement of electroosmotic velocity up to 80 v, where the change in specific conductivity is only 5%, very small amounts of electrode products are formed. This is confirmed by the good reproducibility of observations with the same sample of water. In the measurements of electroosmotic pressure difference where each run is of comparatively longer duration (2-3 hr) the per cent change in conductivity was as follows. In experiments with voltages varying from 10 to 50 v. the increase in the specific conductivity was of the order of 7 3 % , but when the voltage was further increased to 70 v, there was a 20% increase in the specific conductivity. For 90 v, there was a threefold increase in the specific conductivity.
Table I : Electroosmotic Velocity and Cross-Phenomenological Coefficients for Water a t 35'
A+, ' 1
10
20 30 40 50 60 70 80 90 100
110 120
Volume cc sec-1 x 102
0.66 i 0.04 1.40 i 0.10 2.13 i 0.06 2.84 =k 0.16 3.65 f 0.18 4.08 & 0.21 4.83 f 0.14 5.60 f 0.10 4.70 rt 0.70 5.20 i 0.40 4.60 f 0.30 5.70 i:0.20
Heating of the Fluid in the Pores. Since the magnitude of the current passed was of the order of 2-5 ma only, and since the duration of the experiment was very short for the electroosmotic velocity measurements, the chances of heating were very small. If A 4 = 100 v, time = 120 sec, the heat produced for 5 ma current would equal 60 joules. Since the amount of water in the main tube (BAC) is about 150 cc, the maximum temperature rise during the whole course of one single experiment would only be approximately 0.1'. However, if local heating takes place, the viscosity should go down and the electroosmotic permeability should be higher. For experiments of longer duration involving the study of the electroosmotic pressure dif'ference, the effect of heating may have been significant. It was assumed that the field on the disk was uniform in spite of the shape of the electrodes.
Results Results are given in Tables I to V and represented in Figures 2, 3, 4, and 5. The standard deviations from the mean are recorded along with the data.
Table 11: Test for Linear Relation for Purely Hydrodynamic Flow
Disk Disk Capillary Capillary
Method of experiment
Slope or LII/T,U g seo -1 dyne-' cmg
Volume flow Mass flow Volume flow Mass flow
1.59 X 1.54 X 1 0 - j 1.22 X 1.15 x 10-5
Intercept,a g sec-1
2.64 X 0.40 x 7.62 X 1.44 x
10-3 10-3 10-8 10-3
The values recorded were obtained by the method of least squares.
Liz/T,* cc eec-I V-1
x
104
6 . 6 i 0.40 7.0 i 0.50 7 . 1 i.0.20 7 . 1 rt 0.40 7 . 3 f 0.36 6 . 8 f 0.35 6.9 i 0.20 7 . 0 f 0.13 5 . 2 f0.8 5 . 2 zk 0.4 4 . 2 f.0 . 3 4.8 f0 . 2
The values recorded are the mean for five runs corresponding to each value of A@.
Table 111: Temperature Dependence of LI2/Ta
Temp, OK
303 308 313 318 323 328 333
---Membrane LIVT
x
104
3.9i00.2(5) 4.4&0.2(4) 5.1 i 0 . 2 ( 6 ) 5 . 4 f0 . 2 ( 7 ) 5.8 i 0 . 2 (6)
Av
I---LIWT
--Membrane LWT
106
104
x
4.06 4.23 4.55 4.50 4.53
4.37 & 0.18
x
11----. LIWT x 106
3 . 4 f 0 . 1 (5) 2.84 3 . 5 i 0 . 3 (6) 2.73 3.7 f 0 . 2 ( 3 ) 2.73 3.9 & 0.2(4) 2.72 Av 2.75 f 0 . 0 4
a The figures in the parentheses indicate the number of readings.
Volume 70, Number 4 April 1966
1022
R. P. RASTOGI AND K. M. JHA
Table IV : Electroosmotic Pressure Difference for Water a t 35’0 A+l
( A P ) J =a,
V
cm of water
L i d T X 104, cc see -1 v -1
Lii/T X 108, cc sec-1 dyne-1 cmp
10 20 30 40 50 60 70 80 90
5.05 f 0.15 10.08i0.20 16.00i0.44 20.33 f 0 . 3 6 25.46 f 0 . 5 8 31.40f1.50 39.7Ofl.00 46.60f1.30 55.15 f 1 . 0 0
6.6 f0.4 7 . 0 f0 . 5 7 . 1 f0 . 2 7 . 1 f0 . 4 7 . 3 f0.4 6 . 8 =k 0.35 6 . 9 f 0.20 7 . 0 f 0.13 5.2 f 0 . 8
1.59 X 1.59 x 1.59 x 1.59 x 1.59 x 1.59 x 1.59 x 1.59 x 1.59 x
10-6 10-6 10-6 10-6 10-6 10-6 10-6 10-6 10-6
( A P / A + ) J = o X 10
5.05 f 0.15 5.04 f 0.10 5.33 f 0.14 5.08 i 0.09 5.09 f 0.13 5.23 f 0.25 5.67 f 0.14 5.82 f 0.16 6.13 f 0.11
LIZL/11
x
101
4.15 f 0.26 4.40 f 0.32 4.47 f 0.12 4.47 f 0.26 4.59 i 0.26 4.28 i.0.23 4.34 f 0.12 4.40 f 0.08 3.27 f 0.52
a The values recorded in columns 2 and 3 are mean of five readings corresponding to each value of A+ and those in column 4 were determined by the method of least squares.
Table V : Direct Test of Linear Phenomenological Equations“
---
r
Rate of permeation, cm sec-I---------
AP,
Disk no.
1 1 1 2 3 3 4 4 5 6
(J)total
om of water
15 106 114.5 88.5 106 114.5 106 114.5 126 106
(J)A+=o
( J )AP = 0, A+ = 50V
0.27 f 0.02 (10) 1.95 2.08 f 0 . 1 3 ( 2 ) 4.46 Z!Z 0.22(6) 1.53 1.59 f 0 . 0 8 ( 2 ) 2.27 2.46 0.46 f 0.04(4) 2.68 f 0.26(2)
0.39 f 0.02 (10) 0.39 0.39 f 0.02(2) 1.33 f 0 . 0 6 ( 6 ) 0.31 0.31 f 0 . 0 1 (2) 0.46 0.45 0.15 =k 0.005 (4) 0.62 & 0.03(2)
‘The figures in the parentheses
ewtl
0.066 f 0 . 0 4 2.34 2.47 f 0.15 5.79 i 0.28 1.83 1.90 i 0.09 2.73 2.91 0.61 zk 0.05 3.30 =k 0.29
indicate number of readings.
Discussion Test for Linear Relation for Flow in the Absence of Potential Di’erence. A detailed verification of the linear relation between pressure difference and the isothermal fluid flow in the absence of potential difference was necessary because the present investigation basically involves a study of liquid flow through the small capillaries of the disk. Volume flow per second and mass flow in grams per second are plotted against the pressure difference in Figures 3 and 4. For the sake of comparison, the volume flow has also been converted into mass flow by multiplying the volume by the density of water at 35”. Straight lines are obtained in both the cases but these do not pass through the origin. The values of the intercepts, obtained by the method of least squares, were g sec-’ for volume flow and 0.4 X 2.64 X g sec-’ for mass flow. It may be noted that these are different. The Journal of Physical Chemistry
(J)totsi,
0.63 f 0.02(10) 2.24 2.47 =k00.1S(2) 5.74 f 0.19 (6) 1.70 2.00 i 0.09 (2) 2.78 2.94 0.62 f 0.04(4) 3.30 f 0 . 2 9 ( 2 )
sum of columna 3 and 4
I n order to have a further check on the existence of the intercept, similar experiments for volume flow and mass flow were carried out with a single glass capillary. When the corresponding data are plotted in the above manner, the intercept in the case of volume flow is g sec-l as compared to 1.44 found to be 7.62 X X g sec-’ in the case of mass flow. The magnitude of uncertainty of the slope and the intercept was 5 to 6% as obtained by statistical considerations. The result for the two cases has been compared in Table 11. Since the straight lines do not pass through the origin, Poisseuille’s relation apparently is not obeyed. The reason for the occurrence of intercepts may be ascribed to the kinetic energyl0,l1of the flowing liquid through ~
(10) F. H. Newman and G. F. C. Searle, “The General Properties of Matter,” Edward Arnold (Publishers) Ltd., London, 1957, Chapter VII. (11) F. Daniels, J. Williams, P. Bender, R. Alberty, and C. D.
Cornwell, “Experimental Physical Chemistry,” McGraw-Hill Book Co., Inc., New York, N. Y . , 1962, Chapter VIII.
1023
CROSS-PHENOhfENOLOGICAL COEFFICIENTS
O P ( r m . Or W A T Z ‘ R )
Figure 3. Test of Poiseulle’s law for flow through Pyrex sinter.
flow was measured in capillaries for the test of the linear phenomenological relations. Since ( J )total and (J)*+=Oinvolve the same amount of error on this account, the conclusions are unaffected. The values of Lll/T were estimated by the method of least squares. Electroosmotic Velocity. When the electroosmotic flow is plotted against AC$, a straight line passing through the origin is obtained within experimental error. The behavior of the disk near about 100 v becomes ambiguous. When the voltage is increased beyond 70 v, the value of Llz/l’ shows a decreasing tendency. This may be due to decrease in the value of ( potential caused by the appearance of an increasing amount of ions due to electrolysis. The conductivity measurements of deionized water in the apparatus before and after the experiment confirm the existence of ionic species. At AC$ = 70 to 110 v, the specific conductivity of water was found to increase from 5 to 30%. Temperature Dependence of Llz/T. According to the classical theory of electroosmosis7
Liz T
0
i
2
3
4
5
A P (cms
6
7
6
9
Io
11
IP
OF WATCR)
Figure 4. Test of Poiseulle’s law for flow through a single capillary.
the disk or a single glass capillary, which is unaccounted for in the Poisseuille’s equation. It is important to note that although the slope for volume flow and mass flow is identical within experimental error, the intercepts differ considerably. It appears that capillary forces are primarily responsible for this discrepancy. I n the case of mass flow, the capillary forces are utilized in the formation of the drops a t the outlet, and consequently the intercept has a smaller value. The apparatus for electroosmotic velocity measurements and electroosmotic pressure difference measurements were so designed that the use of capillary was avoided. Hence the influence of capillary-active forces is entirely negligible when the pressure difference developed is considerable. It was thus ensured that the fluid satisfied the linear relation. However, the rate of
- DlS
4~qZ
(15)
where D is the dielectric constant of the medium, ( is the ( potential a t the interface, S is the area of the cross section of the capillary, q is the viscosity of the medium, 1 is the length of the capillary, and T is the temperature in degrees absolute. Equation 15 shows that L12/T should depend on the temperature since the dielectric constant D and the viscosity q depend on temperature.12 The manner in which Llz/T depends on temperature, however, is not very clear. From the experiment,
Figure 5. Electroosmotic uressure difference a t various voltages.
(12) R. A. Robinson and R. H. Stokes, “Electrolyte Solutions,” Butterworth and Co. Ltd., London, 1959.
Volume 70,Number 4
April 1966
1024
however, it appears that L12/T has a tendency to increase with temperature. A rearrangement of the above equation shows that the quantity ( L a / T ) ( q / D ) should be independent of temperature for the same system. Table I11 confirms this point. Electroosmotic Pressure Diference. When (AP) J,.o is plotted against A& a smooth curve is obtained which passed through the origin. This may be treated aa a straight line at least in the region up to 60 v as shown in Figure 5. Up to 60 v the conductivity of deionized water used in the experiment changes only by 7.5% during a single run. At higher voltages, however, the conductivity of deionized water changes by 2-3-fold the original value for a single run. I n all probability the { potential at the glasFwater interface changes due to the appearance of ionic species in greater concentration. This would affect the value of L12/T. We have seen earlier that at higher voltages L12/T has a tendency to decrease. Let us consider Table IV, where the experimental values of ( A P / A ~ ) J - have o been
Ths Journd of Physkal ChSmWrv
R. P. RASTOGI AND K. M. JHA
compared with L12/L11 by using the experimentally determined values of L12/T and L l l / T . Up to 60 v, there is moderate agreement between the experimental and the theoretical values considering the uncertainty involved. The agreement is not improved even when a kinetic e n e r d correction (which is of the order of 0.01) is made. Test of Linear Phenomenological Equation. Table V records the values of ( J ) A p = o , ( J ) A , = o , and ( J ) t o t a l in simultaneous sets of experiments for a new set of disks. It is found that the directly observed value of ( J ) t o t s l agrees with that calculated from eq 11 within experimental error. It confirms the linear phenomenological eq 1 in the range of study.
Acknowledgment. K. M. J. is thankful to the authorities of Bhagalpur University, Bhagalpur, India, for the grant of leave for study purposes. The investigation is supported by Indian Council of Scientific and Industrial Research, New Delhi, India.
