NOTES
1593
D satisfy (i,j = 1, 2, . . ., N ) (22) which is consistent with the estimates of these quantities given earlier. By defining di
- ttj < (2/a)
kDI =
Q-lED’Q
(23)
where Q is th,e modal matrix of X, and with a similar definition of E,, the result of the integration in eq 21 may be written as
In summary, the two proposed analytical techniques may be applied without modification to mixtures containing any number of components and do not impose additional experimental or mathematical restrictions. The estimate of the experimental error in the diffusion coefficients is valid within a specified range of conditions that is believed to apply to most meaningful experiments.
Acknowledgment. The author is grateful to the National Research Council and the National Bureau of Standards for a Postdoctoral Research Associateship which supported this work:
(24) Expanding the denominator o,f eq 24 inna power series and using the definition of ED’ and E,, the matrix formulation of the result is ED‘ = (l/S)Ex
+ &(DEx- ExD) + . . .
(25)
The first two illustrated terms in the expansion are sufficient since eq 21 is itself an approximation. The true error in the diffusion coefficients may now be approximated by
+ Ea
ED = Eo
for Electroosmosis by R. P. Rastogi, Kehar Singh, and Shri Nath Singh Chemistry Department, Gorakhpur University, Gorakhpur, India (Receiwed September 30, 1 9 6 8 )
(26)
where
+ &(DE, - ExD)
(27)
+ +(DE, - E,D) + D]
(28)
E, = (l/E)E,
Nonlinear Phenomenological Equation
For a Markoffian system, the flux Ji can be expressed as a function of forces XI, Xz, . . . and the structural factor G , as1n2
and Ea = -eC(l/n)E,
Ji
Terms of the order of e2 have been neglected in comparison to unity and E and the total error separated into a component due to concentration errors, E,, and a component due to errors in the cell constant, E,. As a hypothetical illustration of the error contributions, suppose that 1.00 0.50 cm2/sec (29) D = 0.25 2.00 X 1.00 1-00 Ex = X (30) 1.00 1.00
[
]
]
*[
E = 1X
lo4 sec/cm2
(31)
E = fO.O1 (32) Using eq 27 and 28, the resulting errors in the diffusion coefficients are given by 1.01 0.95
1.05 0.99
and 1.10 0.59
Ea =
*[
]
0.35 2.10
X lo-’ cm2/sec
(34)
=
f(xi,Xz, . . . , G )
(1)
If the structural factor G does not alter, the flux J i may be expressed as
Ji
=
CLijXj j
+ 4C XLijkXjXk + . . i
I
(2)
k
where Lij and Lijk, etc., are phenomenological coeficients. Expressions similar to eq 2 have been found to be valid in the case of chemical reactions3 and thermoosmosis of ideal gases.4 Recent measurements on electroosmosis in Pyrex-acetone systems show that the transport processes involved are nonlinear.6 Precise measurements on electroosmotic velocity in Pyrexacetone systems show that mass flux J is given by eq 2 retaining the terms up to third ordersG It was thought desirable to test this conclusion by examining the electroosmosis of differential liquid through a plug of different material and different geometry. Accordingly, electroosmosis of methanol through a quartz plug was studied. In the present study, electroosmotic (1) J. C. M.Li, J . Chem. Phys., 29, 747 (1958). (2) R. P. Rastogi and K. Singh, Trans. Faraday Soc., 6 3 , 2917 (1967). (3) R. P. Rastogi, K. Singh, and R. 0. Srivastava, ibid., 61, 854 (1965). (4) H.J. M.Hanley and W. A. Steele, ibid., 6 1 , 2661 (1965). (5) R. P.Rastogi and K. M. Jha, ibid., 6 2 , 585 (1966). (6) R. P. Rastogi, K. Singh, and M. L. Srivastava, J . Phys. Chem., 73, 46 (1969). Volume YS, Number 6 May 1969
1594
Table I: Mass Flux J at Various Ahp and A# (35") AP, cm of
108 J ,
108 J .
AP,
A$, V
cma sec-1
cm of methanol
A#, V
20 24 28 30 32 34
200 200 200 200 200 200
3.22 4.51 5.77 6.44 6.75 7.40
37 37 37 37 37
100 200 300 400 500
10.08
24 26 28 30 32 34
300 300 300 300 300 300
2.37 2.79 3.41 4.00 4.59 5.22
38 38 38 38 38
100 200 300 400 500
10.02 8.85 6.89 4.26 1.16
methanol
em* 8ec-1 9.02 7.27 4.49 1.41
\
15 20 25 ao 5
10
A P XKP( d p s em-aj
pressure measuremenk have also been used for correlation. The present analysis is more comprehensive than the previous one and confirms our earlier findings. Experimental Section Materials. ( a ) Methanol. Analytical reagent methanol was refluxed with magnesium and iodine for 3 hr and then distilled in the absence of moisture. The density of the purified sample was 0.7772 g at 35". ( b ) Quartz Membrane. A plug of quartz powder (120 mesh) was fixed in a glass tube with the help of Araldite. The average radius of pores constituting the membrane was found to be 8.5 X lod4cm by using the method employed by Rastogi and coworkers.' The apparatus and experimental techniques described
Figure 2. Estimation of
Ll112
and LIZ using eq 4.
earlier5$6 for the measurements of (J)AP--0, ( J )A+Q? and J were employed. The area of cross section of capillary tube was 8.5 X cm2. For measurement of J , potential difference and pressure difference were applied in opposite directions. All of the measurements were carried out in an air thermostat maintained a t 35 f 8.1". Results The experimental values of J corresponding to variolas magnitudes of AP and A$ are given in Table I. Each observation is the mean of a t least four readings. Discussion The phenomenological eq 2 can be written for the case of electrokinetic phenomena retaining terms up to third order as
J
=
+ T~LIII(AP/T)~ + ( A P / T ) ( A $ / T ) + +Liz2( A$/T) -I-$Liiii( A P / T ) + 3Liiiz ( A P / T ) ( A $ / T ) + +Lnzz(AP/T)(A$/T)' + 4 L i z ~ 2 ( A $ / T ) ~(3)
L i i ( A P / T ) 4-h z ( A $ / T ) L112
since the forces8 are AP/T and A+/T. The data on solvodynamic permeability show that the phenomenological coefficients are LIII = 0 and E l m = 0. IC is also found that LlZz2 = 0 since, when values of ( J ) A P - o ~ are A + plotted against Ad, a straight line obtained. Equation 3 can be expressed as
J
=
+ Llz(A+/T) + L I I Z ( A ~ / (46/a) T~) ( A+/T) + 3Llnz ( A P / T ) ( A d / T ) + + L I I Z ~ ( A P(A+/T)' /T) (4;
hl(AP/T)
+
$I422
1
100
200
300
400
A+ (VOLTS)
Figure 1. Estimation of L11zz and Llia using eq 4. The Journal of Physical Chemistry
600
(7) R . P. Rastogi, K. 6,446 (1968).
Singh, and
8.
N. Singh, Indian J . CPMP..
(8) 6 . R. de Groot and P. Mazur, "Non-equilibrium Thermodynamics," North-Holland Publishing C o . , Amsterdam, 1962,p 417.
NOTES
1595
According to eq 4, a plot of (J
+ (J)~+=o]}/flA+
- [(J)AP-o
against A+ should yield a straight line when AP is kept constant. Similarly, when the same quantity is plotted against A P keeping Ad constant, a straight line should be obtained. This is confirmed in Figures 1 and 2 showing thereby that eq 4 satisfies the experimental data. The phenomenological coefficients were estimated as follows. Ll1 was estimated from the solvodynamic data. Llz and Llzzwere estimated from the analysis of electroosmotic velocity data. L1122 and Llllz were estimated in a similar manner using Figures 1 and 2. The intercept in Figure 2 is equal to L112/!P $L1122/TaA4 and that in Figure 1 i s equal to Lllz/T2 &lllz/Ta~P. LIE could therefore be obtained from either of the intercepts. These values are found to agree within &4%. The values of phenomenological Coefficients obtained are
+ +
Ln/T = 3.87 X lo-' cm5sec-l dyn-l L1z/T = 1.58 X
cma sec-1 V-'
L122/T2 = 2.76 X lo-* cmasec-l L11z/T2 = -6.88 X 10-lO cm5sec-l dyn-1 V-1 Llllz/TS = 19 X 10-l6 cm7sec-l d y r 2 V-I L1122/T9 =
- 13.6 X
The validity of eq 4 was independently checked by analyzing the data on electroosmotic pressure. Electroosmotic pressure can be computed in the steady state for various magnitudes of A+ by substituting the values of phenomenological coefficients in eq 4 and using the condition J = 0. The calculated values of electroosmotic pressure using eq 4 and retaining terms up to second and third order are compared with the experimentally obtained values of electroosmotic pressure in Figure 3. The experimental and theoretical O ~ ~ P V Q B agree when terms up to third order are included. The measurements of (J)AP,o,(J)A+o, and J are fairly precise and reproducible. The departures from mean values never exceed 1% when each observation is repeated at least four times. The uncertainty in electroosmotic pressure measurements is of the order Z from of &2%. Since the mean value of L ~ I obtained Figures 1 and 2 has been used, the computed values of electroosmotic pressure may have an uncertainty of &5%. In view of these limitations, the agreement between calculated and experimental values of electroosmotic pressure may be considered very satisfactory. The above analysis, therefore, confirms that phenomenological eq 4 holds good in the nonlinear region under study.
Acknowledgment. S. N. S. is thankful to Council of Scientific and Industrial Research (India) for the award of a junior research fellowship.
10-la cm5sec-1 dyn-I V-2
For the evaluation of slopes and intercepts the method of least squares was used. Reduction of Quinone in Its Charge-Transfer Complex with the Hydrogen Molecule by Motoyuki Tsuda, Hiroo Inokuchi, and Hideo Suzuki The Institute for Solid State Physics, The University of Tokyo, and the Department of Physics, School of Science and Engineering, Waseda U n h m i t y , Toxyo, Japan (Received October I , 1 9 6 8 )
Quinone-type compounds have been extensively studied for their importance in biological functiom mainly because of their ability to form charge-transfer complexes and their capability for reversible reduction to the hydroquinone form. For example, Tollin and Green demonstrated the occurrence of light-induced electron-transfer reactions between electrically excited chlorophyll molecules and various quinones and detected a semiquinone radical intermediate by the esr method.' Further, Yamazaki and Ohnishi reporte that semiquinone plays an important role in the
4$ (VQUS)
Figure 3. Examination of the validity of eq 4: 0,experimental curve; 0,calculated curve using eq 4 and retaining second-order , calculated curve using eq 4 ~
(1) 0. Tollin and G . 66, 308 (1963).
Green, Biochim. Biophys. Acta,
6 0 , 624 (1962)
~
Volume 75,Number 6 M a y 1060
