Currents Limited by Gas Solubility The discrepancy between Slendyk's polarographic limiting currents in 0.01 M HCI solutions and predictions of the effect of ionic migration is attributed to the formation of hydrogen gas at the electrode and the consequent perturbation of the convective velocity. Neglect of activity coefficient variations in the earlier calculations cannot account for the discrepancy. Diffusion coefficients of some gases in electrolytic solutions could conceivably be measured with an electrode at the upper end of a stagnant capillary b y taking advantage of the limited solubility.
An earlier work
(Newman, 1966) treated the effect of ionic migration on limiting currents. The effect is particularly draniat'ic for the discharge of hydrogen ions from HC1-KCl solutions, where t.he limit'iiig current may be enhanced by a factor of 3.34 over the diffusion-limiting current for a growing mercury drop. The data of Slendyk (1931) confirmed the calculations for 0.001,11 HC1 solutions with KC1 concentrations ranging from 0 to O . l J 1 , but for 0.01J1 HC1 solutions with KCl concentrations ranging from 0 t o 1.11 the data deviated significantly from the theory. The first attempt to resolve this discrepancy involved a refinement of the mass-t'ransfer theory to account for the activity coefficients of the ions. I n advance, one should not expect this explanation to be sufficient., because the data for 0.001.11 HC1 solutions, which agree with the theory, involved ionic strengths as high as some of the 0.01-11 HCl solutions, and t'he activity coefficients should be just as important. A second, more plausible explanat'ion for the discrepancy lies in the 1iniit.ed solubilit,>-of the hydrogen produced by t'he electrode reaction. This solubility is 0.000763N for pure water a t 25°C. ;in assumption of equal diffusion coefficients for H + ions and for Hz would lead to a concentration of H2a t the surface of 0.000531 for the 0.00131 HC1 solutions and an excess of KCI. This is just under the solubility limit, but with the more concentrated, 0.01X HC1 solutions, t'he solubility limit should be exceeded by a considerable amount, because the limiting current for H + discharge would be about ten times higher. For a growing mercury drop, the format.ion of hydrogen gas a t the surface should effectively increase t.he stirring of the solution, increasing t,he limiting current. Slendyk's data support this supposition, since the current increased by more than a factor of 10 for t'he same ratio of KC1 to HCl (Table I). Activity Coefficients in Electrolytic Mass Transfer
The theory of electrolytic mass transfer in dilute solutions inherently assumes that the activity coefficients of all neutral combinations of ions are equal to unity. Since activity coefficients are strong functions of composition in electrolytic solutions, one might expect a good first correction to dilutesolution theory to involve the activity coefficients. Thus the ionic flux might be written
represents an approximation to the mult'icomponeiit'theory appropriate to concentrated solutions, in that it still neglects the frict'ioii interact,ion between solute species, as represented, for example, in the Stefan-Maswell equation. It is convenient, particularly in calculations, to introduce a quasi-elect'rostatic potential, @, b y the definit'ion p, =
RTlnc,
+ r,F@ +
p:
(2)
where species n is a charged ion and p i is a constaiit. The behavior of @ depends on the (arbitrary) choice of the ionic species n. By means of Equation 2, it is now possible to espress the first term in Equation 1 in terms of the (measurable) act>ivitycoefficient's of neutral conibinat'ions of ions:
(3) for uniform temperature. The equations describing the system thus become similar to those solved earlier (Newman, 1966), except for the inclusion of activity coefficients referred to species n and the use of the Sernst-Einst'ein relation ui = Di/RT for the ionic mobility. For the activit,y coefficients of ions in moderately dilute solutions of several electrolytes, one can use, 8or example, the formulas of Guggenheini (1959). The numerical solut'ion of the final set of coupled, ordinary differential equations is then complicated, but st'raightforward (Tewman, 1968). This approach to ionic mass transfer has been used by Smyrl and Newman (1968). The results of such calculations for 0.01 and 0.001X HC1 solutions for a growing mercury drop are expressed in Table I1 as the ratio of the limiting current. to the value calculated b y dilut'e-solution theory, including the effect of niigrat'ion, as outlined earlier (Newman, 1966). The inclusion of the activity coefficients shows a small effect in each case, not sufficient to account for the discrepancy in Slendyk's data. The largest effect occurs in the solutions without KC1, the most dilute solutions, because the ionic strength is more nearly uniform in the presence of KC1, and consequently t,he activity coefficients vary less with position. limitation of Hydrogen Solubility
where p i is the electrochemical potential of species i. The first term on the right thus represents both diffusion and migration. I n this first improvement over dilute-solution theory the diffusion coefficients, Di, are taken to be constant and equal to their values a t infinite dilution, and v can be regarded as the solvent velocity or a mass-average velocity. Equation 1
T o test the hypothesis t h a t hydrogen gas evolution in Slendyk's 0.01J1 HCl solutions enhances the convection and causes the discrepancy with the calculations of the effect of ionic migration, experiments were run with an electrode at the upper end of a stagnant capillary filled with solution and open at the bottom into a cell filled with solution (25OC) and containing the counterelectrode. In the absence of Ind. Eng. Chem. Fundam., Vol. 9, No.
4, 1970 677
~~
Table 1. Galvanometer Deflections for limiting Polarographic Currents in HCI-KCI Solutions
(Slendyk 1931) O.OO1M HC1 0.0001 0.001 KC1 0 46 29 Deflection. 58 0.01i1f HCl 0 0.0001 0.001 0.01 KC1 Deflection* 83 69 54 36 1.18 1.24 Ratio 1.43 a Galvanometer sensitivity of 0.6 Ia/mm. b Galvanometer sensitivity of 6 pa/mm.
0.01 19
0.1 14
0.1 24 1.26
1 21 1.50
u Hg
N HZ
Table II. For Growing Mercury Drop, Calculated Ratios of limiting Currents to Those Calculated by Dilute-Solution Theory @
0.99504 0,95346 0.70711 0.30151 0 a
r
=
CK+/CC,-
0.01M HCI
0.001M HCI
0.99998 0.99962 0.99846 0.99636 0.99514 in bulk solution.
1.00065 0.99983 0.99650 0,99030 0.98655
Table Ill. limiting Current of HCI-KCI Solution in Stagnant Diffusion Cell at 25OC Applied Id?, idF, Run
No.
1 2 6 7 8 9 10 11 12 24 25 26 31 32 33 34 35 36
HCI, M
KCI, M
0.10
0,0333
0.10
0.0191
0.10
0,0568
0.10
0.178
Voltage, Volt
10-%a
0.418 0.418 0.296 0.296 0.296 0.296 0.296 0.296 0.296
2.18 2.16 4.95 5.1 3.9 3.9 3.7 5.0 3.8 3.9 2.6 3.08 3.5 3.4 3.2 2.25 2.8 2.65
0.200
0.253 0.200 0.205 0.181 0.181 0.181 0.181 0.181
Ma
0.196 0.194 0.446 0.46 0.35 0.35 0.33 0.45 0.34 0.35 0.234 0.278 0.315 0.306 0.388 0.292 0.252 0.24
extraneous st,irring, the growing mercury drop and penet'ration iiit,o a stagnant fluid show essent'ially the same behavior (Xewman, 1966). However, in the capillary system used here, a gas blanket will cover the electrode when the solubility limit' is exceeded, and t,he current should be effectively limited by the solubility of the gas produced in the electrode react'ion. Furthermore, the current decreases inversely proportional to the square root of time, and there is assurance that the solution a t the electrode surface is saturated, but not supersaturated. The cathode was a platinum sheet sealed in a stagnant capillary (1.19 mm in diamet'er) with mercury as electrical contact (Figure 1).The anode was a platinum gauze hydrogen electrode. Laboratory-grade cylinder hydrogen was used without. further purificat'ion. The hydrogen pressure a t the anode side rras measured by a n open-end manometer, M (llerian fluid D-8325, density 1.75 used as meter fluid), and maintained a t 1-inch pressure higher than atmospheric. 678
Ind. Eng. Chem. Fundam., Vol. 9,
lo-*
F -
-&FC
V'SC Cm3
No. 4, 1 9 7 0
I I
100
IO
1000
t isec)
Figure 2. Limiting current curves of HCI-KCI in stagnant diffusion cell at 25°C 0
0
A
Run 1 Run 6 Run 8
Solutions were prepared from reagent-grade potassium chloride and hydrogen chloride (specific gravity 1.191). The current flow through the cell was measured from the voltage drop across a known resistor ( R = 1000 ohm-) on a VIDXR-520 digital voltmeter, and the time was counted on a VID-kR-625 digital clock. Some current us. time curves are plotted on log scales and shown in Figure 2. The slopes of most curves are slightly less than 0.5 (ranging from 0.45 to 0.50) and poor in reproducibility. They tend to be flat a t large time, which we believe t o be caused by the stirring of hydrogen gas formed within the diffusion layer. For solutions of HCl (0.1Jf) and KC1 (ranging from 0.0191.V to 0.178;1.1) the observed values of il/i (averaged over the time period where the current is inversely proportional to the square root of time) are shown in Table 111. The measured values vary over a wide range from 0.194 to 0.46 ma-sec1'*/cm2. If this current were limited by transport of hydrogen ions to the electrode surface, the diffusion cm2/ coefficients of H + would vary from 3.18 to 17.8 X
sec, considerably below the accepted value of 9.31 X crn?/sec. On the other hand, if the solubility of hydrogen gas is taken to be 0.0007523f [the solubility (Linke, 1958) is lower than in pure water by 3,6y0 for 0.531 HC1 and about lOy0for 0.5J1 KCl], the diffusion coefficient for dissolved hydrogen turns out to be 0.566 to 3.16 X 10-6 cm2/sec. This is to be compared x i t h the values of 4.09 (Tammanil and Jessen, 1929), 6.9 (Davidson and Cullen, 1957), and 3.83 X loW5 cni2/sec (Xikazyan and Fedorova, 1952, a t 23°C in 0.5.11 H,SOd). Gubbins, Bhatia, and Walker (1966) obtained a value of 3.79 x cni2jsec in I X KCl. Discussion
like hydrogen, it is probably better to use anodic limiting currents from a saturated bulk solution, as used by Mkazyan and Fedorova (1952) on a rotating disk electrode, but the method may be useful for a gas like osygeu, where slow electrode kinetics can cause difficulties (Davis et al., 1967). Care will be necessary to ensure that a uniform gas blanket will cover the electrode and paths of electrolytic conduction will be blocked. literature Cited
AIkazyan, E. A., Fedorova, A. I., Dokl. Akad. ih'auk SSSR 86, 1137-40 (1952). Davidson, J. F., Cullen, E . J., Trans. Inst. Chem. Eng. 35, 51-60
,.
tia57) \ A Y " '
The experimental results support the claim that the solubilit8yof hydrogen is reached or exceeded for limiting current's a t a drol)ping-iiiercury electrode or a stagnant'-capillary electrode in HC1-KC1 solutions for HCl coiicent'rations greater than about 0.00131. With an excess of KC1, one can refine the earlier estimate of the surface concentration of Hz ___ by mult,iplying by 1 / D l f + / D E 1to 2 yield a value of 0.00076X. The effect of migration call increase this by an additional factor of 3. Although t,hese values can be slightly greater than the solubility limit, one suspects, on the basi.; of the agreement with the calculated migration effect,, that' in glendyk's experiments with 0.001.11 HCl solutions either the solution is su1iersaturatc.d and no gas is formed or the convective velocity is not significantly enhanced by any gas which may be present'. On the other hand, with 0.01JI HC1 solutions the solubilit'y limit is exceeded by more than a factor of 10, and the stirring is sufficient to cause the observed discrepancy. Currents in a stagnant capillary limited by gas solubility can 1)rovide a nicthod for measiiriiig the diffusion coefficient of a gas dissolved in an electrolytic solution. For a reactive gas
Davis, R. E.. Horvath, G. L.. Tobias, C. W., Electrochim. Acta 12, 287-97 (1967). Gubbins, I(.E., Bhatia, K. K., Walker, R . D., Jr., A.Z.Ch.E. J . 12,548-52 (1966). Guggenheim, E. A , , "Thermodynamics," pp. 357-61, XorthHolland Publishing Co., Amsterdam, 1959. Linke, W. F., "Solubilities," F'ol. I, pp. 1078, 1081, Van Sostrand, Princeton, W.J., 1958. Sewman. John. ISD.ENG.CHEM.FUXDAM. 5,525-0 (1966). Newman: John; IND.ENG.CHEX FUND. 7, 514-17 (1968). Slendyk, I., Collect. Czech. Chem. Commun. 3,385-95 (1931). Smyrl, W.H., Xewman, John, J . Phys. Chem. 72,4660-71 (1968). Tammann, G., Jesqen, Vitu?;, 2. Anorg. d l l g . Cheni. 179, 125-44 (1929).
JOHK n'EWllAN1 LIMIN HSUEH Cniversity o j California Berkeley, Calif. Qg720 To whom correspondence should be sent. RECEIVED for review January 12, 1970 ACCEPTEDAugust 12, 1970 Work supported by the Petroleum Research Fund administered by the American Chemical Society arid by the U.S. Atomic Energy Commission.
Effects of Frequency on Modeling High-Frequency Electric Discharge Reactors A study was made of the effect of variations in driving frequency on the spatial distributions of electron concentration, electric field strength, and reaction rate in a high-frequency electric discharge.
Development of a model for describing the rates of chemical reactions occurring in high-frequency electric discharges requires a clarification of the specific effects of frequency. However, very few experimental observations have been made t o determine how a variation in frequency might affect reaction rates and product distribution. Eremin et al. (1937) reported that t'he rate of osidat'ion of nitrogen t o nitric oxide in a discharge sustained between two steel electrodes was identical for frequencies of 2i0, 500, and 700 kHz, and 1 hlHz. More recently, Dinan et al. (1969) compared the composition of products obtained from the reaction of toluene vapor in a discharge operating a t 28 AIHz with the data obtained by Streitweiser and Ward (1963) in a discharge operating at 3000 l I H z . Table I summarizes the product distributions normalized t o exclude recovered toluene, polymeric, and noncondensable products. Dinan and coworkers concluded
that the formation of substantial amounts of dimer biaryls, bibenzyl, diphenylmethane, and biphenyl a t the lower frequency was a n indication that free radicals were t h e major intermediates leading t o the formation of condensable products. On the other hand, Streitweiser and Ward had argued that anions were the most likely intermediates. Thus, there may be differences in the products obtained from radiofrequency and microwave discharges. The primary steps for all electric discharge systems are t h e result of electron-molecule collisions. The rate constant for such reactions depends upon the electron temperature or its measure, E / p , the ratio of electric field strength t o gas pressure. The over-all rate of reaction is also proportional t o the electron density, n. The present communication explores the effects of a variation in frequency on the magnitude and spatial distribution of E / p and n. As an example, we consider Ind. Eng. Chem. Fundom., Vol. 9, No. 4, 1970
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