3' in these equations is defined by
0' where 3 is the diffusion coefficient and ('3n 'cj is the mobility ter1n.j Table I contains details of the calculation of (%/c) and subsequently D',.'C'/~ for lithium nitrate and values for these quantities for cesium nitratc.
I1
0.05
0.10
14. Fig. I.-Plot for the computation of the activity coefficients of lithium and cesium nitrates a t 2.5": 0, lithium nitrate; 0 , cesium nitrate.
The plot of D'//c1/2 as. cr/2 used for the computation of the activity coefficients is shown in Fig. 1. ( 5 ) T h e equations for computing (%IC) are given by H S . Harried and B. B. Owen, "The Physical Chemistry of Electrolytic Solntioni." 3rd Ed., p. 121--122, equations (4-$-19), (4-4-30) and (4-,4-21), Relnhold Pulil. Corp., New York, K,Y.,1968. See also refs. 3 and 4.
Here the circles represent the results for lithium nitrate and the dots those for cesium nitrate. T h c graph is a straight line. Individual plots n ~ w used in evaluating the activity coel'ficicnts give11 i n 'rkllllc 11 ~ r l - ~lvhicll lll it, al,I,earetl thy :Lctivity o i ccsiiiiii nitratv is slightly lcss t l x i i i l i t h i u i i i ilitratc,
Similar calculation of the activity coefficients of lithium and potassium perchlorates leads to the results recorded in the last two columns of Table 11. It is safe to conclude from these results that the activity coefficients of the perchlorate are greater than those of the nitrates. The result of this determination t h a t potassium perchlorate appears to have a greater activity coeficient than the lithium salt a t 0.00t5 and 0.01 seems unusual and may be due to experimental error. This investigation was supported in part by the rltomic Energy Cc Jnit-nission uiider contract AT(30-1)-1373.
[COSTRIBUTION FROM THE DEPARTMENT O F CIIEMISTRI', SORTHIVESTERN USIVERSITY;
Voltammetric Membrane Electrodes. I. Basic Theory and Characteristics of Thallous and Cadmium Reduction BY R I C H A R D
c. BOWERSAKD -1RTHUR A f . ~ ~ T I I , S O K ' RECEIVED SEPTEMBER 9, 1957
A uew type of voltammetric mercury electrode is described. I t is constructed in a manner such that linear finite diffusiciii the reducible or osidizable species takes place across a cellophane membrane. The theoretical equations for diffusioii across the membrane are given and a method is outlined for the determination of diffusion coefficieiits in the membraiie. Excellent agreement betweeri theory and experimeut is observed for diffusion currents due to the reduction of thallous ion. A4dsorption in the membrane markedly affects the apparent diffusion characteristics of cadmium a t low conce~itratiotis. The experimental results indicate a "homoge~ieous" phase r:cthrr than a porous ,structure for the cellophnnc membraucs used. rif
Introduction Clark, et a l . , have reported the use of a platinum electrode covered with a film of collodion or cellophane for the determination of oxygen in whole blood. These workers observed t h a t the film prevented poisoning of the electrode while allowing oxygen access to the electrode. An electrode eniploying mercury as the electrode material and cello(1) Taken from t h e research by t h e j u n i o r author in partial fulfill ment for t h e P h D . degree ( 2 ) L. C. Clark, R. Wolf, D . Granger and Z. Taylnr. J . .4ppi. Physiol.. 6 , 189 (19531
phane as an ion permeable diffusion layer has been described briefly by the author^.^ In theory the membrane should act as a well defined diffusion layer when the electrode is placed in a stirred solution. Due to the finite thickness of the diffusion layer, a steady state is obtained. The present investigation was initiated to evaluate theoretically the currents obtained with voltammetric membrane electrodes and to characterkc more fully the behavior of this type of electrode. (3) R . C. Bowers and A. Wilson, "Third National Symr~nsium011 Instrumental Methods of Analysis," Chicago, Illiniiis. 1957
June 20, 193s
Theoretical The current a t an electrode when the rate-controlling step is the diffusion of the electroactive species to the electrode surface is given by
(g)z
i = n F.4D
=
where n is the number of electrons transferred in the electrode process, F the faraday, A the cross sectional area through which diffusion takes place, D the diffusion coefficient and (bC/bx), = 0 the concentration gradient a t the electrode surface. In order to obtain an expression for the concentration gradient a t the electrode surface, the fundamental diffusion equation must be solved. Steady-state Limiting Current.-At potentials where essentially instantaneous reduction (or oxidation) of the electroactive species takes place, the boundary conditions for diffusion across the membrane are C(o, t ) = 0 and C(1, L) = CI. At the steady state, bC/bf = 0, and i t follows immediately that im
(3)
= nFADCI/l
is the concentration of the electroactive species a t the interface of the membrane and the body of the solution and 1 is the thickness of the membrane. Instantaneous Currents.-Experimentally, the steady-state current can be approached from either direction. I n case A, the electroactive species is present in solution a t the time when the potential is applied. Thus, the initial concentration, C(,, o), in the membrane is equal to C1. I n case B, the electroactive species is injected into the stirred solution after a potential, sufficient to cause reduction, has been applied. Under these circumstances, the initial concentration within the membrane is zero. Equation 2 can be solved under the above conditions by using the Laplace transformation. The solution in both cases can be written in two possible forms4which result in the following alternate equations for the instantaneous current Cl
Case -4:
IC(* 0) = Cll 1
or it = nFACI
+2m
m = m
exp( --"r2Dt/12)]
=
1, 2 , 3. . .
(4)
.\/Dm [1 + 2
m = m
.,
(5)
Case B: [C,,.
1
exp( -m212/Dt)
m = 1, 2 , 3 .
0)
=
01
m = m
m = 1, 2, 3.
..
(-l)"exp(
-m2dDt/P)]
(6)
( 4 ) See, for example: J. C r a n k , "The Mathematics of Diffusion," Oxford University Press, London, 1958, p . 17-22.
Equations 4 and 6 are generally the more useful forms since their series terms converge rapidly except a t short times. However, a t times less than 0.19 P/D,the magnitude of the series term in (5) is less than 0.01 and this equation reduces to that for linear infinite d i f f ~ s i o n . ~ , ~ Experimental Reagents.-1 .O M solutions of recrystallized potassium nitrate were used as supporting electrolyte in all experiments. Stock solutions of thallous chloride were prepared from recrystallized thallous chloride. Cadmium nitrate solutions were prepared from reagent grade chemicals and standardized according to the procedure of Wiley.7 The mercury employed in the electrode was purified by treatment with a 10% nitric acid solution followed by distillation at reduced pressure. Electrode.-The electrode was constructed by placing a piece of cellophane membrane, previously swollen in water, over one end of a piece of glass tubing. A polyethylene ring was used to hold the membrane in place, a water-tight seal being obtained between the ring and the glass using Apiezon wax. Prior to assembling, the inside of the glass tube was treated with a silicone. The thickness of the membrane was determined after swelling in water by measuring several thicknesses with a micrometer. The average deviation of a series of such measurements was less than &l%. A small pool of mercury was placed on top of the membrane and electrical contact made with a platinum or tungsten wire. Apparatus.-A jacketed cell was used to maintain a constant temperature of 25.0 zt 0.1" except where otherwise stated. Stirring was accomplished using a magnetic stirrer. The solutions were deaerated with Linde H. P. nitrogen. ,411 current-voltage curves and current-time curves were recorded using a Sargent Model XXI polarograph, the pen response of the recorder being approximately four seconds full scale.
Results and Discussion Residual Currents.-An electrode constructed from commercially available cellophane which has been swollen in water exhibits a somewhat erratic cathodic residual current. This initial residual current can be eliminated by applying a potential of about -1.5 volt us. S.C.E. for a few minutes. After this treatment, the residual current is less than 0.10 pamp. in the potential range of zero to - 1.2 v. v s . S.C.E. in 1.O Jf potassium nitrate. The residual current exhibits a rapid increase as the potential is made more negative than this. In 0.1 M potassium nitrate as well as in 0.1 M tetramethylammonium bromide, the residual current remains negligible out to about - 1.5 volt as.
S.C.E. Current-Voltage Curves.-Typical current-voltage curves for the reduction of thallous and cadmium are shown in Fig. 1. A normal appearing wave is obtained upon scanning the potential from left to right; however when scanned in the reverse direction, a shift of the wave to more negative potentials and an anodic stripping current are observed. Both of these effects are due to an accumulation of the reduced species in the mercury. An additional effect of this accumulation is that the "half-wave" potentials are shifted to more negative values with decreasing rate of potential scan (Table I). ( 5 ) I. M. Kolthoff and 1. J . Lingane, "Polarography," Vol. I , 2nd E d . , Interscience Publishing Co., Kew York, N. Y . , 1952. (6) P. Delahay, "New Instrumental Methods in Electrochemiqtry," Interscience Publishing Co., New York, N. Y . , 1954. (7) R. C. Wiley, I n d . Eng. Chem., Anal. E d . , 3, 1 4 (1931).
0.25
0.50 0.55 0.60 0.65 0.70 US. S.C.E. Fig. 1.-Current-voltage curve of 0.244 X 10-8 Ai? TlCl (1) and 2.36 X I1P Cd(NOs)z (11): (a) potential scan from left t o right; (b) potential scan from right to left. 0.40
0.45
-w
h similar phenomenon has been observed with the rotated mercury electrodeg and the convection mercury electrode.g In both of the above cases adequate theoretical treatments of this phenomenon have been given. A more detailed description and a theoretical discussion of the characteristics of current-voltage curves a t membrane electrodes will be given in a subsequent publication. EFFECTOF
Case
A:
TABLE I it/im RATE OF POTENTIALSCANON “HALF-WAVE” POTENTIAL OF 0.244 X M TIC1 Case B:
Scan rate (v./sec.)
DS. S. C.
0.00250 ,00125 ,000625
E. it/t- =
-0.435 - ,452 - .463
TABLE I1 EFFECT OF COXCENTRATION, AREAA N D MEMBRANE THICKXESS O N STEADY-STATE LIMITIXG CURRENTS A. Concentration dependence Cd (NO,) I
TIC1 i (pamp.)
i/Cl
8.2 0,00993 0.081 ,0446 .36F 8.21 ,0938 ,772 8.24 .289 2.39 8.27 .948 7.87 8.30 B. Area dependence 0.0989 mM. TiCI; I = 0.82 X 10-2,cm. I
A (cm.2)
(+amp.)
i/A
0.408 1.09 2.53
1.31 3.66 7.88
0.321 ,297 .311
CI (inmolar)
i(ramp.)
i/C1
0.0236 0,111 4.70 ,119 0.577 4.83 ,359 1.i5 4.86 ,835 4.09 4.89 3.15 15.3 4.86 C. Thickness dependence I
x
0 5G6 m y T l C l 102,
I
cm.
(pamp.)
0.81 1.62 2.47
8.37 4.67 2.87
1
?n = =
C
I i- 3
+2
il
6.8 7.6 7.1
The deviations in i / A and il are of the order of 3 to 5y0 which is larger than the experimental error associated with the measurement of the current. ( 8 ) T.S. Lee, THISJ O U R N A L ,74, 5001 (1952). (9) I . hI. Kolthoff. J. J. Jordan and S . Prager, i b i d . , ‘76, 5221 (1954).
exp(-mzn2~t/i~, (3)
.
m = m
(9)
(-limexp(--m2**01/12)
..
The above equations predict that the rate of steady-state attainment is independent of the area through which diffusion takes place and of the concentration of the electroactive species. The solid lines in Fig. 2 were constructed from values of the quantity +Dt / 1 2 which is designated I
2.4 A
1
I
‘,
-
R E F E R E N C E POINT
.
2.2.
IHFORETICAL
I
CIJRVF
I
2.0 -
$ 1 2 1.8,
v
‘
0.4
+
l . O ? L _ _ _ -
8
- 0.6 c“
.,81.6 L
31.41
0 0.5 1 0 1.5 5 0 X 10%
m
m = 1, ’3. 3.
m = 1, 2, 3 .
Limiting Currents.-The form of equation 3 was tested by a series of experiments in which the concentration, area of the electrode and thickness of the membrane were varied independently. Table I1 summarizes the results obtained.
CI (mmolar)
This much deviation in current also was found in successive experiments using different pieces of cellophane of the same measured thickness ant1 area, and i t is felt that the cellophane membrane is somewhat non-uniform both with respect to thickness and diffusion properties. In the concentration dependence experiments, the same meinbrane was used throughout and excellent results were observed over a wide range of concentrations. Evaluation of Diffusion Coefficients.-Diffusion coefficients in the membrane can be calculatecl from observed steady-state limiting currents (equation 3). Two assumptions are inherent in this method however. The cross sectional area through which diffusion occurs will be equal to the area of the electrode only if the membrane is considercd to be a “homogeneous” phase.lOfll If access to the electrode is limited to pores. the area through which diffusion actually occurs will be much smaller than the measured area. In addition, i t must be assumed that the concentration of diffusing species a t the interface of the membrane and the solution is equal to that in the body of the solution. The rate of steady-state attainment offers a second method of evaluating the diffusion coefficients. Dividing equations 4 and C, by e r p t i o r i 3 one obtains
”
+a---L --C--L--
0-
’*
0
3.0 3.5 4 0 4 5 5 0
e. Fig. 2.-Theoretical and evperiinental steady-state attainment: case A, 1.19 X 10-3 TlC1; case B, C, 0 = 0 0742 X 10-8 MTlC1, A C = 0 0246 X 10-3 MTlC1; 0 = n’Dt/lZ.
-
as 9. The circles represent experimental values from the current-time curves in the reduction of thallous. The time axis for these experimental (10) J Wilbrandt, J Gcn Physroi , 18, 933 (1935) (11) r Teorell, Disc Faraday SOC, a i , 9 (195G)
June 20, 1955
VOLTAhIhlETRIC h k M B R A N E
points has been adjusted so that one experimental point from each curve coincides with the theoretical curve. The values of 0 corresponding to experimental values of &/i& can be obtained from these theoretical curves. The quantity 0 / t is a constant equal to n2D/12and diffusion coefficients can be evaluated from this quantity. An alternate procedure of calculating the diffusion coefficient in case A is to use only the initial portion of the current-time curves since a t short times
idi
=
T Z F A C ~ ~ ~
(10)
This method requires the same assumptions with regard to A and C1 as discussed previously. However, i t is to be noted that the use of equation 10 does not necessitate a knowledge of I , the thickness of the membrane. Table I11 lists the values of 0 / t and i1diobtained from the current-time curves for the reduction of thallous.
ELECTRODES
2971
TABLE V DIFFUSION COEFFICIENTS OF CADMIUM
I O N AT V A R I O U S
CONCENTRATIONS
A = 0.404 cm.2; 1 = 1.8 X 10-2 cm. Concn. ( m M )
D m X 106 (cm.Z/sec.)
0.0236 ,0715 .119 .359 .598 .835 3.15 6.23
0.97 .97 .99 1.00 1.oo 1 .oo 1.00 0.96
D e / l X 108 (cm.Z/sec.) Case A Case B
0.19 .43 .47 .78 .93 .97 .95 1.10
0.30 .50 .65 1.02 1. o i 1.10 1.17 1.07
trations good agreement is obtained and i t is reasonable to take the value of 1.0 X 10-8 cm.*/sec. as the correct one for the diffusion coefficient of this ion. The low values observed for Dell reflect a slower rate of steady-state attainment than predicted on the basis of D,. It is believed that adsorption of cadmium occurs within the membrane and that this accounts for the discrepancy betweer? TABLE I11 Delr and D,. ATTAINMENT OF STEADY STATE Initially the amount of adsorbed cadmium will A = 0.424 cm.2, I = 2.7 X 10-2 cm. be uniform throughout the membrane. When a Case A (Ct-8 = CI) Ca9e B (Ct-a 0) 1.19 m,M TIC1 0 0246 m M TIC1 potential sufficient to cause reduction is applied, 1 ir I
-
the diffusion coefficient can be expressed by the equation D = .4 ex&)(- E / H T j
(11)
TABLE VI ENERGY O P
A4CT1V.%'~1(iXAND
where E is the activation energy for diffusion and .I =
el2
(12)
'{e\p(~~ ~ / R I
X being the distance between two successive equilibrium positions. In Table VI, the values of E and X[exp(AS+IR)]"2 for diffusion in the membrane are compared with values of these quantities calculated from literature datal3 on the temperature dependence of diffusion currents a t the dropping mercury electrode. It is apparent that the entropy factor is largely responsible for the difference between the diffusion coefficients of these ions in aqueous solution and in the cellophane membrane. Due to its structure, a (13) V (1929)
S e j e d l y , Collectzoiz Cziclriirloo
Chem
C o ~ i ~ r n u n, s1, 319
COSrRIBUI'ION FROX THE
E N I ' K O P Y ~.4CI'OR FOR L>IP-
:"i'SION
Diffusion s y s t e i n
A
x
103 (cm.g/sec.)
T I A -in aqueous solti. 9.4 T1+in cellophane me1ubr;~iie 3.2 Cd'& in aqueous soln. 41 C d + - in cellophctne xncmhranc 6. 6
E(kral.)
Xlexp( A S */R)11/1
(A)
3.8
4.2 4.0 5.0
4.9
6 5
3.0
1.4
considerable fraction of the iiiembrane is blocked insofar as diffusion is concerned eveii though the diffusion is not restricted to rigorously defined pores. The smaller entropy factor is in accord with this model. Acknowledgment. --The authors are indebted to Professor D. D. DeFord for many valuable suggestions and discussions. EVANSTON, ILLISOIS
RESEARCH INSTITUTE
OF
TEMPLE UNIVERSITY]
The Cyanogen-Oxygen Flame under Pressure BY J. B. CON WAY^ .\SD -4. lT. G~ossr RECEIVED DECEMBER 20, 1957 The cyanogen-oxygen flame yields one of the highest flame temperatures obtainable by chemical means. The experimental flame temperature (4640'K.) a t atmospheric pressure is in good agreement with the calculated flame temperature (4835'K.) providing additional evidence in favor of the high value for the dissociation energy of nitrogen. A technique is described for operating this flame at elevated pressures t o give increased flame temperatures. Flashback velocities at various pressures are reported and are very sensitive t o pressure changes -4 generalized plot is presented for calculating theoretical adiabatic flame temperatures for various CO/N? ratios.
cyanogen-oxygen flame. The inflammability of mixtures of cyanogen and air has been studied by Pannetier and Lafitte.* Doliqueg investigated the represents the simplest possible stoichiometric rela- combustion of cyanogen and reported that i t does tion describing the combustion of cyanogen. The not entirely follow the theoretical chemical equaheat of the reaction a t 298°K. is equal to 126j6802 tion. Reislo made a thorough study of the cyanocal. which when coupled with the fact that the reac- gen-oxygen flame, analyzed the gaseous products tion products are extremely stable suggests the pro- of combustion and also analyzed the flame specduction of very high flame temperatures by this trum. A theoretical flame temperature of 4740" reaction. was reported, but no experimental flame temperaThe reactions of cyanogen with air and oxygen ture was measured. Thomas, Gaydon and Brewer" have been studied for many years. Dixon3 stud- have reported recent work on the cyanogen-oxygen ied the explosion rates of various mixtures of cyano- flame in which the flame temperature was measured gen and oxygen a t several pressures. Smithells by determining the vibrational intensity distribuand Inge14 and Smithells and Dent5 described the tion of the cyanogen bands. The vibrational temstructure of the cyanogen-air flame as obtained perature was found to be 3800 f 200°K. for the with a cone-separating apparatus and also indicated stoichiometric composition. The theoretical flame the gaseous composition of the various cones of the temperature for the stoichiometric composition was flame. The kinetics of the oxidation of cyanogen calculated to be 4SSO"K. using the high value have been studied by Hadow and Hinshelwood'j (AH: = 226,000 cnl. per mole) for the dissociation and Pannetier7 discussed the mechanism by which energy of nitrogen. X temperature of 4325°K. was various additives decrease the luminosity of the obtained when the low value (AH: = 170,240 cal. per mole) was used. From these results the au(1) General Electric Company, Aircraft Nuclear Propuliion Department, Evendale, Ohio. thors concluded that the high value is correct. (2) J. W Knomlton and E. J . Prosen, J . Reseavch Tali. B w . SLandAs is the case with all flames the cyanogenu v d s , 46, No. ti (June 1951). oxygen flame temperature is limited by the dis( 3 ) €€, Dixon, J . Chem. Soc., 49, 381 (1886). The equation CZNZk)
+ O ? k ! --+
2CO(g) i S P W
(1)
( 4 ) A. Smithells and H. Ingle, ibid., 61, 204 (1892). ( 5 ) A . Smithells and F. Dent, i b i d . . 65, 603 (1894). ihndnn) (fi)H. J. Hadow and C. pi. Hinshelwood, Pvoc. R o y . SCJC
8132, 375 (1931). t7) 0.I'nnnetier, Re?,.I i i c t . F r a n c . Peiit,lt~,4 . ,418 ( l q l l ~ l ~
(8) G. Pannetier and P. Laffitte, Compl. vend., 226, 341 (1918). (9) 12. Dolique, Bzdi. soc. ckim., [ 5 ] 3, 2347 (1936). (10) A. Reis, Z. ,biiy,Tik. Chem., 88, 515 (1914). (11) S.Thnmns. \ C,. f ; , ~ y ~ l a
