8 t h Annual Summer Symposium-Role
of Reaction Rates
Concept of Polarographic Currents limited by Rate of a Chemical Reaction and Some of Its Applications KAREL WIESNER Chemistry Laboratory, University o f N e w Brunswick, Fredericton, Canada
The concept of rate-controlled polarographic currents is discussed and illustrated by eeveral examples. Both rate constants and equilibrium constants, which are not accessible to measurement by other methods, may be calculated in a simple manner from rate-controlled polarographic waves. The simple, but essentially correct theory, derived from the concept of reaction volume, is used and the statistical meaning of reaction volume is explained. Only examples which can be regarded as fully rate-controlled have heen chosen to achieve greatest possible simplicit?.
In the specific situation being considered i k g is negligibly small with respect to idA. Consequent,ly an expression for i k g may h e very simply formulated i k B = nF X
Assume that B is reducible at tthe dropping mercury electrode and A is either not reducible or reducible at a potential more mgative than B. Obviously the height of a polarographic reduction wave of B will be in the most general case determined by the rate of diffusion of B from the bulk of the solution t o the electrode ( i d B ) and by the rate of transformation of A into B in the electrode environment (ikg). Now further limit the case by assuming that the equilibrium concentration of B is so small that its diffusion current, i d B , is negligible as compared with the kinetic current, i k g ; and the kinetic current i k B is in turn much smaller than the diffusion current of A ( i d A ) or, in the case that A is not reducible, than the hypothetical diffusion current of 9 calculated from the diffusion coefficient of A by means of the Ilkovi6 equation. This case is extremely simple t o deal with and h:ts considerable practical importance, because it is possible to find conditions in which the above limitations are valid for many systems. Further it is possible by adjustment of concentrations, pH, or other variables, to bring the rate-controlled current i k g into the optimum range for measurement., whereas i d B is conipletely negligible and i d A , which does not have to be measured, is several hundred or more times larger than a measurable polarographic wave. Under conditions defined in this manner the concentrations of ,4 in the bulk of the solution and in the vicinity of the electrode are the same, as the extent to which A is exhausted in the interface is negligibly small. Roughly the ratio of the concentration of A in the bulk of the solution and in the vicinity of the electrode is given by the expression id,
id,
-
i 6 ~
(1)
d i e r e k = rat,e constant for the conversion of A into B n = number of electrons required for the reduction F = 96,500 coulombs 4 = average surface area of the electrode 3 p = 5 0.85 ( v z ~ ) ” ~
?n = outflow velocity of mercury in grams per second t = time of one drop in seconds
T
H E purpose of this article is to show the applicability of ratecontrolled polarographic waves to the determination of extremely rapid reaction rates and also to the determination of equilibrium concentrations, which are not accessible t o direct measurement. The simplest form of the theory (11) of fully rate-controlled currents, later found essentially correct by rigorous calculations (6, 7 ) , is used, because it makes possible an easy understanding of the pror’esses involved. Consider the following syqtem:
k [A]
The effective thickness of t,he reaction layer, p , is a statistical quantity and can be derived in the following manner (11). The condition for reduction of each molecule of B, formed a t a distance X from the electrode, is t’hat it must reach the surface of the electrode (the potential of Khich, of course, corresponds to the limiting current) within its lifetime-that is, it must reach the electrode before it is reconverted into A . Clearly X must be a distance that the molecule can travel in its lifetime. The average displacement of a molecule in the tIvo directions perpendicular t o the electrode surface is given by the Einstein formula, in which D is the diffusion constant of the molecule poi1 sidered : A = -\/20r
Consequently molecules of B with lifetime, r, are, on the average, reduced only if they have been formed at a distance from the electrode
1 The factor - results from the fact that only one half of the mole-
2
cules move toward the electrode. If now the individual lifetimes of the molecules are replaced 1 by their mean lifetime 7(k’ is the rate constant for the reaction k
B +. A ) , an average of the average displacements 2 =
dg
and a condition valid for the reduction of all molecules of B are obtained-viz.,
A rigorous treatment of the problem has later given ( 7 )p
=
which differs onlyby the constant factor, derivation. The introduction of this value for p into Equation 1 gives Equation 3, which constitutes a complete solution of the problem under consideration. 1712
V O L U M E 27, NO. 11, N O V E M B E R 1 9 5 5
1713 (3)
This result was corroborated 5 years later in another manner by Delahay ( 5 ) . T h e Equation 1 for i b B further reveals an interesting property which has been pointed out in the first clearly recognized case of a rate-controlled current ( 1 3 ) . Because i k g is proportiona! t o 3 the average surface of the electrode, - X 0.85(mt)2’3,i t is independ-
5
ent of the height of the mercury reservoir. This is due to the fact that ( m t ) is the weight of one drop and is consequently a constant independent of mercury pressure. It’ is well known t h a t a diftusion-controlled current is proportional to the square root of the height of the mercury column. I n this respect a ratecontrolled current differs significantly from a diffusion current and can be easily distinguished from it. As an example of the determination of an extremely rapid reaction rate from the fully rate-controlled current let us consider the case of dissociation of reducible acids, which has been treated approximately and rigorously in the region of ioint diffusion and r:tte control (1-4, 6, 7 , 1 1 ) . I n many cases a polarographic wave of a n undissociated acid (phenylglyosylic, pyruvic, and many others) is located at a more positive potent’ial than the n a v e of the corresponding anion. Since this separation into two waves occurs in a p H region where the undissocintcd acid is present in negligible concentration, the height’ of the more positive wave must be determined by the rate of the reaction k
+
.IC’ I%+eAcH k’
Phenylglyoxylic acid has been recently studied (14) under conditions TT here the limitations specified in this article are strictly v:b!id. I n the p H region between 8 and 10 the more positive wave is at least 200 times smaller than the wave of the anion. I n order t o make it precisely measurable one can increase the concentration of the acid to O.02MJwhich of course brings the diffusion current of the anion out of range. I n the presence of buffer (borate in this case) proton transfer is mediated by the folloTving reactions: Ac’
+ H30+
ki AcH
k:
+ Hz0
Ac’
kz + Hz0 e AcH + ’OH k;
Ac‘
ka + BH e AcH + B’ k;
AcH is here the reducible acid; B H the buffer acid. By using Equation 3 for these three reactions the Equation 4 may be easily derived:
K A= ~ dissociation constant of reducible acid K B = dissociation constant of buffer acid CB = buffer concentration B y fitting the experimental data obtained a t various buffer concentrations and various hydrogen ion concentrations to Equation
1, it has been possible to elucidate the three rate constants in cluestion:
T h e precision of kl is much better than that of kp and k,, these having the nature of second-order correct,ions. This result at the same time s h o w t h a t in the more acidic range studied earlier the recombination with hydronium ions is the only important merhanism of proton transfer. Iiouteck9 ( 6 ) recently achieved a rigorous way of dealing with cases in which diffusion limit8 the current jointly with reaction rate. This solution has been amply verified experimentally, since Kouteck9’s asymptotic solution is again identical in form (except for a constant factor) with the solution arrived a t very early (11) by substitution of Equation 2 for p in an approximate equat,ion ( 2 ) for the case in which the current is rate and diffusion controlled. This treatment in turn has been known for a long time to agree precisely with the experimental data. T h u s the complete clarification of the problem of recombination of acid anions with hydronium ions has culminated in Kouteckfs brilliant mathematical treatment, and in spite of the difficulties of this treat,ment has resulted in very simple equat’ionst h a t can be used easily by chemists. T h e way is now open to a systematic study of recombination rates of various substituted pyruvic (5,4 ) , phenylglyoxylic ( I O ) , and other acids with hydronium ions and also t o a n interesting study of the same reactions in heavy water ( 1 0 ) . These experiments are under way at. the University of New Brunswick. T o illustrate further the possibilities of polarographic ratecontrolled currents, the case of glucose in which i t has been possilile to derive the value of the concentration of the open chain glucose tautomer by analyRis of the rate-conlrolled current of glucose may be mentioned. I t was shown some time ago t h a t the polarographic current of glucose is controlled completely by the rate of formation of the aldehyde tautomer from a- and 8-glucosr ( 1 2 ) . It is now possihle to solve the system of glucose in the following manner (9). T h e mechanism of mutarotation of glucose may be represented by:
Since it is possible t o determine the rate-controlled current due t o the formation of the aldehyde glucose from a and 8 glucose separately, there are two independent equations involving the unknowns k,, k ; and kz, ka. The remnining equations required to calculate all four unknowns are available from well known data. The mutarotation vrlocity expressed by the four unknowns is: h - m , = (kikz’
+ k ; h ) l ( k ; + k;)
T h e over-all equilibrium constant of a- and 0-glucose is li, X k; = 1.740 h-1’ X kz
Consequently there are four independent equations, and the fcur unknown rate constants can be calculated. From these in turn the unknown equilibrium concentration of aldehyde glucose can be calculated. This comes out to 0.0027% of the glucose present This procedure is an approximation, because in reality the glucose syptem should be represented by the following scheme. a
+ laldl Ti
ap
[hydrate]
1714
ANALYTICAL CHEMISTRY
The correction for the influence of hydration of the aldehyde group is, however, not a serious one and experiments designed to elucidate it have been under n-ay for Pome time a t the University of Kew Brunswick (8). LITERATURE CITED
(1) BrdiEka, R., Chem. Listy, 40, 232 (1946); Collection Csechosiou. Chem. Communs., 12, 212 (1947). (2) BrdiEka. R., and Wiesner, K., Ibid., p. 138; Chem. L i s t y , 40,66 (1946). (3) Clair, E., 3 1 . S ~ .thesis, University of New Brunsn-irk, Xew Brunswick, Canada, l l a y 1950. (4) Clair, E., and Wiesner, K.. Sature, 165, 202 (1950). (5) Delahay, P., J . Am. Chem. Soc., 74,3506 (1952).
(6) Kouteckf, J., Collection Czechoclm. Clitm. Communs., 18, 597 (1953). (7) Kouteckg, J., and BrdiEka, R., I b i d . , 12, 337 (1947). (8) Los. J. AI., and Simpson, B., unpublished result.8, University of Kern Brunswick, Kew Brunswick. Canada, 1955. (9) Los, J. hI., and Wiesner, K., J . Am. Chem. Soc., 75,6346 (1953). (10) Wheatley, hlary, unpublished results, University of New Brunswick, New Brunswick, Canada, 1955. (11) Wesner, K., Chem. Listy, 41, 6 (1947). (12) n’iesner, K., Collection Czechoslou. Chem. Communs., 12, 64 (1947). (13) Wiesner, K., 2. Elebtrochem., 49, 164 (1943). (14)
Wiesner, K., Wheatley, Alary, and Los, J. AI., J . A . Chem. Soc., 76, 4858 (1954).
RECEIVED for review June 23, 1955. Accepted September 2 , 1955.
[END OF EIGHTH ANNUAL SUMMER SYMPOSlUMj
Spectrochemical Determination of Boron in Carbon and Graphite CYRUS FELDMAN and JANUS Y. ELLENBURG O a k Ridge National Laboratory, O a k Ridge, Tenn.
Moving plate studies showed that the available material which best matched the volatility behavior of boron in carbon and graphite in the direct current arc was finely powdered iridium. When the discharge was burned in a 76 to 24 (by volume) argon-oxygen mixture, the line pair B 2497/Ir 2543 gave intensity ratios having a relative standard deviation of 1.870 or less between 0.5 and 4 p.p.m. boron. By use of a powder spark method, the sensitivity was extended to 0.25 p.p.m. boron. In this technique, a “sifter” electrode (a porous cup electrode with a perforated floor) was filled with powdered sample and used as the upper electrode in a high voltage spark discharge. The opening at the top of the electrode was closed with a small cork. During the discharge, the sample material sifted into the discharge area and was excited. When this discharge was conducted in argon, the line pair B 2497ICu 2492 gave a relative standard deviation of 1.870 or less between 0.25 and 7 p.p.m. boron.
C
HEMICAL determination of small amounts of boron in carbon or graphite is commonly accomplished by igniting
the sample in the presence of calcium oxide or fusing it with an alkaline oxidizing mixture, and determining the resultant increase in total borate by colorimetry or titration. Because of the large blanks and long processing involved, however, such methods begin to lose accuracy and precision as the boron concentration decreases to the parts per million range. Several attempts have been made t o perform this analysis spectrochemically, but the results achieved appear to have been semiquantitative. In 1936, Gatterer (4)suggested that boron, as well as other impurities in spectrographic graphite, might be determined by evaporating appropriate amounts of solutions containing the salts of interest on flat-topped pure graphite electrodes, and burning to completion. Concentrations were estimated by comparing the times required for the untreated graphite and the standards to produce spectral lines of equal density. This procedure ignored the possibility of the premature escape of boron oxides from the standard electrode, however, and a t best yielded semiquantitative results. S o data on sensitivity, accuracy, or precision were given by the author. Shugar ( 2 3 ) recently published a procedure for determining boron in carbon or graphite used in nuclear piles. He mixed the carbon or graphite with a calcium hydroxide slurry, dried the mixture, ignited it 1.5hours at 850” to 900” C., and exposed it spectro-
graphically, using beryllium as internal standard. The author stated that 0.1 p.p.m. of boron could be “detected” and 0.2 p.p.m. “estimated” by this method. He regarded the method as semiquantitative, however, and “not as precise as colorimetry.” LIitrovic (IS) gave an absolute sensitivity limit of 0.01 to 0.03 y for the detection of boron in carbon or graphite. I n view of the semiquantitative nature of the procedures available, and the ubiquity of boron as a contaminant of spectrographic and other carbon, it was considered advisable t o examine the factors m-hich complicate this analyei-, and if possible, to devise a quantitative procedure. DIRECT CURRENT ARC PROCEDURE PHYSICOCHEMICAL PROPERTIES OF BORON CARBIDE AND THEIR ROLE I N SPECTROCHEMICAL ANALYSIS
The spectrochemical determination of traces (0.2 to 5 p.p.m.) of boron in carbon and graphite, or in their presence, is considerably complicated by the behavior of the boron-carbon system a t high temperatures. At 2500’ to 2600” C., boron, boric oxide (20), borates, and even boron nitride (24) react with carbon to give boron carbide (BdC). This compound is extremely stable and unreactive; it can safely be assumed (see below) that all elemental or combined boron which has not volatilized before the sample reaches the above temperature will react with any carbon present to give boron carbide. Any attempt to devise a spectrochemical procedure for determining trace concentrations of boron in carbon or graphite must therefore provide for the volatilization of this compound. The literature does not appear to contain any quantitative information on the vapor pressure of boron carbide; however, its volatility is known to be very low. Ridgway ( 2 0 ) ,who was the first to isolate pure boron carbide, described its vapor pressure as “inappreciable” at the melting point [2450° C. (6)] : Rusanov ( 2 1 ) confirmed this in stating that traces of boron are still present in graphite which has been heated at 3000” C. for 30 seconds. Steinle ( 2 5 ) observed that when a direct current arc was struck to an anode containing a mixture of less than 1% boric oxide in carbon, the surface of the anode was immediately covered with tiny globules of a molten substance, later shown by x-ray diffraction to be boron carbide contaminated with carbon. As the burning proceeded, these globules coalesced into large drops. The fact that the boron carbide failed to volatilize completely v;hen in a fine state of subdivision is further evidence that its vapor pressure
