Oct., 1956
TRANSFEREKCE NUMBERS IN MOLTEN SALTS
in.which the coefficient k4ka/kskoris not an equilibrium constant. l 3 For the usual limiting cases, the activated complex is in equilibrium with the predominant form of the enzyme and even though the apparent rate constant is a composite of several rate constants, the temperature coefficient of the apparent rate constant is related to the enthalpy difference between the activated complex and the reactants. For the unusual limiting cases, such is not true; the temperature coefficient of the apparent rate (13) A set of numerical values of the concentrations and rate constants which yields this unusuallimiting case is: (9) = (M) = (P) = 1, 108,k r = 108,ks = 1, ka = 108,kr = 10-1, ka a kr = 109,ke = 1, ks 105,k a = 105,kfl = 10-1,kx = 10, and k, = IO’.
1381
constant is not directly related to the enthalpy difference between the activated complex and the reactants. If an uncatalyzed reaction (or a catalyzed reaction in which no significant amount of the catalyst is tied up as reaction intermediates) proceeds via more than one pathway, these pathways are completely independent. This is not true for catalyzed reactions in which a significant fraction of the catalyst (which is a catalyst for two or more of the pathways) is tied up as intermediates. The alternate pathways cannot, under such conditions, be viewed as independent and unusual kinetic consequences of the type outlined may arise in reactions which follow such mechanisms.
TRANSFERENCE NUMBERS I N MOLTEN SALTS BYBENSON R. SUNDHEIM Washington Square College, New York Universitg, New York, N . Y . Received A p r i l 6 , 1966
It is shown that the transference numbers of the species in a pure molten salt measured in any particular experimental arrangement may be calculated in terms of the relative masses and boundary conditions. In particular the transference number measured in the “bubble cell” type of experiment is t~ = MB/(MA MB).
+
A pure molten salt cannot sustain a concentration gradient. For this reason, the experimental arrangement usually employed in measuring Hittorf transference numbers can give only certain results. The forces imposed by the walls of the cell or by gravity cause the transference number of the species to which the electrodes are reversible to appear to be unity.’ It has been suggested2J that an experimental arrangement which obviates the effect of gravity would enable more realistic transference numbers to be determined. Transference numbers in this sense would be based on the mobility of each ionic species “with respect to the bulk of the liquid.” Following suggestions put forward by several authors, e x p e r i r n e n t e r ~carried ~ ~ ~ out electrolyses in cells so delicately constructed that lateral motion of the fluid within the cells could be detected. For example, in the cell pictured in Fig. l it was observed that electrolysis between electrodes reversible to the cations was accompanied by an increase in the volume of the cathode compartment and a corresponding decrease in the volume of the anode compartment. The interpretation heretofore made of this phenomenon was as follows: the volume change in the cathode compartment per equivalent of charge is due to the equivalent volume of the metal discharged at the electrode minus the equivalent volume of salt lost by the electrolysis plus t+ equivalent volumes of salt due to migration into this compartment. After correcting for the volume changes a t the electrode, there yet remains a decrease in volume equal to 1 - t+ = t- equivalent volumes of electrolyte. (1) K. E. Schwara, 2.Elektrochem., 46, 740 (1939). ( 2 ) S. Karpachev and S. Pal‘guev, Zhrr. Fiz. K h i m . , 23, 942 (1949). (3) F. R. Dukeand R. Laity, J . A m . Chem. Soc., 76, 4046 (1954);
F. R. Duke and R. Laity, THIE JOURNAL, 69,549 (1955).
It is the purpose of this communication to demonstrate that simple considerations about the conservation of momentum lead to definite predictions about the relative motions of parts of such cells as measured in any particular arrangement and hence to predictions of the quantities which have been called transference numbers.
Fig. 1.-An electrolysis cell. The arrows show the motion of the ions in the various parts of the cell. For simplicity the figure shows the motion after any necessary corrections for volume changes a t the electrodes are made.
Since a crucial concept in the following is the balance of momentum, we begin with the following thought experiment. Consider two separate masses of different fluids, each of which has any given momentum. Any type of interaction between the two species is permitted. The samples are allowed to collide in any way, with any sort of mixing process. The final momentum of the mixture must equal the sum of the initial momenta. Of course, kinetic energy is not conserved in such an inelastic collision unless heat is included in the kinetic energy balance. I n short, the interaction of the two species according to any viscous dissipative process does not disturb the momentum balance. A second important point is that the application of a homogeneous electric field to any neutral
1382
BENSON R. SUNDHEIM
sample of matter does not lead to any net body force. Although one part of a fluid may move with respect to another, momentum of an electrically neutral set of charges is conserved in a homogeneous field. Finally, the concept of the flux of momentum in a chemical reaction must be discussed. For convenience we limit ourselves here to incompressible liquid systems. Whenever there is a volume change in a chemical reaction the chemical species come into (or out of) existence so t o speak by thrusting their way against the fluid already existing. This process creates motion and hence kinetic energy but no net force and no net momentum. If the reaction is heterogeneous, taking place at an electrode for example, the momentum of the electrode must be taken into account.
Vol. 60
a porous plug (subscript P) inserted in the cell between the electrodes. It floats freely and because of the friction of the moving fluid has the mass average velocity of the fluid between the electrodes. All velocities are measured with reference t o a coordinate system fixed in the laboratory. I n the event that complex species are formed by the ions, respective velocities of the two components denote the appropriate number average velocities. The conservation of momentum then requires
+ M ~ O+G O ~
M.~O;O
A
~
i
+
d
v M B~~ V Bi'
+
+
A
MEVE MGP = 0 (1) The equation of continuity for each species requires that the material lost by a source appear in the liquid. Electroneutrality requires that the boundaries of the two species remain coincident. These conditions lead to the equations -
A
vAi
A
- vA0
--------A
/A\
where q is the strength of the source. It may be expressed in terms of the current density, the dimensions of the cell and the molar volume of the salt and it represents the volume swept out per cm.2 of cross-section per second by material issuing from the source. The mass average velocity, v1 of the liquid between the electrodes is defined by d
Making the appropriate substitutions into equation l, the conservation of momentum is displayed in the form
MAG+ MB% + MA'(^. + VB) + M E I G+ MEVE+ 4
h
-
The masses and the strength of the source are d
given so that equation 4 serves to determine V B in terms of VE. I n each experimental arrangement d
A
6
Fig. 2.-Three types of cells. The arrows show the motion of the ions. For simplicity the figures show the motion after any necessary corrections for volume changes a t the electrodes have been made.
Consider the cell shown in Fig. 1. The electrodes are represented as sources adjusted so as to produce an ionic species a t rates equal in magnitude and opposite in sign. The ends of the cell are open, gravity being considered absent and the cell walls of negligible mass, The liquid is incompressible and slips freely along the cell walls. There are two ionic species and the electrodes are reversible to the species A. The flow pattern is as shown in the figure. The superscript i or o serves to distinguish between portions of the fluid between the electrodes (i) and outside of the electrodes ( 0 ) . The subscript E refers to the electrodes. There is
a further relation between V B and V E is given, thus determining the motion of all parts of the cell and consequently determining the motions upon which transference number calculations are based. The three experimental arrangements commonly used are shown in Fig. 2. The ordinary arrangement of an electrolytic cell, Fig. 2A, is exposed to the leveling effect of gravity. The porous plug is omitted and the experimentally A
A
observed quantities are of course V A ~ - VE. I n this case d
-
-
L
-
VB~
VE
and
.
Mp L
E
MA'
=
Mao = 0
A
vJ3
- VE = 0 - VE = q
-
(5)
2
VA
'A
A
A
=
(vAi
-
VE)/q
= 1
THEOXIDATION OF ARSENIC TRIOXIDE
Oct., 1956
The arrangement of Fig. 2B constrains the electrodes and the porous plug to zero relative velocity. -
A
The experimentally observed quantity is vg - v' after correction has been made for the volume change of the electrodes. 4
VB
-
- u'
=
(fila'
+
d
VB
A
fiJB')
- fifAi (q f MAi
+ Mgi
-
UP,)
- fifHivBi
=
-q M A MAi
+
AfBi
The arrangement of Figure 2C requires that &
A
D E = VB.
The experimentally observed quantity
&
1383
A
is V B - v'. I n addition to the motions described by the above relations certain other effects can have a bearing on the net motion. The extent of frictional drag in cells of type B is one such factor. Electrophoresis in the plug or along the walls is another. The quantities calculated above thus have the character of ideal results. The significant result is that the analysis of the relative motion of parts of an electrolysis cell does not yield information about the structure of the liquid. It is therefore seen that transference numbers in pure molten salts have little significance. It is a pleasure to acknowledge assistance to a portion of this work by the United States Atomic Energy Commission. (4) Several workers have been kind enough to show me their experimental results before publication. At the time of writing no set of experimental resrilts agrees with the results of this study or with the results of any other work.
KINETICS OF THE REACTION BETWEEN CHROMATE AND ARSENIC TRIOXIDE IN ALKALINE MEDIUM. INDUCED REDUCTION OF OXYGEN BY THIS COUPLE1 BY I. M. KOLTHOFF AND MORTON A. FINE MAN^ Contribution from the School of Chemistry, University of Minnesota, Minneapolis, Minnesota Received April 18, 1068
The oxidation of arsenic trioxide by potassium chromate in alkaline medium in the absence of oxygen has been found to be first order with respect to both the chromate ion and arsenic trioxide. The average value of the second order rate constant a t 30' in solutions of p H 9.1 and ionic strength of 1.75 was found to be 1.61 (i0.08)X 10-8 liter mole-' sec.-l. This rate constant is independent of hydrogen ion concentration at p H greater than 9.1 and increases with hydrogen ion concentration a t pH smaller than 9.1. The chromate ion-arsenic trioxide reaction induces the reduction of oxygen (by arsenic trioxide) in alkaline medium. The induction factor at 40" was found to be a function of both the pressure of oxygen and the concentration of the chromate ion but apparently independent of arsenic trioxide concentration. A limiting induction factor of 4/3 was obtained. A mechanism has been proposed to account for the kinetic and the induction factor data.
The initiation of polymerization of styrene, acrylonitrile, methyl isopropenyl ketone and butadiene-styrene mixtures by the alkaline chromate ion-arsenic trioxide system has been successfully accomplished in this L a b ~ r a t o r y . ~An understanding of the mechanism of the over-all reaction (equation 1) between chromate and arsenic trioxide in alkaline medium and of the initiation of polymerization by this system is desirable. 4CrOr"
+ 3Asp03 + lOHSO +4Cr3 + 3Asz05 +
20(OH)-
(1)
There are several experimental techniques available for studying the mechanism of initiation. These methods include: (1) the determination of the stoichiometry of the chromate ion-arsenic trioxide reaction in the absence and presence of the polymerizable monomer, (2) a kinetic investigation of the reaction in the absence and presence of monomer, (3) the identification and quantitative (1) This work was carried out under the sponsorship of the Federal Facilities Corporation, Office of Synthetic Rubber, in connection with the Synthetic Rubber Program of the United States Government. (2) Department of Chemistry, Providence College, Providence, R. I. (3) I. M. Kolthoff and E. J. Meehan, J . Polymer Sci.,9, 327 (1952).
determination of the initiating fragment which is formed in the chromate ion-arsenic trioxide reaction and is tied up in the polymer, and (4) the investigation of other induced reactions involving the chromate ion-arsenic trioxide primary reaction in alkaline medium. With respect to the last method it should be pointed out that the initiation of polymerization with this oxidation-reduction system is an example of an induced chain r e a ~ t i o n . ~ I n the present paper we describe a kinetic study of the oxidation of arsenic trioxide by chromate ion in alkaline medium in the absence of monomer, and the investigation of the stoichiometry of the induced reduction of molecular oxygen by the chromate ion-arsenic trioxide reaction in alkaline medium. The induced reduction of oxygen was chosen since the results will be useful not only for the interpretation of the mechanism of the primary reaction but for an understanding of the effect of oxygen in polymerization initiated by the above system . Experimental Materials.-Baker
A.R. grade potassium chromate was
(4) A. I. Medalia, A n d . Chern., 27, 1678 (1065).
