THE ELECTRODYXAMICS OF SURFACE CATALYSIS* BY A . KEITH BREWER
Perhaps no phase of chemistry has received a more intensive study than has surface catalysis, especially during the past few years. A great mass of evidence has been accumulated which indicates in a most satisfactory manner that a catalytic surface is not uniformly active, but that its ability to promote chemical action seems largely confined to the molecules of gas adsorbed a t certain centers or active points on the surface. While this must be looked upon as a decided step in advance, the fundamental mechanism employed by these active centers in promoting chemical action is still a n unsolved question. The writer while investigating the physical side of the reaction process, namely studying the ionic emission during chemical action and the effect of gases on thermionic emission and on contact difference of potential, has been led to certain conclusions that he hopes will add to our knowledge of the reaction process. The basic principles involved in the proposed mechanism are not fundamentally different from those suggested by Sir J. J. Thomson'. I n brief the mechanism is that the image and intrinsic surface forces combined with kinetic energy of agitation dissociate the gas molecules on the surface into ions; the ions thus formed are driven from the surface by kinetic agitation with a probability distribution of velocities; and that chemical action results from a rearranged combination of the ions thus driven from the surface. I n this manner it is possible to treat chemical action as a special case of thermionic emission. The experimental work, consisting of a study of ionization in reacting gases, and the thermionic emission in the presence of various gases, on which the proposed mechanism is based, has been presented in a series of recent The salient points may be summarized as follows: The ionization, which has been detected in every reaction investiI. gated, is specific for each reaction. The current through the reacting gas is proportional to the number 2. of molecules reacting, but the ratio of ions capable of being removed from * Fertilizer and Fixed Kitrogen Investigations Bureau of Chemistry and Soils U. S. Department of Agriculture. 1 Slr J. J. Thomson "The Electron in Chemistry," Chapter 11'. Trans. Am. Electrochem. Soc., 44, 257 (1923). J. .4m.Chem. SOC., 46, 1403 (1924). 4 Proc. S a t . Acad., 11, 512 (1925). 5 Phys. Rev., 26, 633 (1925). 5Proc. Kat. .\cad., 12, 560 (19261. 7 Proc. Kat. Acad., 13, j94 (192;).
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the reacting gas by a n applied voltage to molecules reacting is exceedingly small, being of the order of magnitude of I to 101~. 3 . The ions present in a surface-catalyzed reaction, such as the oxidation of ethyl alcohol, are liberated only at the electrode surface3,‘where the reaction is taking place, with no detectable ionization occurring in the gas volume. Gas phase ionization was observed during the thermal decomposition of ozone, and doubtless exists during the decomposition of nitrogen pentoxide, although the test was not made in the latter case. 4. The ionization current, as well as the amount of chemical action, is dependent on the chemical nature of the s ~ r f a c e . ~ ~Platinum ~,‘ and gold gave quite similar results in the cases tested, but copper, iron, brass, glass, copper oxide, iron oxide, and aluminum oxide possessed characteristic properties. j. The emission during the osidation of ethyl alcohol and hydrogen on gold and iron and copper oxides, when compared to the thermionic emission from these electrodes in the presence of the gases separately has shown definitely that the ions emitted in the reaction are the same as those emitted thermally. Apparently the chemical action allows ions to escape from the surface a t a lower temperature and with less work than when in the presence of the various gases taken singly; this lowering of the temperature and work of emission is more pronounced for the negative than for the positive ions in the case of the osidation reactions s t ~ d i e d . ~ . ~ , ~ 6 . The ionization current at atmospheric pressures is directly proportional to the applied difference of potential between the electrodes, Le., Ohm’s 5 This fact seemed surprising until the cheniical emission was shown to be a type of thermionic emission where saturation is obtained only a t low pressures. The effect of pressure on the saturation current has been given in a series of experiments by Brotherton.8 Saturation disappears a t pressures as low as a fraction of a millimeter of mercury. j . The temperature-current curves follow the Richardson equationQfor thermionic emission (i = X T k b T). X straight line is obtained by plotting (log i - I,;Z log T) against (1:T.2.303), where b, a measure of the work necessary for the removal of an ion from the surface, is given by the slope of the 8. I n the case of both thermionic and chemical emissions a definite interdependent relationship was shown to exist b e h e e n the temperature and work of emission for the positive and negative ions in various gases and rea c t i o n ~ . ~This , ~ interdependence has been made use of to postulate the presence of a selective intrinsic force, electrical in nature, existing a t the emitting surface. This intrinsic force, it is believed, is one of the controlling factors in thermionic emission in gases, and in chemical action; it is effective to approximately 3 X IO-^ cm. from the surface. 8 9
Brotherton: Proc. Roy. SOC.,105, 468 (1924). 0. 11‘. Richardson: “The Emission of Electricity from Hot Bodies.”
1008
A. KEITH BREWER
Analysis of Results While the above phenomena do not offer a direct explanation of the relationship existing between chemical action and ionic emission, get when the results are analyzed from the standpoint of electrodynamics it becomes possible to draw certain definite conclusions regarding the conditions existing on hot surfaces.
Application of the Concept of Electrostatic Zmage Attraction Since the ions liberated by the various chemical actions had their origin a t the electrodes, they must be treated as charged bodies near a conducting surface. (Little if anything being known about the conditions under which gas phase reactions take place, the discussion will be confined a t present to pure surface reactions.) A charged body, situated near a plane conducting surface, will induce a charge in the surface, such that the body will be attracted to the surface by a force equal to that which would be exerted by another charged body of equal but opposite sign, placed at a similar distance behind the surface, that is, a t the mirror image of the body in the coqductor. The force of attraction acting on a charge q e.6.u. a t a distance r cm from a conducting plane is given by q2/4r2dynes. The charge g is in a n electric field of 300 q/4r2 volts per cm. If q is the electronic charge 4.774 X 1 0 - l ~ e.s.u., the field acting on a singly charged ion at different distances from the surface is given in the following table. Distance from surface in cm.
IO-^ IO-^
Volts per cm.
3.6 3.6 X
Distance from surface in cm. 10-6
IO*
IO-'
Volts per cm.
3 . 6 X 104 3 . 6 X xo6
It will be noted that a difference of potential between the electrodes of volts per cm. will just hold in balance an ion situated a t 6 X IO+ cm from the surface. From this it becomes obvious that to remove all the ions from the surface a fall of potential between the electrodes would be required that is far above the break-down potential of the gas; saturation is, therefore, quite impossible for any appreciable gas pressures. It will be noted that the distance a given applied voltage is able to plumb down into the image layer and remove an ion varies with the inverse square root of the voltage. An important deduction to be drawn from the application of the electrostatic image concept to the straight line relationship existing between current and applied voltages is that the concentration of the ions upon approaching the surface increases a s the inverse cube of the distance. These calculations, however, take account of only the image force. The recent experiments on the intrinsic force would lead one to believe that a still more rapid increase exists very near the surface. Thus, while these considerations throw no light on what is happening a t the surface they do show the condition existing a t a very short distance from the surface, in which region it is believed that the ionic rearrangement resulting in chemical action takes place. 1000
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The enormous magnitude of the image field would make it seem unlikely that any appreciable change in the amount or rate of reaction could be occasioned by the application of any voltage under the break-down potential of the gas. The question, however, might well repay investigation as there is a possibility of detecting such a change especially in reactions possessing a high chemical work function. Application of the Richardson Equation. The straight line relation between voltage and current has made possible the exploration of only the outer region of the image layer (~o-lcm)from the surface. The fact that the temperature-current curves follow an equation of the Richardson type, the development of which is presented later, enables the study to be carried much deeper into the surface region. In the Richardson equation, i = AT*e-b’T, where i is the current per sq. cm. of surface, A and b are constants, and T the absolute temperature, b may be looked upon as a measure of the total work done in removing an ion from some point near the surface to a point outside the sphere of image attraction. This work is w = bk, where k is Boltzman’s constant. If we substitute dC/dt for i, and E for w, we have the usual expression for the rate of chemical action. I t is significant to note that not only is the form of the equation the same, but that the values of the constants are of the same order of magnitude for chemical action as for thermionic emission in gases; in general b is from two to four times larger than E l k . This close analogy has led some physicists to conclude that thermionic emission is due to chemical action between the emitter and the gas present; Richardsong, however, has shown that this cannot be the case, since b is larger than could possibly be expected from a chemical process. Conversely, the fact that b is larger than E;k is in line with the proposed concept that chemical action is a special case of thermionic emission. Thus, for chemical actions, b becomes a measure of the work done in removing an ion, not through the entire image layer, but to a point just far enough from the surface for successful recombination to take place. The escape of an ion from a surface against its electrostatic image attraction should not be affected by the sign of the charge on the ion, since the image force is purely non-selective in character. X comparison of the values of d b and d T for the positive and negative thermionic emission in various gases’ has shown that a selective force also exists at the surface. This force is of such a nature that it allows a more ready escape of positive ions from the metals gold, iron, copper, and platinum, but of the negative ions from iron and copper oxides. The nature of this selective force thus imposed on the image force, as given by its behavior, is that of one possessing the properties of a resultant electrical field effective to a few molecular diameters out from the surface. Such a field would be positive for iron and negative for iron oxide, for instance. I t has been suggested6 that the positive intrinsic field may result from a leakage of the positive field of the nucleus through the electron lattice of the surface atoms (especially those of small atomic
A . KEITH BREWER
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radii), while the diminution and change of sign of the field accompanying the accumulation of oxide on the surface may result from the surface being covered with the negative oxygen ions of the oxide, which, with increasing concentration on the surface, gradually neutralize the positive intrinsic field of the pure metal, and finally, upon high oxidation, actually establish a negative field This intrinsic force, which it is believed is one of the controlling factors in thermionic emission, and chemical action. is doubtless very closely allied to contact difference of potential.
'
Mechanism of Ionization The thermionic and chemical emission data clearly indicate the presence of positive and negative ions in the image region of an emitting surface, also that the concentration of these ions increases very rapidly upon approaching the surface. While the exact process of ion formation cannot be told with certainty, it is possible that the mechanism is not fundamentally different from that found in solutions, i.e., the molecules of gas are dissociated by the electrical forces at the surface as are the molecules of the solute dissociated by the dielectric forces of the solvent.' The strength of the image field in volts capable of drawing an ion to the surface has already been pointed out. This same force will also act, but to a lesser extent, on a neutral molecule having an electrical moment. As the molecule is drawn to the surface, its poles will become more widely separated and just to the extent that the moment is increased, the more strongly nil1 it be drawn to the surface; the forces of attraction and dissociation are, therefore, cumulative. The image force existing between a molecule of gas and a surface does not become comparable with the binding force in a molecule until the molecule is within about its diameter from the surface. At this point, as has been mentioned, the molecule is also operated on by the intrinsic force tending t o draw one ion to the surface and to repel the other outside the region of intrinsic attraction. For metals such as gold, where the intrinsic field is positive in character, the negative ion is held more closely to the surface, while the opposite should be true for a highly oxidized iron surface. Nature of the Ions Very little can be told at present about the exact nature of the ions emitted thermally in the presence of single or reacting gases. The negative ion emission in gases on such surfaces as gold and platinum takes place at several hundred degrees lower temperature than does a corresponding electron emission in a vacuum; positive ion emission completely or almost completely disappears in a vacuum while in the presence of a gas it occurs at even lower temperatures than does the negative. Unfortunately it docs not seem likely that mass spectrograph or mobility measurements will be of much help in determining the chemical nature of the ions emitted in the presence of gases, due t o the shortness of the mean free path and the association of the ions.
T H E E L E C T R O D Y N A M I C S O F S U R F A C E CATALYSIS
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Two general theories have been advanced regarding the nature of the emitted ions. H. A. Kilsonlo has suggested that the positive ions come from the gas through a dissociation of the molecules of gas into positive ions and electrons; the negative ions, on the other hand, he considers to be electrons which are able t o escape from the electrode because of an electrical double layer formed by the gas on the surface. The writer has put forward the hypothesis' that the gas molecules are dissociated into positive and negative ions on the surface and hence arc atomic in character; this, of course, applies only to those ions emitted in the presence of gases at temperatures below which electron emission would be obtained in a vacuum In line with this contention, some indirect evidence has been obtained which indicates that ethyl alcohol dissociates into H+ and C2Hj0- ions. It is possible that molecules with small electrical moments dissociate into positive ions and electrons while it seems probable that highly polar molecules are dissociated into positive and negative ions.
Mechanism of Reaction In order that an ion be able to escape from a surface, it must possess kinetic energy of agitation with a velocity component perpendicular to the surface capable of removing it against the attractive forces to a distance greater than 10-4 cm. from the surface; then it becomes a thermion. However, should an ion whose velocity component is less than this come in contact with another ion of opposite charge to form a neutral molecule, the energy necessary to complete the escape becomes materially smaller. If heat is liberated as a result of the union, then the probability of escape is greatly enhanced because of the increased kinetic agitation. However, should such a new born molecule be formed within less than a certain critical distance from the surface, the chance for its escape is small, because in case the surface force is large the probability of being drawn back to the surface and redissociated is proportionately large. I t follows, therefore, that the formation of molecules takes place at a point i n the image region so far removed from the surface that the chance for redissociation is small. The energy that an ion must possess in order to move out to this critical distance where it can undergo successful combination is given directly by the equation derived later. The order of magnitude of this energy is approsimately half that necessary for the escape of thermions. While definite evidence is lacking, it does not seem necessary for two gases to react chemically that both be equally dissociated at the surface. In the case of the oxidation of ethyl alcohol, for instance, it was very evident that the ions escaping from the surface region came primarily from the alcohol and not from the oxygen. It is possible that the effect of the partial pressure of the various reactants on the rate of reaction is determined by the degree of dissociation of the respective gases. H. A. FYilson: Phil. Trans., 208A,248 (1908).
A. KEITH BREWER
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A Mathematical Analysis of the Reaction Mechanism The suggested mechanism making surface catalysis a special case of thermionic emission is readily amenable to a mathematical treatment, since it employs only the electrical forces of the surface and the kinetic energy of agitation of the gas. The analysis should not only be simpler but more elucidating if the various contributing factors are treated individually. Let us consider a plane metal conductor bathed in a gas, and having its surface covered with a melee of gas ions as has been previously outlined. Let the case be simplified, for the present, by assuming no intrinsic force a t the surface; the escape of the ions from the conductor is impeded only by their image attractions. Since the image force and the kinetic energy of agitation will be the same for the positive as for the negative ions, the treatment will apply equally well to either. From the standpoint of the mechanism outlined, the number of molecules of the reaction product formed per second in a given unit volume situated a t a distance r from the surface will be a function of the number of ions that it contains. To determine the rate of escape of the ions from the surface to a region where chemical action is possible, we must consider the velocity distribution of the ions a t the surface. The number of ions per unit volume with velocity components between u du, v dv, w dw, is:”
+
+
+
n being the number of ions per unit volume. The number of ions per unit volume having velocity components wo to m perpendicular to the surface and any velocity parallel to the surface is:
For a n ion to be emitted thermally, it must possess energy sufficient t o be completely removed from the image layer. However, a n ion does not have to possess energy sufficient to make its escape through the image layer t o be able to combine with another ion to form a neutral molecule giving chemical action, so w, for chemical action is less than wo for thermionic emission. Eflect due to the Intrinsic Force. I n determining the effect of the image attraction on the rate of escape of ions, an ideal surface was assumed devoid of a n intrinsic force. While it may be possible that such a surface exists among metals, it is certainly not true for those that are ordinarily used as catalysts. See, for example, R. C. Tolman: “Statistical Jlechanlcs,” Chapter 5
(I
927).
THE ELECTRODYNAMICS OF SURFACE CATALYSIS
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An iron or copper surface may be oxidized to the point where no effective intrinsic force appears, likewise an aluminum surface covered with a film of oxide possesses a very small intrinsic field. The aluminum surface showed no ability to catalyze the oxidation of ethyl alcohol a t temperatures where gold was an excellent catalyst5; under similar conditions the catalytic power of the iron and copper oxides was very small. Poisoning of catalysts on the basis of the proposed hypothesis may largely be due to the formation of a compound on the surface of the catalyst of such a nature that the intrinsic field of the metal becomes neutralized, thus decreasing its ability to dissociate the gas molecules on the surface. The mathematical treatment for the effect of the intrinsic field is the same as for the image field, with the exception that while the image force is an attraction for ions of either sign so that (work done against the image field) = e41 = w1 > o always, the intrinsic force is an attraction for ions of one sign and a repulsion for ions of the opposite sign, so that the (work done WP 0, depending on the nature of the against intrinsic field) = e42 surface. The field (image or intrinsic) will be called positive when it urges a positive charge towards the surface. Taking account of the image and intrinsic fields, the concentration of the ions in the chemically active region is
5
The Eflect of Pressure. The equation as it now stands, although giving an expression for the rate of reaction, is not sufficiently comprehensive in that it considers only the surface in detail, while the n term alone refers to the reactants. It will be recalled that n is the ionic concentration in unit volume of gas at the surface. This concentration is obviously determined by two factors, the pressure of the gas, and the nature of the gas, Le., its dissociation potential. The effect of gage pressure p on the ionic concentration as given from thermionic studies is = B ,-Fi-wa/kT where B is a constant, F1 and w3 are functions of p only, and not of T. This can be written. = no e - wa/kT where w3 is the difference in the energy content between a n ion on the surface a t pressure p and a n ion on the surface when the pressure is such that the ionic concentration is no. The Eflect of the Nature of the Gas. I n a previous article6 in determining the distance from the surface to which the intrinsic force was effective, the assumption was made that the value of b found from the slope of the curve represented only the work
A . KEITH BREWER
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necessary to get an ion away from the surface, and did not include the energy necessary to complete the dissociation of gas into ions. This assumption doubtless serves as a fair approximation where it was intended to be applied, namely to the dissociation of ethyl alcohol. Such an assumption, however, cannot be given a general application; for highly polar molecules like ethyl alcohol the electrical forces at the surface are sufficient to so separate the ions that but little external work would be needed to remove an ion from its mate. For non-polar molecules, on the other hand, it would seem evident that the energy necessary to complete the dissociation of the molecule into ions would become very appreciable. The work w4 necessary to complete the dissociation of a molecule is expended in removing one ion against the electrical attraction of the other; w4 > o always, by an amount determined by the nature of the gas. The previous expression for n must be modified, therefore, by the factor which is an obvious application of the Maxwell-Boltzman distribution law.
General Expressions for the Rate of Reaction The energy of an ion in the chemically active region is thus made up of four parts: The work done against the image attraction, e$l = wl. (I) ( 2 ) The work done against the intrinsic field, e$* = WZ. (3) The energy dependent on the gage pressure, w 3 . (4) The energy to complete the dissociation, w4. I t will be noted that the first two are determined by the surface and the last two by the gas. The complete expression for the rate R at which a given kind of ion gets into the chemically active region is, therefore:
by putting bk = w1
+ +
+
w2 wg I V ~ . b is a measure of the work which a n ion must do in escaping from the surface to a region where it can react chemically. The above expression for the rate of escape of ions from a catalytic surface may now be applied directly to the rate of chemical action. Let us consider a condition in which c ions of C, d ions of D, . . . . leave the catalyst per sq. cm. of surface per second and enter the chemically active region to form intermediate ionic combination products, S, T, , , , according to the equation cC* dD* eE* . . . . sS tT . ..
+
+
-
+
+
where (*) refers to ions. An ion entering the chemically active region may be either of two types, a simple ion resulting from the initial molecular dissociation, or a complex ion formed by the attachment of the simple ion to a neutral molecule.
T H E ELECTRODYSAMICS OF SURFACE CATALYSIS
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In general each kind of ion will have a different rate R of getting into the chemically active region. The forward reaction will be, therefore: R~= a; -4; ~ d .re, , , . . T d ( c + d - e + . . . I e - (cb, + db,, + eb, . . . I T
-
- a; ~ -
q m ‘Z’e -b’ T
+ + +
n‘ = c d e , , , . is the number of ions contributing to the forward reaction; al is the fraction of the ions of C, D , E, . . . arriving per second in the chemically active region which actually combine to form the intermediate product’s, T, . . . . The ionic combination product S.T. , , , may be converted into the final reaction product in several different ways: ( I ) , S, T. . . . may themselves be the end product; ( 2 ) S. T? . . . may combine with gases C. D, E, . . . in a homogeneous reaction according to the equation s’S
+ t’T + .. . c’C + d’D + e’E + . . . = SS + yY + ZZ + . . .
where X, Y, Z, . . , are the end reaction products which escape from the image region; or (3), S,T, . . . may be drawn back to the surface and again decomposed into C, D? E? . . . . ( 3 ) will always be in process. The rate of the forward reaction may now be expressed by
(IC’, dt
=
a; a: A’ T”‘ eb’ n z ’e -b’ T
-& ~ =
where ai is the proportionality factor referring to the probability of S,T, . . . being converted into X, Y, . . . . By setting E/k = b, the above cspression for the rate of chemical action becomes similar in form to the Xrrhenius equation, where E of the latter refers t o the “energy of activation”. The present development, however, gives the terms an entirely new significance. b, a measure of the work done by the ions in escaping far enough from the surface t o react chemically, may best be termed the chemical work function. A, containing the terms a,a, referring t o the probability that the ions entering the chemically active region will combine to form the initial and final reaction product may be termed the combinatl’onfactor. Since, as has been stated, an ion does not have to escape from the image region to react chemically, the Chemical Work Function is less than the Thermionic Y o r k Function. I t is to be assumed that chemical combination is divided according to the laws of probability between all processes which bring the required ions and molecules together in the chemically active region. The process most favored by chance under one set of conditions need not necessarily be so under all. The back reaction is represented by a n equation identical to that developed for the forward reaction, but with different values for the constants). dCff/dt = a;‘a;‘ -A’’ T”’‘l2 e-Y = -
T”“,’ e -Q/T
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A. KEITH BREWER
where q is the heat of formation of s molecules of S, t molecules of T, Equilibrium is reached when dC'/dt = dC"jdt and is expressed by the equation
I
,
where K is the equilibrium constant and (LlfBi)o represents the probability (a;a:l0 a t equilibrium that the average molecule, situated in the chemically active region, will escape from the image zone rather than be drawn to the surface and dissociated.
Deductions The effect of gases, and the previous treatment of the catalyst on the intrinsic force can only be expressed in a qualitative manner a t this time. It is hoped that the continued study of contact difference of potential will add t o our knowledge of the factors underlying the intrinsic force. This is important since it follows on the basis of the proposed mechanism that the change in w2 due to variations in the intrinsic force from point to point over the surface gives rise to the "chemically active center" phenomena established by H. S. Taylor. It has already been suggested that the poisoning of catalysts may be due to the formation of a compound on the surface which materially changes the intrinsic force. From the argument as presented, it will be seen that if the intrinsic field is uniform over the surface, the Chemical Work Function will be practically independent of the strength of the intrinsic force. Let a case be considered in which the intrinsic field is positive as on gold, then the negative ions will be drawn more closely to the surface while the positive ions will be pushed out. Thus, since the distance r is less for the negative ion than for the positive, the image force will be greater on the negative than on the positive ion, so that (w, w2)+ < (wl wt)-. Since w4 must be the same for either ion it follows that b+ < b- in proportion to the strength of the positive intrinsic field. However, for chemical action to take place both positive and negative ions must enter the chemically active region; hence a concomitant lowering of bf and a raising of b- will have but little effect on the catalytic activity of a surface. For most substances, on the other hand, it is to be expected that the intrinsic force will vary over the surface, or it might actually change in sign from point to point on a surface like partially reduced iron oxide or a heavy metal impregnated with alkali. I n this case the positive ions will enter the chemically active region in greater numbers and with less work from the areas of strong positive intrinsic force, while the negative ions will come more readily from the weak positive or negative areas; therefore, (b+ b-) will
+
+
+
THE ELECTRODYNAMICS OF SURFACE CATALYSIS
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be lower than on a surface of uniform intrinsic force. From this it will be seen that the more irregular the intrinsic force over the surface, the lower will be the Chemical Work Function, and the greater will be the number of molecules reacting per sq. cm. per second, hence the greater the catalytic power. The writer is especially indebted to Dr. W.E. Deming of this Laboratory for his constructive criticisms and his many valuable additions to the present paper.
summary Certain observed facts, namely, the emission of ions during various surface catalyzed chemical reactions following an equation of the Richardson type, and the concentration of these ions increasing as the inverse cube of the distance upon approaching the surface, have been made the basis for the proposed mechanism of surface catalysis, wherein the subject is treated as a special case of thermionic emission in gases. I n brief, the mechanism is that the gas molecules upon approaching the surface are dissociated into ions by the combined image and intrinsic forces of the catalyst. The ions so formed are driven from the surface by kinetic agitation with a probability distribution of velocities. Chemical action results from a combination of the ions whose velocity components perpendicular to the surface are sufficient to carry them out to a region of weak surface forces-the chemically active region. The equation for rate of the forward reaction developed from the point of view of electrodynamics is dCljdt = AiTn'/2 e-b'/T where A is the combinationjactor and b is the complete chemical work junction. The back reaction is expressed by the same equation as is the forward but with different values for the constants. Equilibrium, therefore, is expressed by
