THE PHYSICAL AND CHEMICAL RELATIONS IN FLUID PHASE

T H E PHYSICAL AND CHEMICAL RELATIONS IN FLUID PHASE HETEROGENEOUS REACTION‘ PAUL S. ROLLER2 U.S. Bureau of Mines Non-metallic Minerals Experiment Station, Rutgers University, New Brunswick, New Jersey Received May $3, 138.4

I n reaction between two phases, two factors a priori determine the overall measured rate of reaction. The first is the rate of chemical reaction a t the interface; the second is the rate of transport of reactant, or product of reaction, to or from the interface. For heterogeneous reaction in which one or both phases are fluid, the transport takes place between the interface and the point of measurement. I n the present work, dealing with the fluid-fluid or fluid-solid heterogeneous reaction, a quantitative relationship is derived for the measured rate as a function of the chemical reaction rate and rate of physical transport. No arbitrary postulates as to mechanism are made. The results follow in the first place from a solution of the diffusion equation with chemical reaction a t the interface constituting one of the boundary conditions, I n the second place, diffusion as a mode of transport is generalized to include transport due to stirring or other type of agitation. THE NERNST THEORY

I n 1904, Nernst (21, 2) proposed a theory of heterogeneous reaction which has been referred to widely. I n this theory certain assumptions are made as t o the chemical and physical factors of the liquid-solid heterogeneous reaction. On the chemical side, reaction is supposed to be infinitely rapid, owing to an “infinite force” a t the boundary. On the physical side, a stationary liquid film is supposed to exist through which diffusion into the main body of the stirred liquid takes place. For the solution of a solid, or for reaction between a solid and a solution, the adhering film a t moderate rates of stirring is calculated to be 20 to 50 microns thick (2,29), 1 Published by permission of the Director of the U. S. Bureau of Mines. Not subject t o copyright. Presented before the Division of Physical and Inorganic Chemistry at the Eightyeighth Meeting of the American Chemical Society held at Cleveland, Ohio, September 10-14,1934. 2 Associate Chemist, U. S. Bureau of Mines Non-metallic Minerals Experiment Station, Rutgers University, New Brunswick, New Jersey. 221

222

PAUL S. ROLLER

Regarding the chemical aspect of the Nernst theory, there is no reason for supposing that the rate of heterogeneous chemical reaction is “infinite.” In other words, not all collisions of the reacting molecules will be expected to be fruitful; rather, an activation energy should be required, as for homogeneous reaction. I n many cases, no doubt the second phase will exert a catalytic effect on the rate, but our knowledge of heterogeneous catalysis leads us to expect that a catalytic effect of the second phase, if and when it exists, would hardly be complete. Regarding the physical aspect of the Nernst theory, experimental evidence strongly contradicts the possible existence of a stationary film. Contrasted with the Nernst film of 20 to 50 microns thickness, microscopic observation indicates a thickness of any possible film of less than 0 . 4 , ~ (26) ;solution of a fine sediment indicates the possibility as less than 0 . 2 , ~ (23); flow through capillaries less than 0 . 0 3 , ~(3); reaction in the presence of a monomolecular fatty acid less than 0.002511, or 25A.U. (20) ;while ultramicroscopic examination of water flowing slowly in pipes shows that, though the flow is axial, motion in all three directions persists right up to the surface (10). The evidence of sedimentation may be considered in some detail. It has been found (23) that the volume of a coarse sediment consisting of 100-micron particles is the same before as after centrifuging, and equal approximately to the value common to all particle sizes after centrifuging. The coarse particles in the normally settled state are therefore densely packed and touching. The force necessary to displace completely any possible aqueous film surrounding the particles is merely that of gravity acting on a 100-micron particle, or a compressive force of 20 dynes per square centimeter for a particle of density 3. However, with water stirred a t the moderate linear rate of 50 cm. per second, the forces are much greater. The tangential shearing stress tending to displace any possible film of, say, 25 microns thickness, is ten times as great, or 200 dynes per “ square centimeter. If the flow is directed toward the surface of the solid, the compressive stress is two hundred and fifty times as great, or 5000 dynes per square centimeter. Theref,ore it is impossible that a stationary film could exist in the presence of the forces evoked by ordinary stirring, a result that agrees with the other evidence already cited. In most liquid-phase heterogeneous reactions the chemical effect is obscured, though only superficially, by conditions of physical transport. However, in many cases the chemical effect predominates, as shown in the following results: (a) The non-dependence of rate of reaction on rate of stirring, as for rate of solution of glass in alkali (25), of metals in acid (5), of olefins in sulfuric acid (9), of mineral sulfides in dilute sulfuric acid (24), of crystallization from supersaturated solution (19), of hydrolysis and saponification (11); (b) the frequent wide difference in rate of solution

RELATIONS IN FLUID PHASE HETEROGENEOUS REACTION

223

of certain of the crystalline faces of the self-same substance (22, 30); (c) a temperature coefficient appreciably greater than that for diffusion (1, 5, 18). The Nernst theory, which concentrates all attention on a hypothetical and apparently non-existent film, naturally has no explanation for these results. On the contrary, these results and the more general facts of the heterogeneous reaction are accounted for in the relationships to be derived, which are founded simply on known physical and chemical laws. RATE O F REACTION WITH TRANSPORT BY DIFFUSION

Let a plane solid of area F react with a prism of quiescent solution3 of depth L, of mea equal to that of the solid, and initial concentration co. In the body of the solution, the ordinary diffusion relation (equation l), in which D is the diffusion coefficient, is obeyed.

Out of all the molecules in solution, only those in the immediate neighborhood of the solid-fluid interface are in a position to make impact with the solid surface. For a gas, for instance, by kinetic theory out of all molecules which are three mean free paths distant from the interface, less than 1 per cent will strike the surface on their next collision. For a solution, the reacting molecules probably will be confined to those within one or two molecular distances from the solid surface. For both gas and solution the number of impacts per second is proportional to concentration, c, in the neighborhood of the interface. The number of gram-moles, m, reacting per unit area per unit time is proportional to the number of impacts and is given by equation 2, in which k is the specific reaction velocity constant. 1 am _ _ = F at

kC

The rate a t which dissolved molecules disappear by reaction a t the solid surface equals the rate a t which they diffuse from the neighborhood of the interface to the interface. Therefore, by definition of diffusion and by equation 2,

a t x = L. a With partial pressure replacing concentration, the same derivation applies to reaction between a gas and solid, or gas and solution, provided i t is of the first order.

224

PAUL S. ROLLER

At the surface of the solution, ac -

==o

ax

(4)

a t x = 0. The solution of equation 1 with the boundary conditions imposed by equations 3 and 4 and the condition that c = c a when t = 0 is analogous to the solution of the equation for the radiating sphere in the theory of heat conduction (12). The concentration a t point x and time t is found to be m

c =

'

B, e - 5 ' m t

cg

Pm

cos - x

1

(5)

L

Dm is given by equation 6 and B,,, by equation 7. kL D

pmtan Bm = -

B,

=

4 sin &, sin 2Pm

2Pm

+

(7)

As a rule interest will be centered in the average concentration, E , throughout the solution. E is obtained by integrating equation 5 between 0 and L, and dividing by L. There is thus obtained

In equation 8 Om is given by equation 6 as before. defined by equation 9 as follows:

B,

is however now

Having determined E against t, Pm may be calculated by equations 8 and 9. Knowing pm, the chemical reaction velocity constant IC, which is the variable of interest, is then obtained from equation 6. 8 To simplify the calculation by equations 8, 9, and 6 of the chemical velocity constant k, table 1 has been constructed. This gives the values of the first two terms in equation 8, NUMERICAL SOLUTION OF EQUATION

and

(also PI and Pz), for values of kL/D from 0 to co and for Dt/L2 up to 1.6.


225

It is seen from table 1 that for Dt/L2sufficiently great, the series of equation 8 converges very rapidly. For Dt/L2 = 0, the second term is at the most 11 per cent of the first, while the sum of the second and all remaining terms is a t the most 22 per cent of the first. For Dt/L2 = 0.1, these figures become respectively 1.8 per cent and probably less than 3.6 per cent, while for Dt/L2 = 0.2, they become 0.4 per cent and probably less than 0.8 per cent, respectively. For an error smaller than 1 per cent, all terms but the first may therefore be neglected, provided Dt/L2 2 0.2. Table 1 is made use of as follows. At a given value of Dt/L2, one proceeds down the column to a value of

equal to the known value of E/Zo. The desired quantity kL/D is then obtained by reading across to the first column. For L one may substitute V / F , the ratio of volume of the vessel to surface area of the solid. I n view of the fact that is an average concentration, the same substitution appears possible when the vessel is non-prismatic. Knowing the diffusion coefficient and depth of the vessel, 7c is readily obtained from the known value of kL/D. RELATION TO THE USUAL FIRST-ORDER EQUATION

The course of most fluid-phase heterogeneous reactions, it has been found, is represented by the usual first-order reaction velocity equation. The equation is, for the case of reaction between a solid and solution,

On integration equation 10 gives

In equation 11 K is the measured reaction velocity constant, V the volume of the vesstl, and F the surface of the solid. Under the condition that all terms but the first may be neglected, that is for Dt/L2 2 0.2 for an error smaller than 1 per cent, equation 8 assumes the form of a first-order reaction velocity equation, and, with V / F substituted for L, may be written D

'

2.303 V

log Pi=--t F Bld

n

t

2 0.2

(12)

B,in equation 12, as shown in the fourth column of table 1, is close to unity, especially for those values of k L / D that are themselves less than unity (which values it will be seen are of most interest).

226

PAUL S. ROLLER

I

-

SI3

I

;,

I I g Y

~ddddodddddd .

u

d SI?

888ssp .0 s 0a0a0a0 0 0 0 0 0

dddddoooooo \

8 II u

S I3

-

d


227

With B1assumed nearly equal to unity and Dt/L2 5 0.2, the right-hand sides of equations 11 and 12 are substantially equal. Therefore

Equation 13 is exact, subject to the stipulated conditions, which are easily satisfied. In this equation K , the measured velocity constant, is given indirectly as a function of k , the true chemical rate constant, D, the diffusion coefficient, and L (or V / F ) ,the depth of the vessel. EXPLICIT SOLUTION FOR MEASURED VELOCITY CONSTANT

Pl2 in equation 13 is determined by equation 6. The latter is a transcendental expression which does not allow of explicit solution for @. In table 1 are given the indirectly solved values of PI, which are seen to extend over a narrow range, for k L / D from 0 to a0 . It will be highly desirable (especially for the purpose of later generalization) that PI2be expressed explicitly if possible, even though an approximation be introduced thereby. From equation 6 it is seen that P I 2 is a function of kL/D. I n any substitution for p12,it is necessary that PIzbe zero for ICLID zero, finite for ICL/D infinite, and that there be numerical agreement a t least over a certain range. It is proposed as meeting these conditions that PI2be set equal to (1 When the substitution is made, equation 13 becomes K

=

D/L (1 - e - k L l D )

(14)

Equation 14 is readily solved for the chemical rate constant IC, giving k

- 2.303 D/Llog (1 - K.L/D)

(14’1

According to equation 14, when k 3 D / L , the measured rate constant K is determined by D/L. On the other hand when k < D / L , K = k, or the measured rate constant is closely equal to the true chemical velocity constant. To test the validity of the functional substitution for PI2, table 2 has been constructed comparing k L / D calculated from equation 14’ with the correct value of kL/D obtained from table 1, for different assumed values of K L / D , i.e. /312 by equation 13. It is seen that kL/D varies over an infinite range, while the variation of K L / D is restricted to the narrow range 0.0 to 2.44. Agreement in the exact values of kL/D and values calculated from equation 14’ are within 10 per cent for K L / D less than 0.4, and within 35 per cent for K L / D less than 0.75. K L I D between 0 and

228

PAUL S. ROLLER

0.75 covers about one-third the total range in values of K L / D . However K L / D lying in the low range of values is most significant, because it is in the low range that k plays an important r6le in determining the magnitude of K. With increase in K L / D , the effect of k in determining the value of K becomes smaller and smaller compared to that of diffusion. Bearing this consideration in mind, the explicit approximation represented by equation 14 may be considered as fairly satisfactory. I n the end, the approximation represented by equation 14 is intended not so much for pure diffusion as a mode of transport as for other modes of transport for which the validity appears to be more extended, as will shortly be discussed. TABLE 2 ICLID, exact and as determined bv equation I.$', against K L / D kLID

KLID

0.00

0.23 0.43 0.74 1 .oo 2.44

Exact

Equation 14'

0.00 0.25 0.50 1 .oo 1.5

0.00 0.26 0.55 1.35 m

m

RATE O F SOLUTION AND CRYSTALLIZATION

Solution or crystallization differs from simple chemical reaction just considered in that molecules simultaneously condense a t and are liberated from the solid surface. The mole rate of condensation at the interface is given by equation 2. The mole rate of liberation of molecules from the surface is determined by the equilibrium condition that it equal the rate of condensation from the saturated solution, or kc,. The rate of solution or of crystallization a t the interface is the difference between the rate of condensation and of liberation, and is given by am 1/F - = k(c. at

- c)

(2')

at x 3: L. From this point the derivation is exactly analogous to that already considered (c, - c merely replacing c as variable), and the same final expression for K is obtained (equation 14).

RELATIONS I N FLUID PHASE HETEROGENEOUS REACTION

2%

GENERALIZED EQUATION FOR MEASURED VELOCITY CONSTANT

Transport of reactant or product of reaction will usually occur by modes independent of and in addition to diffusion. Concentration and convection currents are frequent natural accompaniments of the diffusion process. Artificial transport is secured by mechanical stirring, agitation, etc. These various modes of transport usually are much more effective than is diffussion itself. Unless they are taken into account, any relationship between the measured and the true chemical velocity constant can be only of very limited scope. It is observed that D / L in equation 14 has the dimensions of a velocity. It may be defined therefore as the rate of group molecule transport by molecular diffusion, i.e., as the coefficient of transport by diffusion. An obvious extension of equation 14 is to replace the diffusion coefficient of transport, D / L , as simply a special case, by a general coefficient of transport, S. Equation 14 for the measured rate constant then becomes K

= S(l

- e-kls)

(15)

B y analogy with equation 14 it may be expected that the limit of validity of equation 15 will be determined by the value of KIS. How close K / S may approach to unity before the equation palpably fails can not be told beforehand, and is to be answered only by experiment. It may be noted a t this point that with transport by mechanical stirring, the generalized equation held, within limit of error as seen below, practically for all values of K / S that were encountered. While an exact limit for K I S can not now be given, it appears that with transport by mechanical stirring the equation holds for K / S equal to 0.9 a t least, and probably also for greater values. TRANSPORT BY MECHANICAL STIRRING

In the measurement of liquid-phase heterogeneous reaction rate, the most common method of inducing molecule transport in the liquid is by mechanical stirring. The coefficient of transport obviously will increase with the rate of stirring. For fast reactions, which are determined by transport, Jablczynski (13) found that the rate varied approximately with the 035th power of the rate of stirring, while Van Name and Edgar (27) .found the power to be 0.80. For the intermediate rates of solution and reaction considered below, the transport likewise is found to be proportional to the 0 3 t h power of the rate of stirring. Therefore, transport by diffusion being neglected against that effected by the stirring, the coefficient of transport S is given by S

3

As0.8

(16)

230

PAUL S. ROLLER

where s is the rate of stirring in R.P.M., and A a constant which depends on experimental conditions and on the physical properties of the fluid, namely the viscosity and the density. The result for the exponent of s in equation 16 agrees with that found in heat transfer with fiuid flow in pipes, where the transfer coefficient, as summarized by McAdams (17), varies with the 0 3 t h power of the rate of flow of fluids4 It has been held (15, 16) that the rate of reaction varies with the first power of the rate of mechanical stirring. Although this result might obtain under exceptional stirring conditions, as in the combined gas-feed and mechanical stirring of Huber and Reid (11)) it is contrary to the generally recognized result of an exponent less than unity. The occasional apparent linear dependence probably arises from the combination of inaccurate, insufficient, or non-uniform data with the fact that a plot of y = x o . 8shows but small curvature. King and Braverman (16), for instance, find that the rate of solution of a zinc specimen rotating axially in acetic acid becomes linear above a certain speed and up to 5500 R.P.M. Nevertheless their plot for zinc and hydrochloric acid shows continuous curvature throughout, and incidentally, as far as can be estimated from the plotted points, the plot for hydrochloric acid appears to be fitted tolerably well by equation 17 below. If any linearity existed it should have applied to the faster reaction with hydrochloric acid, rather than to the reaction with acetic acid. It is to be concluded that the apparent linear result with acetic acid is due probably to the entrance of some secondary effect with increased high-speed stirring, such as heating, vibration, vortices, or some other disturbing factor not present under uniform conditions. Substituting, then, from equation 16 into equation 15, the following relationship connecting K and rE with the rate of mechanical stirring is obtained: R = As0.8(1 - e - k / a 8 0 . a ) (17) An apparent exponent of less than 0.8, in accordance with equation 17, is due to the progressive influence of the chemical reaction velocity on the measured rate constant. For instance, Brunner (2) considered that the rate of solution of benzoic acid in water was proportional to the two-thirds 4 The film theory has also been widely carried over to heterogeneous heat transfer. Recently Kennard (14) has examined the atmosphere adjacent t o a hot solid surface and found no evidence of any film, The case of heterogeneous heat transfer is analogous t o that of heterogeneous chemical reaction, and the results obtained here apply also to heat transfer, K being redefined as the measured coefficient of heat transfer and k as the velocity constant of heat transfer at the interface. I n view of an exponent of 0.8 for the fluid velocity, IC would appear t o be large against the coefficient of physical transport.


231

power of the rate of stirring. However, it will be seen below that the exponent for Brunner’s data is 0.8 as usual, but the rate of solution is slow enough to affect measurably the value of K and thus to result in an apparent exponent of less than 0.8. APPLICATION

In heterogeneous as in homogeneous chemical reaction one is interested in the value of the chemical reaction veloci’ty constant, k. Separation of the physical and chemical factors that determine the measured constant K is secured in general by equation 15. When the transport is effected by mechanical stirring, equation 17 applies. To determine I%, it merely is necessary to evaluate A by measurement of K a t different stirring speeds. For the calculated value of k to be accurate, its contribution to the value of the measured constant K must be sufficiently great. I n other words, K must be measured a t sufficiently high stirring speeds. A criterion as to whet,her one is in a range of stirring speeds in which k exerts an appreciable influence on the magnitude of K is found in the ratio K/SO.~. If this ratio decreases sufficiently with increase in stirring speed, the range of speeds is then proper to an accurate determination of k. EXAMPLES

The oxidation of mercury by iodine in potassium iodide solution, measured by Van Name and Edgar (27) at different stirring speeds up to 300 R.P.M., is a reaction in which the decrease in the ratio Kobed./So’S is so small that accurate determination of the true reaction constant k is not possible. The data for this reaction are given in table 3.6 It is seen from column 3 that the decrease in the ratio with decrease in stirring speed is less than 2 per cent over the whole range of stirring speeds from 170 to 300 R.P.M. However, a large proportion of the reactions that have been reported in the literature show a sufficiently rapid decrease of K o b s d . / S O . * with increase in stirring speed to permit application of equation 17. Reactions for which accurate data are’available are the following: (a) Benzoic acid in water (30), (b) oxygen in water (4), (c) magnesium in 1/8 M hydrochloric acid ( 6 ) , (d) thallium in 1/2 M nitric acid @), (e) zinc in 0.02 M copper sulfate (7). These typical reactions are considered in tables 4 to 8 inclusive. In column 2 of these tables is shown the decrease in K o b a d . / S o J with increase in s. In the third and fourth columns are shown K o b s d . and the value of K calculated by equation 17 for assigned values of k and A . 6 I n all examples, the author’s value of the measured rate is recalculated where necessary so as t o conform t o the precise equation (II), or t o its analogue, for K o b d . . K and k are expressed in centimeters per minute.

232

PAUL S. ROLLER

For benzoic acid in water (table 4),the same value of IC satisfies excellently the results of three different observers, though naturally the values of A differ for the different experimental conditions. The fifth column shows that there is no systematic deviation in the percentage difference between the calculated and observed values of K , the differences being positive and negative at random. TABLE 3 Rate of oxidation of mercury by iodine in potassium iodide (87) 8

IN R . P . I .

Kobsd.

170 210 240 300

0.574 0.679 0.759 0.885

9.45 9.45 9.45 9.30

TABLE 4 Rate of solution of benzoic acid in water k = 0.157 8

IN R.P.M.

&bad.

x lo3

Kobad.

A 3 53 122 204 365 470

2.18 1.13 1.10 1.34

1 .oo A

PEIRCENTAQE DIE'P'ERENCE

0.0035 0.0348 0.0618 0.0810 0.1015 0.1090

+32 -29 -20 $14 $9 -6

= 0.00147(30)

0.0052 0.0270 0.0513 0.0943 0.1120 0.1030

0.75

Koalcd.

= 0.0011(1)

450

0.046 0,088

0.042 0.096

202

0.118, 0.131

I ::;:: 1

-2.5 +1.5

The least accurate data are those for benzoic acid; this is reflected in the irregular decrease in K / s 0 * for * this reaction (table 4), contrary to that for the other reactions. The percentage difference averaged for all the experimental points for benzoic acid is f 1 3 per cent. The most accurate measurements appear to be those on the solution of zinc in copper sulfate (table 8). The average percentage difference is only h 1 . 2 per cent. For all the reactions the average difference is 2 ~ 6 . 2per cent.

233


Table 9 summarizes the values of k and A for the reactions considered. It is seen that k varies between 0.078 and 2.80, or over a thirty-five-fold range. The variation of A , from 1.1to 12.0 X is more restricted.

TABLE 5 Rate of solution of oxygen in water (4) k = 2.06; A = 0.00177 8

Kobsd.

I N R.P.M.

loJ

1.82 1.64 1.63 1.55 1.54

60 93 120 145 180

Kobsd.

I

0.0480 0.0621 0.0749 0.0835 0.0977

Kcalcd.

PERCENTAQE DIFFERENCE

0.0462 0.0639 0.0749

+3.7 -2.9 0 -1.1 +3.6

0.0844 0.0942

TABLE 6 Rate of solution of magnesium in 1/8 M hydrochloric acid (6) k = 2.80: A = 0.010

Kobsd.

a IN R.P.M.

8C.8

15.7 10.5 9.7 7.9 7.2 7.7

50

100 200 400

750 855

IINR.P.M.

32 74 120 193 454

I

0.36 0.41 0.67 0.95 1.43 1.72

0.23 0.40

0.68 1.08 1.50 1.59

4-33 +3 -1 14 -5 +8

-

Rate of solution of thallium in I/$ M nitric acid ( 8 ) k = 0.078; A = 0.0044 . *K 8c.8

x 103

2.96 1.84 1.42 1 .oo

0.55

Kobsd.

0.0474 0.0578 0.0654 0.0678 0.0733

I

Koalcd.

0.0472 0.0596 0.0649 0.0688 0.0731

PERCENTAQID DIPPERENCE

+0.4 -3.1

+0.8 -1.5 +0.3

THE FORM O F THE CURVE

I n figure 1, KCalcd. is plotted against R.P.M. for reactions a to e. The experimental points fall irregularly on either side of the curves. The curves all are concave downward. The degree of curvature shows con-

234

PAUL S. ROLLER

siderable variation. For reaction b it is slight, the curve appearing almost as a straight line; for reaction d it is very pronounced up to 150 R.P.M., after which the curve becomes asymptotic. For the other reactions the curvature is intermediate. I n the same range of stirring speeds, the form of the curve is determined by the ratio k / A shown in table 9. For reaction d the ratio is only 0.02 X lo8, and the curvature is high; for reaction b the ratio is 1.16 X lo3, TABLE 8 Rate of solution of zinc in 0.02 M copper sulfate (7) k = 1.50:A = 0.012 8 IN

R.P.M.

54 94 295 500 600 680

Koalcd.

13.8 11.5 8.8 7.0 6.4 5.6

0.336 0.436 0.836 1.007 1.061 1.028

’

0.292 0.440 0,834 1.005 1.055 1.088

PERCENTAGE DIFFERENCE

$13.0 -0.9 +0.2 $0.2 +0.6

-5.8

TABLE 9 Summary of values of chemical reaction velocity constant, k , and of A k

REACTION

A X 101

b

x lo-’

cm.,per

man,

(a) Benzoic acid in water (30)... . . . . . . . . . . . . . . . . . . . . Benzoic acid in water (1). , . . . . . . . . . . . . . . . . . . . . . . Benzoic acid in water (2).. . . . . . . . . . . . . . . . . . . . . . . (b) Oxygen in water (4).. . . . . . . . . . . . . . . . . . . . . . . . . . . . (c) Magnesium in 1/8 M hydrochloric acid (6).. . . . . , (d) Thallium in 1/2 M nitric acid (8).. . . . . . . . . . . . . . . (e) Zinc in 0.02 M copper sulfate (7). . . . . . . . . . . . . . . .

0.157 0.157 0.157 2.06 2.80 0.078 1.50

1.47 1.10 5.50 1.77 10.0 4.4 12.0

0.11 0.14 0.03 1.16 0.28 0.02 0.12

and the curvature is very small. Reactions a and e have about the same intermediate ratio and curvature. TEMPERATURE COEFFICIENT

By equation 17 the temperature coefficient of K is determined by that of k and of A . A varies with temperature chiefly through its dependence on the viscosity. By analogy with the heat transfer coefficient (based simply on the Reynold’s number and considered as a coefficient of transport), A should be equal to A’4°.8,where 4 is the fluidity. This result accords with the measurements of the temperature coefficient of zinc in 0.02 M copper


235

sulfate a t 100 R.P.M. (reaction e) by Centnerszwer and Heller (7). These = 1.97 and q5%i/q5:.8 = 1.72; authors find &/KO = 1.83, while also &0/K26 = 1.47, while q 5 ~ ) / q 5 2 6 = 1.64 and 4::/q5:; = 1.48. On the other hand, the results of King and Braverman (16) and Van Name and Hill (28) for rapid reactions in which alcohol and cane sugar have been added to the solutions, indicate that the measured constant

,

varies as the first power of the fluidity. If these measurements are accurate, and if a possible effect of added reagent on the chemical velocity constant is neglected, then A = A'$. In any case Icr

+ io/kt

> qy:,(F

')/q;"

(Or".

236

PAUL S. ROLLER

Applying this inequality to equation 17, and remembering that in the limit K = k , the following relation is found to hold kt + io/kr

> Kt + io/Kg > q ~ ' ~ " " / q ~ ' * (Or')

( 18)

The inequality 18 states that the measured temperature coefficient will lie between the coefficient of viscosity and of true chemical reaction. This is the result that has been found generally, with leanings towardviscosity coefficient or chemical coefficient depending on the ratio I~/AsO.~. SUMMARY

.

1. The postulates of the Nernst theory of fluid-phase heterogeneous reaction are criticized. In particular, independent experimental evidence is brought forward against the assumed existence of a stationary film. 2. In this paper, the measured rate of fluid-phase heterogeneous reaction is considered to be simply the resultant of two simultaneous processes -of chemical reaction a t the interface and of physical transport to or from the interface-with no arbitrary assumptions as to mechanism. 3. To determine the functional relationship between the chemical and physical factors, the ordinary diffusion equation has been solved with one of the boundary conditions that of first-order reaction a t the interface. The solution permits accurate calculation of the true chemical rate constant, knowing the value of the diffusion coefficient and the dimensions of the vessel. The solved equation is brought into relation'to the first-order rate equation that is usually found to apply. An implicit, and finally an explicit equation, connecting the usual measured rate constant, true chemical velocity constant, and diffusion coefficient are deduced, and the range of validity indicated. 4. By regarding diffusion as a special mode of molecule transport, the explicit equation is generalized to the following expression for the measured rate constant K , K = S(l

-

e+/B)

where k is the chemical reaction velocity constant, and S the coefficient of physical transport of reactant or product of reaction to or from the interface. 5 . For transport by mechanical stirring of the liquid phase, it is found that S = AsOJ3,where A is a constant and s is the stirring speed. Exponent 0.8 agrees with that commonly found for the fluid velocity in the expression for the coefficient of heat transfer. 6. The values of K , measured as a function of the stirring speed for five typical reactions, are compared with the calculated values, and excellent


237

agreement obtained. IC varies thirty-five-fold between 0.078 and 2.80 for the different reactions, while A lies between 1.1 and 12 X 7. The form of the curve of K against R.P.M. is considered in terms of the ratio k / A . 8. It is shown that the temperature coefficient of K must lie between the chemical coefficient and the viscosity coefficient, depending on the ratio k/Aso.8. REFERENCES

(1) BRUNER AND ST. TOLLOCZKO: Z. physik. Chem. 36,283 (1900). (2) BRUNNER:Z. physik. Chem. 47,56 (1904). (3) BULKLEY:Bur. Standards J. Research 6,89 (1931). J. chim. phys. 9,228 (1911). (4) CARLSEN: Z.physik. Chem. 137,352 (1928). (5) CENTNERSZWER: CENTNERSZWER AND ZABLOCKI: ibid. 122,455(1926). JABLCZYNSKI, HERMANOWICZ, AND WAJCHSELFISZ: Z. anorg. allgem. Chem. 180, 184 (1929). (6) CENTNERSZWER: Rec. trav. chim. 42,579 (1923). (7) CENTNERSZWER AND HELLER: Z. physik. Chem. 161A,113 (1932). (8)CENTNERSZWER AND LEWI: Z. Elektrochem. 37,603 (1931). J. Am. Chem. SOC.62, 3757 (1930). (9) DAVISAND CRANDALL: (IO) FAGE AND TOWNEND: Proc. Roy. SOC.London 136A,656 (1932). (11) HUBER AND REID: J. Ind. Eng. Chem. 18,535 (1926). (12) INGERSOLL AND ZOBEL:Mathematical Theory of Heat Conduction. Ginn and Co., New York (1913). (13) JABLCZYNSKI: Z. physik. Chem. 64,748(1908). (14) KENNARD: Bur. Standards J. Research 8,787(1932). (15) KLEIN: Z. anorg. allgem. Chem. 137,56 (1924). (16) KINGAND BRAVERMAN: J. Am. Chem. SOC.64,1744(1932). Heat Transmission. McGraw-Hill Book Co., New York (1933). (17) MCADAMS: (18) MARC:Z. physik. Chem. 67,470 (1909). (19) MARC:Z. physik. Chem. 61,385 (1908). KAYA,AND NAKATA:Bull. Chem. SOC.Japan 6, 133 (1931). (20) MIYAMOTO, (21) NERNST:Z. physik. Chem. 47,52(1904). (22) RINNE:Centr. Mineral. Geol. 1904,116. KORBS: Z. Krist. 43, 433 (1907). GROSS:Z. Krist. 67, 145 (1922). GLAUNER: Z. physik. Chem. 142,67 (1929). (23) ROLLER:J. Phys. Chem. 36, 1202 (1932). E Ranorg. : allgem. Chem. 87, 319 (1914). (24) R O S E N K R ~ N ZZ. (25) SCHMIDT AND DURAU:Z. physik. Chem. 108, 128 (1923). WOLF: Z. angew Chem. 36,138 (1922). (26) SPANGENBERG: Z. Krist. 69,383 (1923). Am. J. Sci. 29,237 (1910). (27) VANNAMEAND EDGAR: (28) VANNAMEA N D HILL: Am. J. Sci. 36,543 (1913). (29) WAGNER:Z. physik. Chem. 71,401 (1910). (30) WILDERMANN: Z.physik. Chem. 66,445 (1909).

THE PHYSICAL AND CHEMICAL RELATIONS IN FLUID PHASE

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