Mar., 1955
SODIUM DIFFUSIONFLAME METHODFOR FASTREACTIONS
261
THE SODIUM DIFFUSION FLAME METHOD FOR FAST REACTIONS. I. THEORY O F THE EXPERIMENTAL METHOD’s2 B Y JOHN F. REED3 AND B. 8. RABINOVITCH Contribution f r o m the Department of Chemistry of the University of Washington, Seattle, Washington Received October 1 , 1064
A critical analysis of the diffusion flame method for sodium atom reactions has been made. When the streaming velocity is considered in the differential equation of continuity and the flow of sodium into the system used as a boundary condition, flame shapes are distorted from spherical and calculated rate constants lowered from those calculated in the conventional manner. Consideration of halide back diffusion leads to an interpretation of the lower v / D limit which is determined by experimental variables and the reaction rate constant. Heller’s lower limit is found to be a special case. Halide distribution in the flame is briefly discussed.
The reaction between metal vapor, usually so- lowing paper.6 The general equation of continuity dium, and alkyl halides in the gas phase has been for any species in an isothermal system is given by studied by numerous workers4 and demonstrated V ( D ~ V P~ )v ( p i v ) = api/at (1) to be essentially the removal of the halogen atom where D i is the ordinary diffusion coefficient of the from the alkyl halide by the sodium atom in a bimolecular step. The reaction occurs with an ac- ith species in the gas mixture, p i is the pressure of tivation energy generally less than 10 kcal./mole the ith species, Qi is the rate of production of the ith and flow systems have been used to study the ki- species in an element of volume and V is the mass averaged velocity of the gas mixture at the point, netics of the reactions. The system used is frequently patt,erned after the i.e. v = yivi diffusion flame method so brilliantly pioneered by i Hartel and P01anyi.~ The usual measurements made on the system when it has attained the steady where V i is the velocity of the ith component, and Ti state are the halide pressure, the total pressure, the is the weight fraction of the ith component. temperature of the reaction zone, the diameter of In the steady state, bp/bt = 0. For both sothe flame and the pressure of sodium as it is fed into dium and halide in the flame, Qi is given by - kpp‘, the system. This latter quantity is identified with where p and p’ are the respective pressures and k the vapor pressure of sodium at the temperature of is the specific rate constant. Since the pressure in the carburetor from which the sodium is swept by the system is constant, the diffusion coefficient is carrier gas. To relate the measurements to the taken to be constant in space. The steady-state specific rate constant, the hydrodynamic equa- differential equation for sodium transport is thus tion of continuity has been used with approxima- given by tion~.~ The original treatment of Hartel and PolDv2p - v ( p V ) - kpp’ = 0 (3) anyi was semi-quantitative. Later, in an impor- A similar equation may be written for alkyl halide. tant extension, Heller6 evaluated experimentally a The solution of these equations for various approxinumber of aspects of the flow and placed some re- mations and applications will now be considered. strictions on the allowed range of variation of some Halide Back DBusion. No Wall Reaction.-An of the experimental parameters. important aspect of the flow of sodium is the A number of aspects of the above treatments problem of alkyl halide back diffusion into the nozwarrant further examination. It also appears de- zle from which the sodium streams. As a model consirable to attempt a solution of a more accurate rep- sider that sodium streams down a uniform nozzle resentation of the equation of continuity applied to tube in a gas mixture traveling with constant velocthe transport of sodium and halide. In the present ity from the source (carburetor), a t z = 03, to the paper, a number of features of the experimental nozzle a t z = 0. The velocity is given as V = method are investigated. The main conclusions -vk where v is a constant scalar and k is the unit apply to reactions between other materials as well. vector along the positive z-axis. The cross-section The General Equation of Continuity.-In the of the tube is taken constant and equal to w-8 where flame, sodium is depleted chemically by a bimolecu- ro is the radius. The steady-state equation of conlar step in which sodium halide is formed and a free tinuity for alkyl halide is radical liberated. Processes wherein sodium is depleted by additional reactions, e.g., with halogenated free radicals, are discussed later and in a folAs a useful simplification, the pressure of sodium (1) Presented before the Division of Inorganic and Physical Chemisin the tube is considered constant and equal try of the American Chemical Society a t the Los Angeles meeting, to the pressure in the carburetor, p”, and wall reMarch, 1953. action is neglected. The latter problem is discussed (2) Abstracted in part from a thesis submitted by John F. Reed in partial fulfillment of the requirements for the degree of Doctor of below. An analytic solution for the halide presPhilosophy a t the University of Washington. sure as a function of z may be found (App. I) (3) Department of Chemistry, Loyola University, Chicago, Illinois.
+
(4) H. V. Hartel and M. Polanyi, 2. phyaik. Chem., B11, 97 (1930); reviews have been given by M. Polanyi, “Atomic Reactions,” Williams and Norgate. London, 1932; C. E.H. Bawn, Chem. SOC.Ann. Reps., 89, 36 (1942); E. Warhurst, Quart. Reus., 6, 44 (1951). (5) W. Heller, Trans. Faraday Soc., 83, 1556 (1937).
p’ = p~’exp[-(a
+ b ) z / 2 ] ; a = u/D’;
b2 = a2
+ 4kp*/D’
(5)
(6) J. F. Reed and B. 8. Rabinovitch, THIEJOURNAL, t o be submitted.
JOHN F. REEDAND B. S. RABINOVITCH
262
where PO’is the pressure of halide a t z = 0. Since all the halide that enters the tube is consumed, the net flow of halide into the tube is found by integrating the amount, of reaction over the entire volume of the tube. Use of the sodium carburetor pressure gives the maximum flow of halide into the tube. From the net flow of sodium out of the tube, the effective pressure of sodium as it leaves the nozzle, pelis then given by (App. I). pe = (1
- 2kpo’/tJ(a + b ) ) p *
(6)
This is an upper limit to depletion since sodium pressure was set constant at the largest value, p ” . Several conclusions may be drawn from equation 6. In line with qualitative reasoning, it is evident that the sodium pressure is depleted less for slow reactions, low halide pressure a t the nozzle, low diffusion coefficient of halide into the gas mixture and high streaming velocity. I n particular, it is seen that a fixed value of the v/D‘ (or v / D ) ratio does not suffice to specify the depletion of sodium due to back diffusion of alkyl halide. Not only does one need to know the collision yield (as k ) , but also the halide pressure a t the nozzle and, further, the actual streaming velocity. These questions arise because the conclusions of Heller6 have frequently been assumed to imply that only the v / D ratio (for reactions with collision yields of the order of 50-5000) is needed to effectively evaluate back diffusion. Heller had selected a lower allowable limit of v / D of 5.5 cm. -l on the basis that it is a t this value that the halide pressure in front of the nozzle, po’ is reduced by streaming t o about 10% of the halide pressure outside the flame; for some experimental conditions, this would make the halide and sodium pressures about equal a t the nozzle. However, even under such circumstances, back diffusion and reaction could be sufficient to deplete the sodium pressure significantly. It is evident that a fixed lower v / D limit is too arbitrary. The finding here is that a number of additional factors enter explicitly into the determination of sodium depletion due to back diffusion. Application of the above equation to some actual data6 indicates that with an arbitrary allowable depletion of sodium of 2% (Le., p e / p * = 0.98) a v / D ratio of as low as 1.0 em.-‘ could be used in some cases. By varying the parameters of a run, such as the total pressure or the streaming velocity of gas out of the nozzle, various suitable lower limiting values may be found. I n particular, the above equation can be used to evaluate the net pressure of sodium as it is fed by the nozzle when the flow parameters and reactivity are known for any run. The upper limit for v / D given by Heller is 12 cm.-l and corresponds to an arbitrary, practical limitation on the allowable depletion of halide in the flame just outside the nozzle due to streaming from the nozzle; Heller also suggests that turbulence in the flame becomes appreciable a t higher values. In a later section a distribution function for halide in the flame, for the case v # 0, is given (equation 15). This is based on the approximation of a constant velocity averaged through the flame and has only qualitative validity. If a precise distribution function for halide in the flame
VOl. 59
were known then, for this factor, values of v / D considerably in excess of Heller’s upper limit would obviously become acceptable. I n the absence of such a function, the errors are limited by restraining the upper limit of v / D as much as is experimentally feasible. Halide Back Diffusion. Wall Reaction.-The extent of wall reaction in the nozzle tube affects the sodium pressure as it issues from the nozzle. If wall reaction is insignificant, then the above calculations on the net sodium pressure a t the nozzle are unaffected. The sodium nozzle pressure is less if wall reaction takes place. Some conclusions may be obtained as follows. The steady-state equation for halide in the nozzle tube is B’v2p’
+
aP‘
- kpp’
=
0
(7)
If a gradient in the direction perpendicular to the wall exists due to wall reaction, then a solution to equation 7 may be found in terms of the coordinates x and p . A satisfactory solution (App. 11) is given by p‘ = B‘ exp[ -(A
+ v/2D)z]Jo(np)
(8)
where B’ is a constant of integration, X2 = (v/ 20’)2 n2,n is a separation parameter of the differential equation and Jo(qp) is a zero-th order Bessel function. On the assumption that reaction at the wall is equal to the net amount of halide diffusing to the wall in the steady state, the ratio of total amount of gas phase reaction to wall reaction, w, reduces t o
+
w = kp*/D’n2
(9)
The magnitude of n depends on the boundary value of the halide pressure a t the wall relative to that a t the center of the tube. If this ratio is designated E, (1 8 0), n is determined by the first root of
> >
Jo(nro) =
E
(10)
where rois the radius of the tube. Heller has implied that wall reaction is rapid. Unfortunately, there is no clear information on the magnitude of wall reaction, whether indeed i t i s digerent from zero, so it is impossible ‘to specify 0, then for some reasonable the value of a. If E values for the parameters of equation 9 wall reaction would be approximately 2000 times that of the gas phase. Such an immense amount of reaction would reduce the sodium pressure tQa negligible value and essentially no sodium would issue from the nozzle. It would appear therefore that wall reaction does not occur on every collision. For 6 = 0.9, wall reaction could still be as much as 80 times the gas phase reaction, which would reduce the sodium flow out of the nozzle to a relatively small value. Polanyi and Harte14 suggest that heterogeneous reaction occurs with alkyl fluoride but this seems unsubstantiated. If wall reaction in the nozzle tube, which must take place on an alkali halide coated surface, were significant with respect to gas phase depletion, then the entire treatment of sodium flames might be in error; inasmuch as there may be some nucleation of sodium halide in the flame, the presence of which would provide surface for heterogeneous reaction.
-
SODIUM DIFFUSION FLAME METI-IOD FOR FASTREACTIONS
Mar., 1955
The tacit assumption seems always to have been made in the literature that either heterogeneous reaction is negligible, or else nucleation does not take place in the flame. Explicit experimental evidence on the magnitude of heterogeneous reaction is highly desirable. Halide Distribution in the Flame.-Cvetanovic and Le ROY,’ in an interesting extension, have examined the problem of halide distribution in the flame on the basis of diffusion as the sole transport process for both sodium and halide in the flame. The pressure of halide was obtained as an analytic function of the sodium pressure and the radial coordinates, r. This halide pressure function was not suitable for an analytic solution of the sodium steady-state aquation. However, it was possible to determine the effect of using such a variation of halide pressure in space on the value of a computed rate constant. An undetermined parameter of their result was the pressure of halide a t the nozzle, i.e., a t r = ro; in their computation it was necessary to assume a value. This pressure was taken by them to be zero, although, as they point out, other boundary values could be used. Utilizing the pressure distribution function given by Cvetanovic and LeRoy, the pressure of halide a t the nozzle may be related to the sodium flow into the system.8 The calculation shows that under the usual experimental conditions the halide pressure is not significantly reduced a t the nozzle if the one t>ransport is by diffusion. The actual halide pressure a t the nozzle as shown experimentally by Heller is of course appreciably reduced, but due to streaming, a process which is usually neglected in the transport equation. It thus appears desirable to consider this type of transport since under the conventional conditions of large excess of halide over sodium this is the process which results in halide depletion in the flame. For the case where the amount of sodium in the flame is much smaller than halide, reaction may be neglected. Then for the halide steady state D‘v2pf - V(p’V) = 0
(13)
Introducing as a simplification for V a constant average velocity in the flame, V = ak, the above becomes D’v’P’
- Vdp/dz
=
0
(14)
A solution to this equation possessing cylindrical symmetry and satisfying the boundary conditions p’( m ) = p’*, p’(ro) = po’ < p’* is = p f * - (p’* - pol)? exp [ ( z - r
+ ro)or];
where
01
= V/D’
(15)
This function is compatible with a pressure of halide a t r = ro that is significantly reduced, even to zero. This pressure function, however, when averaged over the flame gives a value only slightly less than p’*. The pressure fall with distance is significant only in ,the region of the nozzle. Use of this function in the sodium equation of continuity (7) R. J. Cvetanovic and D. J. Le Roy, Can. J . Chem., 29, 597 (1951). (8) For furtlier details, see J. F. Reed, P1i.D. Thesis, University of Washington, 1953; available on microfilm, University Microfilms, Ann Arbor, Michigan.
263
would consequently not change the results much since the halide pressure is relatively constant, a t p‘*, throughout the flame. Hence the halide distribution obtained in this approximation agrees only qualitatively with the experimental finding of Heller. Failing a solution to the equation incorporating a space dependent function of velocity it is felt that a reasonable practical approach to the nature of halide distribution is acceptance of a correction factor based on Heller’s data. His data, however, appear to be unique for the conditions used. The nozzle size (or cross-section of the flux out of the nozzle) in particular does not enter expressly into his suggested correction. By a dimensional argument, this can be taken into account*; this analysis yields
where f is the correction factor to be applied to yield the average pressure of halide in the flame, rH the radius of Heller’s nozzle, r0 the radius of the nozzle of any other system, a n d f ~the correction factor deduced by Heller. The values of f~ are found in Heller’s work as a function of flame size and of
VOlD.
Very recently, a calculation of the effect of atmosphere reactant distribution in the flame on rate constants has been developed by Smithgin which a diffusion model is used and an analytical solution obtained. Sodium Distribution in the Flame. Pressure at the Nozzle.-Polanyi and Hartel4 and following workers have assumed in their treatment of the sodium diffusion flame that the effect of streaming of gas out of the nozzle is negligible for the transport of sodium in the flame. To this approximation, the steady-state equation of continuity for sodium is Dv2p
- kpp’
=0
(17)
This equation was solved by using a spherical Laplacian in the radial coordinate, r, and by setting the pressure of halide, p’ (present in large excess), constant in the flame. The solution subject to the boundary condition p ( a )= 0 is p = A exp( - C r ) / r ; C2 = k p ’ / D
(18)
From the second boundary condition, p(ro) = pa, there is obtained p =
poro
7exp[ -C(r
-
TO)]
(19)
Polanyi and Hartel used the nozzle radius for ra and set po equal to the carburetor pressure of sodium, p*. Some question arises with regard to this latter assumption inasmuch as the locus of points for which r = ra corresponds not to the nozzle but rather to a sphere of radius ro, and the pressure of sodium on this hypothetical sphere may correspond to some value other than p * . A different method of evaluating the constant A may be used as suggested by the work of Garvin and Kistiakowsky.‘O Since all of the sodium is consumed in the flame, the total amount of sodium reaction per second in the steady state is set equal (9) F. T. Smith, J . Chem. Phys., 22, 1005 (1954). (10) D. Garvin aqd G. B. Kistiakowsky, ibid., 20, 105 (1952).
JOHN F. REEDAND B. S. RABINOVITCH
264
to the flow rate of sodium into the system in moles per second, bo. When this is done the constant A is found to have the value A =
bo RT exp (Cro) 4 r D (1 (30)
+
(20)
VOl. 59
where v is a constant scalar, and k is the unit vector along the flow axis designated as the z-axis. The equation of continuity for sodium now becomes DV’P
- vdp/dz - kpp‘
=0
(24)
A solution for p , satisfying the boundary condition
Further, the quantity to be taken for po may be that a t infinity the pressure of sodium is reduced to evaluated from the relation A = pore exp(Cro) and is zero, is“ PO = rovop*/4D(1 Cro) (21 1 A’ where vo is the average streaming velocity of the p = exp(az - p r ) ; a = v/2D; p 2 = a2 + kp’/D carrier gas a t the nozzle. (25) I n general, Cro < 1. For a nozzle with ro = 0.1 where r is the radial coordinate (App. 111). The cm. and with vo/D = 10 cm. -l, somewhat representA’ is evaluated as before by setting the constant ative of values normally used, it is seen that pa is approximately 0.25 p”. The evaluation of con- flow rate of sodium into the system equal to the tostant A by this method obviates the need to ex- tal amount of reaction per second. Thus plicitly associate any particular pressure with the points r = ro. From the form of this last equation, it may appear that po would exceed p* a t high val- The functional dependence of p on the coordinates ues of vo. Of course this cannot occur and when is identical with the previous case except for a facvelocities of this magnitude are encountered, it is tor exp(az), the effect of which is essentially to reapparent that the original differential equation of move some of the sodium from behind the nozzle continuity must allow for transport by streaming. and place it in front, which is the effect that If the pressure at the edge of the flame is the lim- streaming actually has on the symmetrical distriiting detectible pressure, pl, a t r = R, the specific bution. The solution gives the pressure contour an rate constant is obtained as oval shape with an apparent center of the oval displaced along the flow axis. When the velocity parameter, v, is reduced to zero, the solution for p For purposes of comparison, the rate constant cal- reduces to that obtained previously on the assumpculated by use of equation 22 is designated k@) tion that the velocity was negligible. I n order to put the pressure dependence in a form when po is set equal to p’, as done by Polanyi and and Hartel, and k 1 when po is taken from equation analogous to that used by Polanyi and Hartel, a quantity ?ro is defined corresponding to the points 21. Sodium Distribution in the Flame. The Case for which T = ro,x = 0. Such points lie on a ring of Constant Streaming Velocity.-Measurement of at the nozzle. Then the flame diameter has conventionally been made “head-on” to the flame, along the flow axis. Observations of the flame perpendicular to the flow (27) axis however show that the flame is not actually The flame “height,” P, is a measured quantity, corspherical, but rather slightly oval, with the major axis responding to the maximum distance from the xalong the flow axisandwith theapparent center of the axis to the edge of the flame along the p l contour. flame displaced along the flow axis. Since the dif- This measurement is made a t distance 2 along the fusion velocity of sodium a t the nozzle and the z-axis in front of the nozzle. The distance from the linear streaming velocity of the issuing gas mixture nozzle to the flame edge a t the point in question is a t the nozzle may be shown explicitly to be of the R. Thus same order of magnitude under the usual condiR2 = P2 + Z a (28) tions,* it is not surprising that some effect occurs due to streaming. Streaming is taken into account Further explicitly in the equation of continuity by a term (ap/az)R,Z 0 (29) . v ( p V ) Unfortunately, the nature of the variation Using this condition on the pressure function given of V with the radius vector in the flame for a nozzle source is not known, and even for some assumed by equation 27, there results 2 = aR2/(1 pr) simple ideal functional forms, e.g., r-2, the differen(30) tial equation of continuity appears analytically The vaIue of the specific rate constant in terms of R insoluble. Perhaps the simplest manner of introduc- and Z is designated W2)and is given by ing a term to account for the effect of streaming is by using a constant velocity in the flame. This constant velocity is to be interpreted as an average over the entire flame, and to arise from the streamEquations 28, 30 and 31 give three relations ing of gas out the nozzle. This velocity should be among the variables /c,(~)v (contained in a),P, R and directed along the flow axis corresponding to origi- 2; all other quantities will be known in a given nation of the momentum flux in the nozzle. To de- experiment. Now since P is measured, only one termine the effect of including such a term in the additional measurement or quantity is needed to equation of continuity, the velocity is first included find the value of the specific rate constant. If the as a parameter with an unspecified value, Le., set *(11) H. A. Wilson, Phil. Mag.. [ti] 24, 118 (1912), has treated a similar equation but with no reaction term. V = vk (23)
+
~
I
+
1
SODIUM DIFFUSION FLAME METHOD FOR FASTREACTIONS
Mar., 1955
parameter Y is specified by means of some averaging , process,* then the rate constant can be calculated from the usual measurements made on the system, i.e., from P. However, if instead an additional measurement is made, namely, 2, then 2) is determined. Recourse may thus be had to the experiment for the evaluation of D. Thus k is calculable from the data of the experiment alone. It is rather obvious that the parameter v is dependent on the linear streaming velocity of gas out the nozzle, vo; the higher the value of vo, the greater will be the value of Z and consequently of v. Comparison of the Various Computed Rate Constants.-The various computed rate constants have the relation k(0) > k(1) >
k(2)
(32)
The difference between the last two is small, but the difference between the first two is significant, IC(') being generally of the order of one-half to onetenth k(0). The treatment by which is derived gives an oval rather than spherical shape to the flame. This last treatment corrects in principle an outstanding discrepancy of the treatment of sodium atom reaction data. It may be noted that a n unreasonable result that follows from the conventional derivation of k(0) is that the size of the flame and total amount of reaction is independent of the amount of sodium fed to the flame. Calculation of po in equation 21 and r0in equation 27 introduces the flow parameter vo into the expressions for k(') and ld2). Activation energies computed in the conventional manner from rate constants a t a single temperature vary by as much as 2.5 kcal. depending on the extent of approximation in the treatment of the equation of continuity. In general, under usual flow conditions, the activation energy computed from k(1) and Id2)is about 1 kcal. higher than that computed from k(0). I n making comparisons of rate data, the differences tend to disappear. The absolute magnitude of sodium atom-alkyl halide rate constants in the literature should be corrected in the direction of ,decreased reactivity, and the conventional activation energies raised correspondingly. Some Other Factors.-One factor which has not been treated concerns the observation of the flame edge. It has been assumed that the edge of the flame can be distinguished a t a previously determined limiting visible pressure of sodium. However, observations are generally made through a finite thickness of emitting sodium and it is the integrated intensity over the observation path length that is detected by the eye. This can simply be accounted for by assuming that emission intensity is proportional to sodium concentration and, by integrating the point intensities over the path length through which observation is made, the actual intensities can be found. This should be done for both the limiting visibility determination and for the length through the flame edge. Correcting the observations in this way would normally correct diffusion flame data so as to make the rate constant higher than conventionally reported. A rough calculation on this basis indicates that activation energies would be lowered by a few tenths of a kilocalo-
265
rie when the limiting pressure calibration measurement is made through a thickness of about 6 mm. This correction disappears when differences in activation energies are calculated or compared. A correction neglected in computing a rate constant for sodium reactions is concerned with depletion of sodium by halogenated species, other than the parent molecule, that appear in the flame, e.g., halogen-containing free radicals. In the simple case where the radical contains n halogen atoms which are all removed by sodium rapidly, relative to the initial process, it is obvious that the rate constant for the primary process computed in the usual way will be in error by a t most a factor of n 1. An approximate treatment of the situation for radicals where there is incomplete reaction is discussed elsewhere.6 The diffusion coefficient of sodium into the gas mixture enters the calculations. The approximation to this quantity has been the binary diffusion coefficient of sodium into the carrier gas. Use of calculated12 ternary diffusion coefficients for sodium in halide-carrier gas mixtures offers a more precise quantity. It may be noted that the treatment given for the flame with a constant streaming velocity corresponds exactly to a system wherein there is diffusion from a point or spherical source into a moving stream. Acknowledgment.-Thanks are due to the Office of Naval Research for their support. We are indebted to Professor G. H. Cady for his generous cooperation.
+
Appendix I. Halide Back DiiTusion. No Wall Reaction.(4)
This is a linear, second-order differential equation with constant coefficients. A solution satisfying the boundary conditions p'( a) = 0,p'(0) = po' is P' = PO' exp[-(a
+ b)z/21;
+ 4kp*/D'
(5)
+ b)z/2]dz pe/p* = 1 - 2 k p o ' / ( ~+ b)v
(6)
a =
v/D', b2 = a2
The net flow of sodium is given by zr20pev/RT = nro2p* v / R T pe
p*
i
':*
--
-
PO'K
expl-(a
11. Halide Back DiiTusion. Wall Reaction.D'v~P'
+ V ~ P / ~-Z kpp'
=0
Set P' = d Z ) R ( P )
(7)
Substituting into the above equation and dividing through by p'
By setting the terms in p equal to -n2, and those in (12) C. F. Curtiss and J. 0. Hirschfelder, J . Chem. Phus., 17, 550 (1949); C. R. Wilke, Chem. En#. Prog., 46, 95 (1950).
/
TAKESHI ABE, ERWIN SHEPPARD A N D IRVING S. WRIGHT
26G z equal to kp/D‘ be solved giving
+ n2,the resultant equations may +
VOl. 59
The ratio of halide pressure at the wall to that at the center of the tube is given by E
R ( p ) = Jdnp); + ( z ) = exp (-vz/ZD’)[A’ exp(Xz) B’ exp( -h)](8)
where X2
= (v/2D’)a
since Jo(0) = 1. It is seen that this treatment is identical to the previous one if wall reaction is neglected, in which case n = 0. 111. Solution to Equation (24).-
+ + kp* D n2
A’ = 0 from the boundary condition +( m ) #
00
Then w = gas phase reaction wall reaction
-
skPP’dV
I
Som [p] -D’
kp
Jomc
-2D’nro
P
Znrodz
P=TO
+ ( z ) J d w ) 2 wdp dz
JOm
dz) [T] Jo(np) p3T0
dz
The integrals in dx in the numerator and the denominator are the same and thus cancel. From the properties of the Bessel function, the above ratio reduces to w = kp*/D’n2
is solved by setting p = exp (ax)+(r), where r is the radial coordinate. Then with CY = v/2D, separation of variables is accomplished giving a solution
+
where ,B2 = a2 kp’/D. Since 0 p ( m ) = 0, then B’ = 0. Thus
3
CY,
r
> x , and
A’
p = - exp (az - p r )
(9)
(25)
ELECTROKINETIC STUDIES ON FIBRINOGEN. V. T H E DETERMINATION OF T H E ISOELECTRIC POINT BY THE STREAMING POTENTIAL METHOD1 BY TAKESHI A B E , ERWIN ~ SHEPPARD~ AND IRVING S. WRIGHT The Department of Medicine, New York Hospital-Cornell University Medical College, New York 81, N . P. Received October 1, 19.54
Streaming potential experiments were performed to determine the zeta potentials for 0.09% fibrinogen solutions (97% clottable) in contact with Pyrex capillaries as a function of pH and ionic strength. It is observed that the slopes of the l vs. pH lines decrease as the ionic strength increases and that there are no discont,inuities in these lines. Within the limits of the ionic strength range investigated, 0.004-0.013 (phosphate and acetate buffers), it is found that the zero for fibrinogen solutions varied from pH 5.85 to 5.95, respectively. These pH values are the isoelectric points for fibrinogen a t these ionic strength levels. The pH of the isoelectric point extrapolated to zero ionic strength occurs a t 5.80. It is found that the thickness and tenacity of fibrinogen layers adsorbed on the capillary wall will depend on the pH of the initial coating and the pHsequence of subsequent buffer streaming. The zeta potentials observed for the fibrinogen-solid interface are due to the thickness and electrical surface properties of the adsorbed protein layer and to the concentration of electrolyte in solution.
Introduction The evaluation of the isoelectric point of proteins by the streaming potential technique was first considered feasible by Briggs4 in 1928. I n our investigation of the electrostatic forces involved in blood coagulation,6 the streaming potential method was used to calculate the zeta potential for fibrinogen solutions, in contact with capillary surfaces, as a function of pH. It was found that the zero streaming and zeta potentials occurred a t pH 5.8 for both human and bovine fibrinogen obtained from Cohn’s fraction I.6 It was stated that this pH was also (1) This ia paper number 6 in the series “Electrostatic Forces Involved in Blood Coagulation.” This work was supported, in part, by the Office of Naval Research, U.9.Navy under contract N6onr-264, T. 0. 10 and also by grants from the Kress, Lasker, Hyde and Hampil Foundations. (2) Department of Physiology and Pharmacology, Wayne University, College of Medicine, Detroit, Michigan. (3) To whom inquiries concerning this paper should be addressed. (4) D. R. Briggs, Proc. Sac. Esptl. B i d . Med., 26, 819 (1928). ( 5 ) E. Sheppard and I. S . Wright, Arch. Eiochem. Biophys., 62, 414 (1954). (6) E. J. Cohn, L. E. Strong, W. L. Hughes, Jr., D. J. Mulford, J. N. Ashworth, M. Melln and H. L. Taylor, J . Am. Chem. Sac., 68, 459 (1948).
the isoelectric point for the adsorbed proteins. When Laki’ described a method to purify bovine fraction I from 4 0 4 5 % clottable to better than 90%, the determination of the isoelectric point for this purified fibrinogen was undertaken in order to compare the results for the two degrees of purity. Since it has been shown8-12 that the pH of the isoelectric point is dependent on the ionic strength of the solution, p , the zeta potentials of fibrinogen solutions were determined as a function of pH a t several low ionic strength levels. It was also of fundamental importance to investigate whether the zeta potential for fibrinogen was due entirely to the adsorbed protein on the capillary wall or to an equilibrium between the adsorbed and the soluble material or electrolyte in .the bulk of the solution. Finally, the nature of the adsorbed layer on the capillary wall with respect to its thick(7) K.Laki, Arch. Biochem. Biophys., 32, 317 (1951). (8) E. R. B. Smith, J . Biol. Chem., 113, 473 (1951). (9) G . Scatchard and E. S. Black, THISJOURNAL, 63, 88 (1949). (10) R . A. Alberty, ibid., 63, 114 (1949). (11) L. G . Longsworth and C. F.Jacobsen, kbid., SS, 126 (1949). (12) S . F. Velick, ibid., 63, 135 (1949).
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