FREDJ. EDESKUTY AND NEAL R. AMUNDSON
148
the maximum error is probably less than 0.01. A qualitative examination of the X-radiograms of all of the orthovanadates of this isomorphous series reveals no significant difference in the relative intensities of the various members of the series. Since the parameters x = 0.19 and z = 0.35 give satisfactory agreement between the calculated and experimental intensities for both neodymium orthovanadate and ytterbium orthovanadate, it is believed that the parameters (x = 0.19 f 0.01, z = 0.35 f 0.01) will give satisfactory results for each member of the isomorphous series except scandium orthovanadate. The observed intensities of the diffraction lines of scandium orthovanadate and the calculated intensities for the parameters x = 0.20 and z = 0.32 are given in Table IV. It will be noted that the calculated intensities of the reflections that were not ..I
1.1
,,kI
L *.1
S.1
I
I
I
I
I
I
I
(.I
1.0
1.1
7.2
7.3
1.4
1.5
ao,
Fig. 2.-Lattice
A.
constants of the orthovanadates with the zircon structure.
Vol. 56
observed experimentally are less than one per cent. of the intensity of the strongest line. There is also good general agreement between the calculated intensities and the intensities measured by the Geiger counter of the Norelco spectrometer. The structure of the orthovanadates of this isomorphous series is the zircon type structure. The interatomic distances are given in Table 111. The trivdent metal atom is surrounded by eight oxygen atoms; four of the oxygen atoms are nearer than the other four. There is a tetrahedron of oxygen atoms around each vanadium atom. In scandium orthovanadate this tetrahedron is elongated in the direction of the c-axis; in the other orthovanadates of this series the tetrahedra are almost regular. In Fig. 2 it will be noted that a graph of the lattice constants of all of the orthovanadates of this isomorphous series is a straight line (except for scandium orthovanadate) within the probable accuracy of the available data. The unit cell of scandium orthovanadate is elongated in the direction of of the c-axis. An explanation for the fact that the unit cell of neodymium orthovanadate was found to be larger than the unit cell of praseodymium orthovanadate* has been found. The previously proposed process of solid solution2 has been confirmed in this investigation. In the X-radiogram of one of the samples of cerium orthovanadate (CeVO,) some of the stronger lines of cerium dioxide were observed. When this sample was heated for a longer period of time a t a higher temperature the cerium dioxide lines disappeared and the lattice constants of cerium orthovanadate became larger.
MATHEMATICS OF ADSORPTION. IV. EFFECT OF INTRAPARTICLE DIFFUSION IN AGITATED STATIC SYSTIEMS BYFREDJ. EDESKUTY AND NEAL R. AMUNDSON Department of Chemical Engineering, Univeraity of Minnesota, Minneapolis 1.6, Minnesota Receind November 0. 1060
The solution of the problems osed in this paper are rather special in that the types of isotherma and kinetics employed have only limited application. Por the linear isotherm there are cwea where this assumption is valid. The cwe of linear kinetics has not been investigated experimentally to any extent. For more general cases numerical methods of solutions of e uations proposed here must be sought and the use of digital computers should rove to be of great help in this endeavor. Tiis is largely because of the iterative nature of the approximate solution to the &fusion equation. T h s problem is being investigated at the present time and will be reported upon later.
The problem of adsorption of a solute from a solution by an adsorbent in a static non-flow system is an old one. The major emphasis in the past has been on the equilibrium attained between the adsorbate on the solid and that in the solution. There is evidence according to Amis,‘ that adsorption equilibrium obtains always in a finite time although a period of years may be required. Many equstions have been proposed for adsorption kinetics, a brief review of which is given by Amis. Kinetic effects may arise in a variety of ways and it is not always clear just what effects are being measured. (1) Edward 8. Amis, “Kinetics of Chemical Change in Solution,” The Maomillan Co., New York, N. Y.,1949.
For example, the distribution of solute in the solution exterior to the adsorbent may not be uniform giving rise to concentration gradients. These may be minimized or eliminated completely by agitation of the whole system. Even moderately high agitation rates may still not eliminate these gradients in the neighborhood of the particles themselves and this condition manifests itself in the so-called film resistance to mass transfer a t the particle surfaces. High agitation rates will reduce the film resistance to a point where there is practically DO kinetic effect at the particle surface. This is tantamount t o assuming that the concentration of solute at the particle surface is identical with that in the main
MATHEMATICB OF ADSORPTION
Jan., 1952
body of the solution. These two effects are dependent upon diffusion of the solute in the solution exterior to the adsorbent. For particles of some size an appreciable portion of the adsorbent surface is in the interior of the particle. Adsorption can only take place on this surface after the adsorbate has diffused through the solution existing in the pore volume of the solid. This diffusion may be much slower than diffusion in a solution alone because of the increased length of the tortuous path caused by the internal structure of the solid. Eagle and Scott2 in a recent paper made some mention of this aspect of the kinetics. Finally, the rate of adsorption itself can be a limiting factor and this is in general the most difficult of the kinetic effects to handle analytically. Even if the rate of adsorption is so rapid that equilibrium is attained practically instantaneously, the equilibrium isotherm is generally complicated enough to preclude exact mathematical 'methods and some recourse to approximate numerical methods must be made. Laidler* has examined to some extent under what conditions adsorption kinetics will prevail. In this paper the problem of adsorption in a static system is considered in the following way. A vessel contains W grams of porous adsorbent and V liters of adsorbate-free solvent. At time 1 equal to zero adsorbate is admitted to the vessel
(Z ?E)
+ ZG + ha) .+ kaa)
D(P
(I
C6
= concn. 111 main body of solution, moles per liter
= concn. in main body of solution initially, moles per liter = concn. of solution in void volume of solid, moles per liter Q = init. concn. of solution in void volume of solid, moles per liter D = diffusivity, cm.* per second h = Laplace transform of e H = Laplace transform of C k mass transfer coefficient at particle surface kl,k, = constants in adsorptionisotherm or kinetic expansion K = kl/kt,equilibrium constant for adsorption kinetics n = amount adsorbed on solid. moles Der liter of atmarent volume no = init. amount adsorbed on solid, moles per liter of apparent volume N = Laplace transform of n = Laplace transform parameter corresponding to t p = radius variable in particle, cm. r = external radius of particle, cm. R 1 = time in seconds V = volume of external solution, liters W = weight of adsorbent, grams
c
i i :
--
pR2-y. fractional void volume B = (k 4W/VP 8' = Dp/Rt r(z) = gamma function a y =-kl CUD A =R 0
If this inequality is not true the quadratic factor will have two complex roots and by an extension of the theorem of Laquerre one can show that g(z) will not have more than two complex roots and possibly it may have none, all the remaining roots being negative. It is obvious of course that g( z ) has no positive roots since the infinite series contains terms all of which have the same sign. Consequently in terms of w it is seen that there will be no imaginary roots, there may be two complex roots, and the real roots are symmetrical with respect to the origin since &(a) is an even function. It would be desirable to make a more definite statement about these roots but it is not possible at this time. Under these circumstances each case must be treated as a special one. (5) 8. Patterson, Proc. Phvs. Boc., 59, 50 (1947). (6) R. A. Ebel, Ph.D. Thesis, University of Minnesota, 1949. (7) P . R. Kasten and N. R. Amundson, Ind. En#. Chem., 42, 1341
(1950).
W.D. Munro and N. R. Amundson, ibid., 42 1481 (1950). (9) E. C. Titohmarsh, "Theory of Functions," Oxford University Preaa, London, 1939. (8)
FRED J. EDESKUTY AND NEALR. AMUNDSON
152
For the case of the infinite mass transfer coeBcient the quadratic factor reduces to a linear factor, 2n 3(@ l),and the zero of this equation is always negative. Therefore in this case the zeroes of g(z) are always negative. Consequently for the well agitated system, which the authors consider to be the most important one, the roots may be calculated with the assurance that all have been accounted for. Case 2. Non Equilibrium.-For this case Equations l, 2, 4 and 5 are still valid but a new assumption concerning the relation between adsorbed material and solution concentration must be made. In general the kinetic relation will replace the equilibrium isotherm and will be of the form bn/bt = F(c,n), where F(c,n) is a complicated function depending upon the adsorbate, solvent and adsorbent. Although a numerical method could be developed to treat a general function it is the purpose here to develop mathematical formulas and the most general relation of this type which can be used is
+ +
z(P)
- [-
+
~ D P p(1-
Vol. 56 e)]
sin A -IA cos A
r s+
p.)
(18)
where p = aW/pV. Equation 17 is the inverse transform of the concentration in the solution. The residue a t the ole p = 0 can be shown to be (Cm p (ace R J ) / ~ P CY (1 PI),where K -kl/kz. This consists of the sum of two terms one coming from the first and the other the second term in Equation 17. There is also a pole a t p = -kl k2a. The residue a t this pole can be found by evaluating the limit in the prescribed manner. Two terms are again obtained which a r e . negatives of each other and hence their sum zero. The remaining poles occur at the zeroes of Equation 18 and the sum of the residues for these poles can be obtained from Equation 11. Direct calculation will show that Coa dace 120) +
+
+ +
+
-
++ Q(1+ rA R [Co(Pn + ki + ha)- o(pn + kra)- %kzl sin -f ePn' (pn + k~ + kw)Z'(pn) -c
dr, t ,
= Kp
+p)
m
n-1 introduced for convenience only. The complete system to be solved for this case where sin An consists of Equations 1, 2, 4, 5 and 16. This probz'(pn) = An(3P + pne) G( pn 1 lem can be handled also by means of the Laplace transformation. Let L[n(r,t)]= N(r,p) = N . G(pn) 38'An+ En I- An'(38' pne)' The transforms of Equations 1 and 16 are, reP"(3P' spectively
CY being
+ + + Pn4 - P"'1 + (pa kik*a + 1
5
dzh
+ 2dh
dT* -r dr =
1
(ph
- co)
E n = - - R'
[l 2DA. 8' = Dp/R'
+ -J1j (PX - no)
and 1 (pN a
- no) = k J
and pn is a zero of Equation 18 with Am*D pn(pn kr 4- h a )
- kzN
1
d*h + 2- dh - -ah drq r dr
+ + k2a
-i-
R'
If N is eliminated between these two equations the following results
=
+ +
5
(Pn
+ noko - 31 [o(p +p kza) + kza
+
1
where a = p ( p k~ k d / D ( p k ~ 4 . The solution of this equation which remains finite a t T = 0 is A h = - s i n r m +
In Equation 13 it is necessary to calculate
+ + h a ) - e~(pn+ kza) - nokr][An
[Co(pn kl
++ ++
CO(P h a ) nokp p(p ki kza)
The constant A may still be determined from Equation 8 with the result that the complete solution is
pn
+ kl + kzaP"(Pn)
COS A
m
- sin An] ePnt
Upon using Equation 18 this can be reduced after substituting for the cosine to
+ + - + + + GI Substitution into Equation X gives 1
[Co(pn
kr
kza)
(pn
kl
eo(pn h a ) h)G(pn)
- nok21Anpnepnt
Equation 19 solves the kinetic problem for the type of kinetics assumed here completely. A mass transfer coeficient has been assumed to exist but the case for infinite mass transfer can be easily obtained by allowing -0 with subsequent simplification of the equation defining the concentration and the roots over which the summation is to be made. It would be highly desirable t o make a statement concerning the roots of Equation 18 but it is not possible a t this time. However preliminary investigation seems to indicate that for the case of no film resistance all the roots in p are negative.
