1272
AI. L. FRANKLIN AND TEDB. FLANAGAN
The Self-Diffusion of Water in Uranyl Nitrate Hexahydrate by M. L. Franklin and Ted B. Flanagan" Chemistry Department, University of Vermont, Burlington, Vermont 06401 (Received December 7 , 1070) Publication costs assisted by the National Science Foundation
The self-diffusion of water in uranyl nitrate hexahydrate (UNH) has been investigated by following the course of isotopic exchange with DzO. The extent of exchange exhibits a square-root depeiidence upon time. This suggests that the slow step is solid-state diffusion. The rate of diffusive exchange of UNH with H21*0is comparable to that with D20 and therefore the entire water molecule diffuses. The pressure and temperature dependences of diffusive exchange have been examined. The temperature dependence can be expressed as D =4X exp(-7600/RT). Possible mechanisms for diffusive exchange in UNH are considered.
Introduction
bond of the structural water to the nitrate oxygen is weak.s I n 1961, Barrerl pointed out that there was a notable Because Kraft observed that diffusive exchange does lack of measurements of the self-diffusion of water in occur in UNH and because its structure is known in nonzeolitic crystalline hydrates. The situation is undetailJ6this hydrate was chosen for a detailed investichanged today with regard to direct measurements of gation. I n addition to determining the time course diffusion but nmr techniques have recently given some and temperature dependence of exchange it was of information about translational diffusion in h y d r a t e ~ . ~ * 3 interest to investigate its pressure dependence since The availability of such data from nmr increases the this dependence has not been hitherto examined in the desirability of having direct diffusional data for purhydrates. poses of comparison. The only detailed study of diffusion of water in crystalline hydrates (nonzeolitic) Experimental Section has been made by KraftU4 He surrounded light water Materials. UNH single crystals were grown using a hydrates with DzO vapor and noted their increase of modification of the procedure of Taylor and Mueller.6 weight as a function of time. Potassium aluminum A slightly acidified aqueous solution of the salt was alum was investigated in some detail and Kraft found allowed to evaporate slowly at room temperature until
D
=
0.56 X 10-7exp(-6000/RT)
This gives D = 2 X 10-l2 cm2 sec-' (25'). He did not observe exchange with magnesium sulfate heptahydrate and copper sulfate pentahydrate. He noted that uranyl nitrate hexahydrate exchanged about 100 times as fast as potassium alum. Wei and Bernstein5 found exchange of DzOwith boehmite powder (a-alumina monohydrate), but in this case, exchange of protons and deuterons takes place rather than water molecule entities. In connection with their detailed nmr studies, O'Reilly and Tsang2 have measured an approximate diffusion constant for water in potassium ferrocyanide trihydrate powder by weighing material stored in a desiccator over DzO. They obtained a value of D of 3 X cm2 8ec-I at room temperature assuming that the particles were uniform spheres and that isotropic diffusion occurred. In UNH the uranyl group is surrounded equatorially by a near-hexagon of four oxygen atoms from two nonequivalent bidentate nitrate groups and two equivalent water oxygens.6 The additional four water molecules are structural water since they are not coordinated to the uranyl group but are hydrogen bonded to nitrate ions and to other water molecules. The hydrogen The Journal of Physical Chemistry, Vol. 76, No. 0, 1071
single crystals had grown. Crystals of a convenient size (-2 mm = dimension alone the c axis and about 8 mg weight) were chosen, removed, and stored in a desiccator over an H2S04 (50% by weight) solution. This solution maintained a vapor pressure sufficient to prevent dehydration and yet low enough to avoid deliquescence of the hydrate. The powder was prepared by grinding crystals of UNH in a water-saturated atmosphere (to prevent dehydration) into a fine powder. The powder was stored as described and small samples ( 5 to 10 mg) were employed for each run. DzO (-9870, Columbia Southern Chemical Corp.) was inserted into the vacuum system and after approximately 30 exchange runs the DzO (-10 ml) was replaced. Hz180 (11.8% l 8 0 ) was obtained from Merck Sharp and Dohme, Canada). Apparatus. A quartz helix balance (Worden Quartz (1) R . M . Barrer and B. E. F. Fender, J . Phys. Chem. Solids, 21, 12 (1961).
(2) D. E. O'Reilly and T . Tsang, J . Chem. Phys., 47, 4072 (1967). Phvs., 28, 565 (1969). (3) S. P. Gabuda and A. G. Lundin, 502.'. (4) H. Kraft, 2. Phys., 110, 303 (1938). (5) Y. K . Wei and R . B. Bernstein, J . Phys. Chem., 63,738 (1959). ( 6 ) J. C . Taylor and M. H. Mueller, Acta Crystallog?., 19, 536 (1965).
1273
SELF-DIFFUSION OF WATERIN URANYL NITRATEHEXAHYDRATE Products, Houston, Texas) of sensitivity 1 cm/mg was used to monitor the exchange reaction. It was mounted with a water jacket (k0.1") within a conventional high-vacuum system. All helix balances employed were calibrated prior to use. The temperatures of the sample and DzOsupply were maintained by well stirred miter-ice-methanol baths to temperatures w-20", and below this by COz-acetone baths. These temperatures were generally held to 10.3". Procedure. I n order to avoid dehydration of the UNH samples the system was evacuated while simultaneously cooling the sample to -78". After approxiTorr was generally mately 10 min, a pressure of obtained. Following this procedure the sample was adjusted to the desired temperature and DzO was admitted to the reaction vessel to initiate the exchange run. Dehydration did not occur prior to admitting the DzO. This was shown by noting that in several blank runs rehydration did not occur following the preliminary procedure. It has been shown that rehydration occurs rapidly under the blank conditions employed.' The weight change, AMa, of the sample following complete diffusive exchange with DzO is
I.
0.
0.
8 z \
f 0.
0.
tl!2~iin"O)
Figure 1. Representative diffusive exchange run on a single crystal of UNH (4.6 mg, p r = 0.18, 25').
A M , = (O.l117)W(% HzO/100)
where W is the weight of the sample and % HzO refers to the weight per cent of exchangeable water in the hydrate. For a 5-mg sample of UNH the helix movement following dehydration is 0.0802 em. Since the helix can be read to *0.001 cm, the error in the final reading a t complete exchange is 11.3% and correspondingly greater percentage errors at smaller percentage exchange. For the experiments with HzlsO (11.8% lSO)the total movement of the helix is 0.0094 em (5 mg sample) so that the error in the final reading is 110.6%.
Results Figure 1 shows a typical diffusive exchange run on a single crystal of habit shown in Figure 2. The temperature of the run is 25" and the vapor pressure of water employed relative to that of liquid water is p , = 0.18. At this and at lower temperatures, only four of the six water molecules undergo diffusive exchange-the structural water molecules. The exchange of these structural water molecules goes to completion. At 35" only the four structural water molecules are lost during dehydration in vacuo but in contrast t o diffusive exchange, a t lower temperatures, e.g., -30" only three water molecules are lost.' The diffusion times for individual single crystals ( 4 3 mg) are long; for example, 24 hr is required at 35" for M , / M , to reach 50%) where M , is the weight of the sample at time t and M , the weight a t t = C D . The fraction of exchange, M,/M,, can be seen to follow a tl'' relationship (Figure 1) which is a character-
Figure 2. Typical habit of single crystals employed for diffusive exchange runs.
istic of many diffusion-controlled solid reactions. The relationships were found to be linear to about 50%. A single crystal was coated with water-impermeable plastic on the (100) faces. If diffusion is isotropic, the diffusion problem reduces from three to two dimensions. The appropriate equation for two-dimensional diffusion in a parallelopiped of sides ul and uZ1is
(7) M. L. Franklin and T. B. Flanagan, to be submitted for publication. The Journal of Physical Chamistry, Vol. 76, No. 9, l Q Y l
1274 which, a t small times, reduces to
The habit of this coated crystal was approximated by a parallelepiped of active sides al and a2 where these are the dimensions of the crystal along the b and c axes, respectively (Figure 2). The calculated value of D is 8 X cm2 sec-l ( O O , Pr = 0.4). Other single crystals, which mere uncoated, gave similar diffusion constants with the assumption that diffusion does not occur in the a direction. The water molecules lie in sheets parallel to the (100) planes and so diffusion might be expected to be anisotropic; i.e., diffusion in directions perpendicular to the (100) planes is insignificant compared to diffusion parallel to these planes. The direct experimental evidence for complete anisotropy which is based on the measured values of D for coated and uncoated crystals is tenuous because of some irreproducibility in these runs using individual single crystals. The long linear region in the M , / M , plots, however, supports the assumption of anisotropic diffusion. A series of consecutive diffusive exchange runs were made on the same single crystal of UNH. Two points of interest emerged. The rates of these consecutive runs were comparable and an isotope effect was noted. The first proves that diffusive exchange does not disrupt the diffusive path lengths of the single crystal. The isotope effect was in the direction such that H20 diffuses into a crystal containing DzO faster than the reverse process. A comparable isotope effect has been observed and examined extensively in another hydrate systeme7 It will not be discussed here save to mention that an isotope effect supports bulk diffusion as the slow step because significant isotope effects would not be expected if exchange of entire water molecules occurred as the slow step at the surface. As would be expected, diffusive exchange occurs much faster for powder than for single crystals as shown in Figure 3. The powder consists of roughly cubic particles of average size about 0.005 cm. Using eq 2 this gives D = 5 X 10-'0 cm2 sec-l ( p r = 0.35, 0') in good agreement with the value found for the single crystals. Since the diffusional behavior of powder is similar to that observed for the single crystals except for the convenience of shorter diffusion times exhibited by the former, the powder was extensively employed for the determination of the diffusion parameters. Pressure Dependence. The pressure dependence of the rate of diffusive exchange was examined at -29' for powder and at 0' for single crystals. These relatively low temperatures were used because the rate of exchange is slow at these temperatures and consequently self-dilution is not a factor. Self-dilution describes the error introduced by the dilution of the DzO with The Journal of Physical Chemistry,Val. 76,No.9, 1971
&I. L. FRANKLIN AND TEDB. FLANAGAN
"-1
L
2
0
4
6
8
t"z (mi n
1 0 1 2 1 4
'',)
Figure 3. Representative diffusive exchange run with UNH powder ( p p = 0.35. OD).
HzO in the gas phase (Appendix). Figure 4 shows that the rate of diffusive exchange decreases with pressure for both the single crystal and powder. The pressure range is quite limited because above p , = 0.7 and below pr = 0.1 the samples deliquesced and dehydrated, respectively. Temperature Dependence. The temperature dependence of diffusive exchange was examined at a constant value of pr. Other possibilities would have been to hold p or p / p g constant, where p , is the equilibrium vapor pressure of the hydrate. Kraft4 chose to hold p / p s constant. Investigators employing nmr techniques coated the crystals with plastic in order to prevent dehydration;2 under these conditions the water vacancy ++ water vapor equilibrium is presumably not established. Surface exchange-the precursor to bulk exchangemust proceed via adsorbed water; it is therefore reasonable to maintain p r constant. Since UNH would dehydrate and deliquesce at the extremes of the temperature range employed, it would not have been feasible to maintain p constant. A value of p , = 0.4 was chosen for convenience because this is halfway between the two limiting values of p , which can be employed. The Arrhenius plot for the diffusive exchange is shown in Figure 5 as log k us. 1/T where k is the slope of fW,/M, against t1l2 plots, The energy of activation for diffusion, Ed, is twice that determined from the slope of the line in Figure 5 and from this value and the diffusion constant for the single crystals D can be expressed as
D
=
4 X
exp(-76OO/RT)
(3)
Diflusive Exchange w i t h Hzl80. I n order to deter-
1275
SELF-DIFFUSION OF WATERIN URANYL NITRATEHEXAHYDRATE
t
I .o
\
5.0
'4
0,s \
\
\
-
\ \
'A\ A \
-
Y x
A 0,s \
CI
T
\
3.0
A
8
\4
z-
\
-
\
0.4
\
f
\ \ \
\
\
Ill
A \
\ \
A I
0
I
0.1
0.2
I
I
I
I
L
0.3
0.4
0.5
0.6
0.7
p, Figure 4. Pressure dependence of diffusive exchange plotted as DK against p , where D is the diffusion coefficient and K is a constant: A, UNH powder, -29"; A, UNH single crystals (0') normalized to the powder data a t p . = 0.4.
Discussion
O.OI!
I
3.6
I
1
1
3.8
4.0
I/T
4.2
I
4.4
I
4.6
XI03
Figure 5. Arrhenius plot for diffusive exchange for UNH powder plotted as log IC against 1/T, where IC is the slope of M t / M , against t 1 / 2 .
mine whether the entire water molecule diffuses, in contrast to only the hydrogen isotopes, a run was performed with HzlsO. The rates of diffusive exchange are comparable for the two water isotopes (Figure 6 ) which proves that the entire water molecule diffuses.
When solid-state diffusion is carried out using a gassolid technique like that employed here, it must be established that the rate-determining process is solidstate diffusion rather than a surface step. Intuitively solid-state diffusion mould be expected to be the slow step in a system such as this since diffusion is relatively slow in comparison to, for example, gas-metal systems (where D for hydrogen in face-centered cubic metals is typically -10-7 ernz sec-' ( 2 5 O , H in a-Pd/Hs) and consequently slow surface steps can often be an experimental problem. If a surface step were slow, the interior of the hydrate would be expected to be uniform with respect to its isotopic distribution. The rate of exchange would then be first order with respect to time. This dependence is not observed (Figure 1) and, therefore, diffusion within the solid is rate controlling. It is not unexpected that only the four structural water molecules, i.e., those not directly coordinated to the uranyl ions, undergo diffusive exchange. These are the only waters which are lost upon dehydration in vacuo above -20" and below ~ 1 4 0 " .Equilibrium ~ is therefore not established between the two types of water ( 5 3 5 " ) . More surprising, perhaps, is the fact that there is no proton-deuteron exchange between the structural and coordinated water within the solid. Below about -20" only three water molecules are lost upon dehydration7 but all four undergo exchange; this J. W. Simons and T. B. Flanagan, J . Phys. Chem., 69, 3773 (1965). (8)
The Journal of Physical Chemistry, Vol. 76, No. 9, 2971
1276
M. L. FRANKLIN AND TEDB. FLANAGAN
suggests that the four structural water molecules are in equilibrium but the fourth cannot leave the lattice in vacuo below -20”. It is generally believed that vacancies are the defects responsible for diffusion in molecular solids.9 Since the bonding of the water molecules in crystalline hydrates resembles that in molecular solids, it seems likely that water vacancies are the defect by which diffusion occurs. A possible role of vacancies in the kinetics of dehydration of crystalline hydrates has been noted by Garner,lo who suggested that prenucleation corresponds to the formation and aggregation of vacancies. The diffusion constant found in the present work must reflect both the concentration and mobility of water vacancies. If an equilibrium concentration of vacancies exists within the hydrate at each temperature and pressure, then K1
HzOa
HzOg
(4)
I‘ -I- HzO,
(5)
K2
Hz0h
where HzO,, & o h , and HzO, represent the total water, D20, in the following regions: adsorbed on the H2O surface, in UNH as exchangeable water, and in the gas phase, respectively. The concentration of vacancies is therefore
+
T’
KIK~HZO~/PD,O
(6)
Experimentally p , has been maintained constant for the temperature dependence studies and therefore ~ D , O = p,po where PO is the vapor pressure of pure D2O. Since the temperature dependence of p o and K I should be comparable, eq 6 reduces to
V
=
B exp(-AHf/RT)
(7)
where B is a constant and AHf is the enthalpy of formation of water vacancies according to eq 5 . If an equilibrium vacancy concentration is attained at each temperature, the observed value of Ed should be AH, E , where E , is the activation energy for mobility of a vacancy. Jostg has pointed out that the energy to form a vacancy in a molecular solid can be taken as the lattice energy and this can be approximated by the latent heat of sublimation. I n the case of crystalline hydrates the heat of dehydration is analogous to the latent heat of sublimation of molecular crystals but it is not as reliable an estimate of AH! as AHBis for molecular crystals because when the water is returned to the surface following formation of a vacancy it does not form a new layer of crystal lattice but rather forms an adsorbed layer. In metals, the energy of vacancy migration is generally assumed to be of the same order as, but less than, that for vacancy formation. l1 Sherwood12 has assumed that the same situation holds for molecular solids. Garner lo has also suggested that the activation
+
The Journal of Physical Chemistry, Val. 76, No. 9 , 1971
energy for water vacancy migration should be of the same order of magnitude as the heat of dehydration. Sherwood13has collected the meager data available for diffusion in molecular crystals and he noted that generally Ed is of the order of twice the latent heat of sublimation. An important exception is ice. Delibaltas, et aE.,14have found that Ed is 15.7 kcal/mol and Granicher15has again equated AHf to the latent heat of sublimation, $.e., 12.2 kcal/mol. This leaves only 3.5 kcal/mol for the migration energy of activation. Delibaltas, et al.,14have pointed out that this is surprisingly small since 3 hydrogen bonds must be broken for a water molecule to jump into an adjacent vacancy. If Ed is about twice AHf then Ed should be 25.4 kcal/ mol, because the enthalpy for dehydration of UNH to the dihydrate is 12.7 kcal/mol Hz0.16 For potassium alum the corresponding value for Ed would be 26.2 kcal/mol since the enthalpy of dehydration per mole of water is 13.1 kca1.l’ O’Reilly and Tsang2 report two values for Ed for potassium ferrocyanide trihydrate, Le., 5.2 (proton relaxation) and 10.4 kcal/mol (deuteron line width) ; no explanation is offered for the discrepancy but the average of these two values is again too small if Ed is to be equal to twice the enthalpy of dehydration. The disagreement can be resolved if it is assumed that an equilibrium vacancy concentration does not exist a t each temperature, then Ed = E,, and, if E , is considerably less than the enthalpy of dehydration, agreement with the observed value of Ed could be obtained. Under these conditions diffusive exchange would occur only via the “grown-in” vacancies. This hypothesis is, however, untenable because the available vacancies would soon be destroyed and diffusive exchange would not go to completion. Vacancies must be both created and destroyed a t the surface for a vacancy mechanism to control diffusive exchange. An equilibrium concentration of vacancies which obtains only near the surface is a possible alternative to the assumption of a uniform vacancy equilibrium. Diffusive exchange would then be proportional to the concentration and mobility of vacancies in the vicinity of the surface. I n this case a steady-state treatment can be applied to the vacancy concentration at and (9) W. Jost, “Diffusion,” Academic Press, New York, N. Y., 1952. (10) G. P. Acock, W.E. Garner, J. Milstead, and H. J. Willavoys, Proc. Roy. Soc., 189, 508 (1946). (11) P. Shewmon, “Diffusion in Solids,” MoGraw-Hill, New York, N. Y., 1963. (12) J. N. Sherwood and D. J. White, Phil. Mag., 15, 746 (1967). (13) J. N. Sherwood, Proc. Brit. Ceram. SOC.,9, 233 (1967). (14) P. Delibaltas, 0. Dengal, D. Helmreioh, N. Riehl, and H . Simon, Phys. Kondens. Mater., 5, 166 (1966). (15) H. Grhnicher, Z . Kristallogr. Kristallgeometrie, Kristallphys., Kristallchem., 110, 460 (1958). (16) W.H. Smith, J. Inorg. Nucl. Chenz., 30, 1761 (1968). (17) D. A. Young, “Decomposition of Solids,” Pergamon Press, Oxford, 1966.
1277
SELF-DIFFUSION OF WATERIN URANYL NITRATEHEXAHYDRATE adjacent to the surface, and this again leads to the result that Ed = AHf E,. I n fact, this alternative is not likely in view of the experimental result that the rate of diffusive exchange is unchanged in a single crystal which has previously undergone exchange. The flux of H20within the crystal is equal and opposite to the flux of DzO; therefore the rate of creation is equal to the rate of destruction of vacancies. The loss of HzO from the surface hydrate layer does not occur simultaneously with the addition of DzO to the resulting vacancy because otherwise vacancies would not be formed internally and they are necessary for diffusive exchange. The creation and destruction of vacancies at the surface are therefore separate events. The fact that their rates become equal almost immediately, i.e., there is no observable dehydration or rehydration of the crystal, indicates that vacancy equilibrium is attained. This is consistent with the observation that the rate of diffusive exchange decreases with increase of pressure (Figure 4). The observed pressure dependence indicates that the equilibrium concentration of vacancies is governed by eq 6. Since the rate does not change during these runs until M , / M , 6 0.6 the vacancy concentration rapidly adjusts to the ambient pressure. This behavior is reasonable if the relative time scales of vacancy mobility and diffusive exchange are considered. The equilibrium fraction of vacancies is of the order of N , / N = exp( - AGf/RT) S exp(-AHf/RT) = 7 X (0”) if AHf is taken as 7100 cal/mol (see below). Diffusion of a water molecule into a given lattice site requires that a vacancy be at the latter site (probability = N , / N ) , whereas the diffusion of a vacancy to a given site requires that the latter site be occupied (probability E 1). Therefore the time scale for vacancy diffusion is -1.4 X lo6 times faster than that for diffusive exchange (0”). This supports the assumption that water vacancy diffusion is extremely rapid compared to the observed rates of diffusive exchange, The basic problem therefore remains-how are the observed values of Ed to be reconciled with estimates of AH? E,? It is of interest t o reexamine the estimated values of AHf and E, specifically for the case of crystalline hydrates. For UNH, four hydrogen bonds (one of these is weak, the H bond to the nitrate oxygen) must be broken to form a vacancy. The strength of these H bonds can be roughly estimated from the heat of dehydration of UNH to the dihydrate.’e Three H bonds must be broken for this process since only two of the bonds involve bonds to removable water molecules. This requires 12.7 kcal and therefore an average of 4.2 kcal/bond is found. (This neglects any modifications in the heat of dehydration upon cooling to 0°K and differences between AH and A E but these corrections are generally small.1E) I n order to form a vacancy plus a gaseous water molecule 16.8 kcal/mol is required, and when this water is returned to an adsorbed layer,
+
+
9.7 kcal/mol is recovered, the latent heat of vaporization of liquid water, The formation of a vacancy therefore requires 7.1 kcal/mol. It is not unreasonable to expect the adsorbed mater to be liquid-like.lg The maximum energy for mobility of a vacancy can be estimated from the energy required to break three hydrogen bonds (one weak), Le., 12.5 kcal/mol. It is difficult to estimate how much smaller this may be when the relaxation of water molecules around the vacancy is considered. Because of their asymmetry, the uranyl ions and the coordinated nitrate ions are not expected to re1ax.l’ I n any case, E d still remains significantly larger than AHf E, for TJNH and similar considerations would lead to the same impasse for potassium alum4and potassium ferrocyanide trihydrateS2 The value of Doobserved here, 3 X lo-* cm2 sec-l, is similar to that which has been found for ice.14 It can be compared to a theoretical value calculated by”
+
ya2v exp[(A&
+ ASd/R]
(8)
where y is a geometri: factor taken as 1, a is the jump distance which is 2.7 A in UNH,6 AS, is assumed to be 1,” A& is taken as the entropy of dehydration of UNH to liquid water,20and Y for the low-energy translational mode of water in UNH has been determined by Rush and coworkerP from neutron scattering to be 5 X 1012 cm2 sec-’. In view of sec-’. D Ois then 180 X the uncertainty in the value of A& this agreement is not bad and indicates that a vacancy mechanism may obtain with a surprisingly low value for Ed. The unresolved problem of the anomalously small value of E d observed here for UNH and elsewhere for ice14deserves further investigation.
Acknowledgments. The authors are grateful for financial support by NSF Grant GP-9560. The authors thank Dr. RI. H. Mueller for assistance in identification of the faces of the UNH single crystals employed in this research.
Appendix The most serious source of potential error can be termed “self-dilution” and arises from the dilution of the surrounding water vapor isotope with that from the sample in the nonstirred system which has been employed. An exact solution of the diffusion problem is impossible because of the complex geometries involved. I n all cases, however, the sample dimensions mere small compared t o the reaction vessel and it can be assumed that the sample is a continuous point source in (18) D. Eisenberg and W. Kauzmann, “The Structure and Properties of Water,” Oxford University Press, New York, N. Y., 1969. (19) H. A. Resing, Advan. Mol. Relaxation Processes, 1, 109 (1968). (20) W. M . Latimer, “Oxidation Potentials,” 2nd ed, Prentice-Hall, Englewood Cliffs, N . J., 1962. (21) J. J. Rush, J. R. Ferraro, and A. Walker, Inorg. Chem., 6 , 346 (1967).
The Journal of Physical Chemistry, Vol. 76, N o . 0,1971
M. L. FRANKLIN AND TEDB. FLANAGAN
1278 an infinite medium. problem as
Crank22gives the solution for this
The integral 4A can be evaluated as
KS
1
C(r,t) = ___ 8Dnrt'/' e X P [ g ]
-r 2
at'
exp[4D(t - t ' ) ] (t - t')'lZ ( W where C(r,t) is the concentration of diffusing substance a t any distance, r , from the source at time, t. The diffusing substance is generated a t the rate cp(t), where (d(t) =
d(iWc/lWw) KSt-'/% =dt 2
K converts the fraction reaction per second to molecules of water per second and S is the observed slope (in seconds-'/').
Substituting this into eq 1A gives
r
-v2
I
If y = l/(t - t') and b = r2/4D,eq 2A reduces to C(r,t) = which can be further reduced to C(r,t) =
dY
if x is substituted for ty, eq 3 simplifies to
KS
exp[-xb/t] -dx=
C(r,t) = 1 6 ( D ~ ) ' / ~ i(z - 1)"'
KS 16(Dn)'/'t
t
exp(-xb/t] (x - 1)'/' dx
The Journal of Physical Chemistry, Vol. '76, No. 9, 19'71
(5A)
A maximum value for this effect can be estimated from the data on powder at 0"; ie., this is the highest temperature used in the Arrhenius plot to evaluate Ed (Figure 5 ) . For a typical sample K = 3.6 X 10'6 see-'/' at 2.0 mm of D20 molecules H20 and S = pressure. The value of D for HzO(g) is estimated as 102 cm2 sec-l from D = l/2XF and X = k T / ( v z n a 2 p ) = 4.6 X cm. Inserting these values into eq 5A taking r = X and t = 60 sec, c(X, 60) = 3.9 X 1014 molecules H20 compared with 7 X 1016 molecules of HzO D20in the gas phase under these conditions of temperature- and pressure. A value of r has been taken equal to X because molecules one mean free path away from the surface will be those which will collide with the surface to undergo exchange. Self-dilution is concluded to be insignificant under these conditions because the concentration of H2O in the gas phase is only 0.5% of the total. At lower temperatures and longer times the effect will be even smaller. Experimental tests for the importance of selfdilution have been made by moving the position of the D20 reservoir closer to the sample, i.e., the diffusion path of the HzOfrom sample to bath is thereby reduced and the diffusion should therefore be faster as the concentration gradient is increased. There was no effect upon the measured exchange rate noted for a fivefold change in the overall bath to sample distance.
+
(22) J. (a)Oxford,
Crank, "Mathematics of Diffusion," Clarendon Press, 1956.
