March, 1961
DIFFUSION TO
A
PLANE WITH LANQMUIRIAN ADSORPTION
473
Thermal decomposition of uranyl nitrate dihyTABLEVI11 X-RAYDIFFRACTION AND INFRARED ABSORPTION PATTERNSdrate a t atmospheric pressure produced either r-UOa between 250 and 400' or y &UOa a t 500" as the OF URANYL HYDROXYNITRATES
+
Tetrahydrate d
I/Io
7.15 6.52 6.30 5.19 4.70 4.49 4.33 3.83 3.56 3.24 2.16
70 60 85 35 45 40 100 35 35 70 50
Trihydrate d
I/Io
6.33 5.68 5.58 5.28 5.20 5.01 4.81 4.08 3.75 3.52
55 55 100 80 75 70 70 70 70 75
-Hydrate? -AnhydrousWave Wave no., Charac- no., Characom.-' teristic ern.-' teristic
3390 1626 1613 1515 1381 1266 1026 943 845 803 749 742
1613 1515 1381 1266 1026 943 845 803 749 742
final product. From 525 to 550" the final product was p-U03. Amorphous anhydrous uranyl hydroxynitrate was found as the only intermediate. The equations in Table X summarize the reactions of uranyl nitrate dihydrate at atmospheric pressure.
TABLEX DECOMPOSITION REACTIONS URANYL NITRATE DIHYDRATE AT ATMOSPHERIC PRESSURE 250' UOz(NOs)z.ZHzO -* UOz(N0s)z 2Hz0 UOz(N0s)t -+ yUOo 2NOz '/zOz U02(NOs)z*2HzO-* UOz(0H)NOa HzO HNOs UOz(0H)NOS + ~ U O I HNOI 300-450' UOt(NOs)~.2HzO + UOz(NOJz 2H20 UOz(N0s)a -* r-UOa 2x02 '/zOt UOZ(NOS)Z*~HIO + UOs(y 8) 2N02 '/z500' 02 2Hz0 2N02 '/eOa 4525-550' UOz(NO~)z-2HzO 8-UOs 2Hz0
+ +
+
+
bands for the tetrahydrate as compared to the trihydrate. The trihvdrate, anhydrous uranyl hvdroxvnitrate and a-ura&um ikioxide monohydrate were identified by infrared and X-ray diffraction as intermediates in the den.itration of uranyl nitrate dihydrate under vacuum. Equations that account for these reactions are shown in Table IX.
+
+
-+
+
+
+ + + + + + +
The results presented show that by proper choice of conditions the thermal decomposition of uranyl TABLEIX nitrate dihydrate will produce a product that is DECOMPOSIT[ON REACTIONS-URANYL NITRATEDIHYDRATEprimarily @-UO,,7-UOa or amorphous uranium trioxide. The &phase is produced by the rapid UNDER VACUUM decomposition of the dihydrate at temperatures 250' Primary Reactions above 500" a t atmospheric pressure, 7-phase over 'I~O~(NC)~)Z*~HZO -* UOz(N08)2 2Hz0 the temperature range of 250-450". DecompositJ02(N08)o UOs(A 2N02 l/~Ot tion of the dihydrate in the temperature range of Secondary Reactions 300-450° under vacuum always produces amorphous uranium trioxide. Above 500" the product is UOz(NOs)z.ZHzO UOs(0H)NOa HzO contaminated with U308. Below 300' some aHNOu 'I~02(OE)NOa t 3Hz0 UOZ(OH)NOS.~HZO phase uranium trioxide is formed. UO~(OFI]NO~~3HpO + ~u-UOs.Hz0 2Hn0 Acknowledgment.-The authors are indebted to "02. Drs. C. H. Ice and R. C. Milham for technical ada-uO~*&o+ UOs(A) &O vice, Drs. W. R. Cornman and J. W. Nehls for Xray diffraction and infrared absorption patterns, 300400' 'I102(NOa)r*2HzO+ UOz(N0a)t 2He0 and Mrs. B. S. Russel' for chemical analyses. UOz(N0s)z + UOa(A) 2N02 '/zOz
-
+ +
+
(Y)
-+
+
-
+ +
+
+
+
+
+ +
DIFFUSION TO A PLANE WITH LANGMUIRIAN ADSORPTION BY W. H. REINMUTH Department of Chemistry of Columbia University, New York, N. Y. Received September 13, 1960
A theoretical treatment is given of semi-infinite linear diffusion to a stationary plane with Langmuirian adsorption at the boundary. The frmtion of the surface covered is a function of two variables C*/a and aZDt/I,Z where C* is the solution concentration of surfactant, a is the solution concentration which would correspond to half coverage, rmis the surface concentration of surfactant at full coverage, D, the diffusion coefficient of the surfactant and t, time. Comparisons of exact the0 with approximate treatments are given and discussed. A formal solution to the same problem at an expanding plane is inxded.
Many workers have concerned themselves with the effectsof surfaceactive agents on electrode procewes.' Delahay and Trachtenberg2 in particular Reilley, .*Re (1) For a review m: w, H. Reinmuth in c. cent Advancea in Analytical Chemistry and Inatrumentation." Interscience Publiehers, h a . , New York, N. Y.. 1980. (2) P. Delahay and I. Trachtenberg, J . Am. Chem. Soo., 79, 2365 (1957); 80, 2094 (1958).
emphasized the influence of the rate of diffusion of these species on the observed results. Delahay and Fikea later attempted to solve the differential equations describing semi-infinite linear diffusion to a plane boundary with Langmuirian adsorption by an unstated method with the aid of an electronic (3) P. Delahay and C. T. Eke, ibid., 80, 2628 (1958).
W. H. REINMUTH
474
Vol. 65
It is also assumed that the adsorbed species does not react a t the boundary or pass through it so that the material flux a t the boundary equals the rate of adsorption drt/dt = D(dC/dx)z=o (3 1 Equations 1, 2 and 3 can be combined to give an integral equation for rt in the form
- rt) = C* - ( r D ) - ' h
art/(r,
It is convenient to define three dimensionless parameters p =
rt/rm;$
= C*/a;e = 4 ~ a 2 D t / T ~ ~( 5 )
Equation 4 can then be written in the form d(1
- EC) =
$
- 2 Joe
(dddp)(O -
p)-'/z
dp
(6)
Because equation 6 contains only two parameters in addition to p, it is apparent that its solution must be of the form P = f($,@
(7)
However, Delahay and Fike3 present theoretical results in the form P = f($,t,D/rm)
0
1.0
3.0
2.0
el/2. Fig. 1.-Fractional coverage of the boundary a8 a function of time for representative values of $. re is surface concentration a t equilibrium.
computer. However, the form of their solutions, as we mentioned,l indicated that their method was invalid. Testa and Reinmuth4 also made a numerical solution. More recently, in the course of other work, we have applied a method which gives solutions in series form. The purpose of the present paper is to describe that method and some of the results achieved by its application. For the sake of completeness the formal solution at an expanding plane is included, although no calculations have been performed for this case.
Theory It is assumed that the adsorbing species obeys Fick's laws of linear diffusion and has a constant diffusion coefficient, D; further, that the initial concentration of the species in solution, C*, is homogeneous. It can then be shown readily from Duhamel's theorem5 that at the electrode surface C = C* - (D/T)'/z
sot
(dC/d~)(t-
T)-'/zdT
(I)
where C is the solution concentration at the boundary a t any time t; x is the distance from the boundary; and 7 is an integration variable. It is assumed that at the boundary Langmuir's isotherm is obeyed in the form Tt/ rm= C/(C a) (21 where rt is the instantaneous surface concentration corresponding tfo solution concentration C, a is the isotherm constant, and rmis the surface concentration corresponding to complete coverage.
+
W.H. Reinmuth. unpublished work. H.S. Carlslew and J. C. Jaeger, "Conduction of Heat in Solido,"
(4) A. C. Testa end
(5)
Oxford University Press, Louden, 1947,p. 67.
(8)
The functional dependence of their results must be incorrect a priori because no combination of their stated variables can yield a dimensionless p . Since these authors give no indication of their method of solution, it is unclear whether the difficulty could be resolved by redefinition of their variables. The results given by the same authors for the expanding plane electrode are invalid for the same reason. The method of solution of eciuation 6 is given - in Appendix I. In principIe, the solution for an expanding plane electrode, the commonly used approximation of the dropping mercury electrode, follows the same lines as that for the stationary plane, with the modification that a different form of Fick's law must be applied. The details for this case are given in Appendix 11. ~~
Results and Discussion The series solution converges rapidly for small values of 8. Results for three representative values of $ are given in Fig. 1. They were all calculated with the aid of a ten term expansion. As might be expected, the rate of attainment of equilibrium is more rapid as $, the motivating force to adsorption, becomes larger. Results computed by the present method were within calculational error of those obtained by numerical solution of equation 6.4 The fractional coverage of the surface a t equilibrium depends on $. For given equilibrium coverage, however, the time required to reach an appreciable fraction of that coverage is inrersely proportional to the bulk solution concentration of the surfactant. For example, assuming $ = 1 ie., (half coverage at equilibrium), 17A.2 as the area of an cni.2/sec. as the adsorbed molecule, and 5 X diffusion coefficient, the times required t o reach quarter coverage for 10-8, and 104M solutions are 2 X lo+, 2 , 2 x 102, and 2 X lo4sec.,
March, 1961
DIFFUSION TO
A
PLANE WITH LANGMUIRIAN ADSORPTION
respectively. The extreme times required a t low concentrations are determined largely by the rate of diffusion to the electrode rather than by the adsorption equilibrium. This may be shown by assuming the same molecular area and diffusion coefficient as above but assuming a = 0. The time required for quarter coverage with a 10-6M solution is still 1 x 104 set. It is of interest to compare the results of exact theory with those obtained by approximate methods. When equilibrium strongly favors the adsorbed form, the solution concentration of the surfactant at the boundary is reduced nearly to zero until the surface is completely coated. By substituting for equation 2 the condition C+O)
= 0
475
1.0
/
i
k
< 0.5
' ; . k
(9)
and solving the resulting set of equations 3, 4, 9 it can be shown that in this limiting case =
$e1/2jT
(10)
This is the same as the first term in the expansion for the case in which Langmuir's isotherm is obeyed. In Fig. 2, curve A exact theory (solid line) is compared with the approximation of equation 10 (dashed line) for the condition I) = 10. The agreeI ment is excellent up to about half-coverage. The 0 approximation becomes even better as $ becomes 0 0.5 1.o larger, and, even for smaller #, indicates the initial (e$) 1'2. slope of the coverage us. time relation. Fig. 2.-Comparison of exact theory with approximate For small values of I)Langmuir's isotherm can be methods. Curve A: solid line, exact theory; dashed line, replaced b,y a linear approximation complete adsorption; both for $ = 10. Curve B: solid ~
FITrn = C/a
(11)
line, exact theory; dashed line, linear isotherm; both for
$ = 0.1.
The solution to the problem under this condition has been given by Delahay and Trachtenberg.z of the latter on the surface. If desorption is not In the notation of the present work their result is onmnloto srt srooo&hlo nntontialc thic Pnn ho talcon p =
p - $exp(8/4T)erfc(8/4~)'/~
(12)
I n Fig. 2 curve B exact theory (solid line) is compared with this approximation (dashed line) for the cortdition = 0.1. Again the agreement is excellent at short times and discrepancy becomes appreciable only at about 50% of equilibrium coverage. For smaller $ the approximation is improved. It should be noted that there is some ambiguity involved in the choice of constants when approximating Langmuirian adsorption with a linear isotherm. In the present case, for example, with $ chosen equal to 0.1 for each isotherm the predicted equilibriurn coverage becomes 0.091 for Langmuirian adsorption but 0.1 for linear adsorption. As Fig. 2 indicates this leads to discrepancies between the two a t long times. If the I)'s are so chosen that the limiting values are equal, then the discrepancy shows up at short times. Appropriate choice of parameters for approximation is therefore dictated by where the error is of lesser importance. Methods for experimentally achieving the mathematically assumed conditions merit some discussion. It would be a difficult operation a t best to introduce an initially clean surface into a solution of surfactant without introducing convection. Hawever, if the surface is a conductor, the same effect can often be achieved by applying a large anodic or cathodic potential to the electrode until time zero. Under such conditions desorption occurs in the presence of electrolytes due to preferential adsorption
+
of a0 for the expansion (see Appendix I). The mathematical treatment given here can be readily adapted to cases in which the adsorbate undergoes chemical or electrochemical reaction at the boundary or the adsorption process itself is slow. Preliminary results for these cases6indicate that the convergence of the series is much less rapid, however. Detailed discussion will be given elsewhere. Appendix I Stationary Plane.-Equation 6 of the text can be solved by assuming the solution to be of the form m
I.1
=
(A1 )
aI,gi/z= 0
From this it follows that m
ja,e"z
dpjde = (2e)-l 1
Substituting equations A1 and A2 into equation 6 and defining the new integration variable, v = p / e , gives =
?/(I
-
5 lo1 -
- e-va
jalei/2
1
~ I / ~ - l ( l v ) - l / z dv
(6) P. Levy and
W.H.Reinrnuth, unpublished work.
(A3)
476
ABBASLABBAUF AND FREDERICK D. ROSSINI
Combining equations A8, A9 and A10 with equation 2 of the text yields
But so1
(1
ZJ~/’-’
- u)-’/z
dv =
(A4)
B(jj2, I/))
= (1
where B is the ,B function. Therefore
evqi
+ $1
=
$evz
0
- (1 -
Vol. 65
-
- Jl [7(ap/a~)+ 2 p / ~ 1 ~ 2(an - S)-I/* da
..
T)$
(All) j ~ i ~ ~ / * B1/z)( j / i ,
where = (2ism/r2y9
ff
(A51
(A12)
The solution of equation A l l is assumed to be of The coefficients uj can be readily evaluated by equating terms with equal powers of 0 in equation the form A5. The first coefficient, a,, is determined by the initial state of the system and is zero if there is no adsorption at time zero. In this case the second coefficient is Substitution into equation A1 1 and simplification gives a1 = J./s (A61 m
and following coefficients are given by the recurrence formula j a j E ( m 1/21 =
+
-Uj-1
j- 1
+
OD
iUiaj-iB(i/t,
1/z)
(A71
(2 1
2
3~/2)bip’B[(st+4)/14~1/11 (A141
where p = &14, and it is assumed that bo = 0. Appendix I1 Expanding Plane.-For an expanding plane the Equating coefficients of equal powers of p allows the calculation of the b’s. analog of equation 1 of the text is C = C*
- (3D/7x)’/l
’(dC/bh)(y -
s o
7)-’/¶ dr
bi = 2 $ / 7 ~ (AS)
where = p r a ,h =
xt2ir
(A91
and following b’s are given by the recurrence formula bi(2
+ 3j/2)B[Oj+4)/~
The analog of equation 3 of the text is dr/dt = D(aC/dA)y’n
- 2r/3y8”
(AX)
-bj-1
1/11
j-1
(2 (A10)
2
+
+ 3i/2)bibi-t~[(ai+*,/lr,
1/81
HEATS OF COMBUSTION, FORMATION, AND HYDROGENATION OF 14 SELECTED CYCLOMONO~LEFIN HYDROCARBONS’ BY ABBASLABBAUF AND FREDERICK D. ROSSINI* Chemical and Petroleum Reaearch Laboratory, CarnegiG Institute of Technolow, Piltsburgh 13, Pennsylvania Recswsd
&p(6dW
10, 1960
Memrementa were made of the heat of combustion, in the liquid state at 25O, of 14 selected cyclomono6lefin hydrocarbons, with 5 to 8 carbon atoms per molecule. From these and appropriate other data were calculated values of standard ap.proPmte, for the liquid state at 25”. The relation between heats of formation, hydrogenation and isomerization energy content and molecular structure of these oompoun& m hscuaeed. Valuee were calculated for the heat of formation, for both the liquid and gaseous statm at for all the 1-normal alkyl cyclopentenes and all the 1-normal alkylcyclohexencs.
hydrocarbons. Accordingly, the present investigaI. Introduction Preceding have provided essentially tion was carried out to obtain experimental data on complete data and information leading to the cal- 14 selected cyclomonooleh hydrocarbons and to culation of values of heats of formation, combustion, analyze the data in terms of the relation of energy hydrogenation and isomerization as appropriate, content to molecular structure. for all aliphatic monoolefin hydrocarbons. BeII. Apparatus and Experimental Procedures cause of the scientific and technical importance of The experimentalvalues of this investigation are bmed on the cyclomonoolefin hydrocarbons, and the lack of the absolute joule aa the unit of energy. Conversion to the data on them, it was desired that similar data and defined thermochemicalcalorie is made by using the relation 1 calorie = 4.184 (exactly) joules. For internal consistency information should be obtained on this class of with other investigations from the Laboratory, the molec(1) This investigation waa supported in part by a grant from the National Science Foundation. Submitted by Abbaa Labbauf in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Chemistry at the Carnegie Institute of Technology. (2) University of Notre Dame, Notre Dame, Indiana. (3) Sr. M. C. Loefaer and F. D. Rossini, J . Phys. Cham., 64, 1530
(1960).
(4) H. F. Bartolo and F. D. Rossini. ibid., 64, 1685 (1960). (5) J. D. Rockenfeller and F. D. Rosaini, ibid., 65, 267,(1961).
ular weight of carbon dioxide was taken as 4 . 0 1 0 g./mole. In this investigation, the chemical and calorimetric apparatus and procedure were the same as described by Browne and Rossini,E except that a new desi of Combustion bomb was used. The new bomb, made of g u m , is of the inverted type,.= shown in Fig. 1. The rise of temperature in each expenment was near 2 O , with the final temperature being (6) C. C . Browne and F. D. Roasini, ibid.. 64,727 (1960).
