March, 19GO The entropy of solution a t constant pressure is (S2 - s : ) p = R ( b In a z / b In r P )( b~In z , / b In T)sat
Although the “Henry’s law correction,” a In &/a In zl is nearly unity in the poorer solvents, it is distinctly less in the better solvents. For solutions in C& and CCl,, Walkley and Hildebrand4 determined directly the variation of the partial vapor pressure of iodine with composition. For the other solutions we havc evaluated this factor from the solubility data by the method described by Shinoda and Hildebrand.b Figure 1 shows values of -R In x2 at 25” plotted against the entropy of solution of solid iodine a t both constant pressure and constant volume for representative iwlutions relatively free from complicating factom, such as polarity, donor-acceptor interaction (slight in the case of CHBr3) and geometrical asymmetry. The slope of the line for the entropy of imlution a t constant volume is 1.10, only slightly greater than 1.00, which it would be if my originztl definition of a regular solution, as one in which the entropy of solution a t constant volume is -R hz z2,were strictly valid. The Line extrapolates to 9.0 e.u. for the entropy of fusion of iodine, a t 25”, not far from the value, 8.0, extrapolated from the heat of fusion at the melting point and the heat capacities of the solid and liquid phases. I wish to acknowledge the support of the Atomic Energy Commission in obtaining many of the data upon which these findings are based. (4) J. Walkley anLd J. H. Nildebrand. THIBJOURNAL,63, 1174
NOTES
37 I
the “steady-state approximation” (method I) wc get the relationships (eq. 15 and 14 in ref. 1)
k’rom (1) ro/a is calculated and from (2) Cl/CB is calculated. Then the concentration distribution is calculated from the steady-state logarithmic relationship between C/CB and r/ro in the range (c = c1, T = a) to (c = 0, T = T O ) . For evaluating experiments we have to calculate Dt/a2 for given values of A / r a 2 , S/CB and ro/a or alternatively C ~ / C B by using (2), and a stepwise procedure in calculating the integral in (1). For the method introduced by the present author (method 11) we get the correspondiiig equations (eq. 28 and 20 in ref. 2).
Equations 3 and 4 are applied just as (1) and ( 2 ) and the concentration calculated from a second degree relationships between C/Q and r/ro in the range previously indicated. Thus the calculations are not more troublesome than for method I. The absence of logarithmic factors simplifies the work (1959). (5) K. Shinoda and J. H. Hildebrand, ibid., 61, 789 (1957). in method I1 and the integral (3) can be exactly evaluated by introducing (4). However in practice it is often more convenient to use the differential DIFFUSION ’WITH RAPID IRREVERSIBLE form of (3) and (1) and calculate D for a series of I~OBILIZATION time intervals t + 1 At. It is of course important to compare the accuracy BY BEBTILO ~ o e s s o ~ of the two methods. It has been indicated2 that Ths Swedish Inslilule for Teztile Research. Gothenburg. Sweden method I1 gives a satisfactory accuracy as the Received October 18, 1969 “affinit.y” s/cB is large. This seems also to be A critical discussion of methods for calculating valid for method I as specially demonstrated by diffusion coupled with rapid irreversible immobiliza- Crank (Fig. 49 for the case Dt/a2 = 9/16, S/CB = tion has been given by Crank.‘ By a convenient 10, A / s a 2 = 21. It was confirmed by the present numerical procedure he calculated and tabulated author that method I1 was just as good as method I values of diffusion constants for some combinations at the larger values of S/CB corresponding to dyeing of “input” parameter values. He also examined experiments. But Crank also has compared the incidentally the conditions under which the “steady- results from method I with his “exact” calculations state approximation” method is useful for practical for the case (Fig. 49 Dt/a2 = 9/16, S/Q = 2, A / r a 2 = 5. We have made the corresponding evaluations of such problems. Another appi.ordmation method has been worked evaluation using our method I1 and the results obout by the present author.2 Crank has referred to tained are compared in Table I. this method but not discussed its applicability. TABLE I We have found that this method is more useful than “Exact” value’ Method I’ Method 111 the steady-state method in some cases, as will be ro/a 0.317 0.222 0.314 demonstrated here. Fractional total concn. at We consider dausion coupled with immobilizat / a = 0.314 0.664 ,714 .667 tion by absorption or rapid reaction into a circular r / a = .50 ,737 ,778 ,737 cylinder (fiber) from a limited bath. This process r / a = .75 .806 ,834 ,810 is governed by the parameters Dt/a2 (reduced dift / a = 1.00 .856 .874 ,859 fusion constanl,), A/lru” (bath volume/fiber volApparently the results from method I1 are in ume) and S/QL (relative concentration immobilized), the symbols in ref. 1 everywhere used. By excellent agreement with the “exact” results, while method I differs significantly, aa also pointed out by (1) F. Crank, Tmnu. Paradoy Soc., I S , 1083 (1957). Crank. (2) B. Olofason, J . T e d . Inut., 4T, T464 (1956).
+
372
Vol. 64
XOTEB
TABLEI1 VALUESOF Dud4 CALCULATED FOR AN INFINITE BATH E = “Exact” values, I = calculations by method I, I1 = calculations by method I1 7 . -
a / “I
E
Plane sheet I
1
-
--------Cylinder-
E
I
zo/a = 0.75
C 8
2 5 10
I1
0.041 .072 .167 .323
rda
0.031 .063 .156 .313
II/I =
m
0.039 .070 .164 ,320
0.038
i.aw
0.162 .289 ,666 1.290
0.138 ,241 ,548 1.049
0.366 0.653 1.496 2.907
m
0.035 ,061 ,141 .273
0.274 0.475 1.057 2.036
0.191 0.382 0.955 1,911 II/I = 1.104
0.026 0.033 .052 .059 ,130 .139 .260 .272 II/I = 1.019 rda
0.118 ,205 ,460 .887
r d a = 0.25
0.281 0.352 0.564 0.(id3 1.406 1.477 2.813 2.883 II/I = 1.OOo
I1
re/a = 0 . 7 5
0.101 0.129 ,202 .233 ,504 .545 1.009 1.066 II/I = 1.033
m/a = 0.25
1 2 5 10
Sphere-I
E
d a = 0.50
0.125 0.156 .250 .281 ,656 .625 1.250 1.281 II/I = 1.Ooo
m
---
= 0.75
0.029 0.036 .057 .064 .142 .150 .285 .293 II/I = 1.006
.067 .152 .295
za/a = 0.50
1 2 5 10
I1
0.50
=
0.083 0.111 ,167 ,202 ,417 .475 ,833 ,931 II/I = 1.094 r d a = 0.25
0.258 0.469 1.102 2.156
0.219 ,370 ,828 1.543
0.141 0.214 ,281 ,394 ,703 ,934 1.406 1.835 II/I = 1.281
zs/a = 0
1 2 5 10
0.650 1.158 2 660 5.167
0.500 1 .000 2.500 5.000 II/I = 1
m
0.625 1.125 2.625 5 125 000
To get a general idea about the applicability of the two apprloximation methods the conditions applied in their evaluation should be considered. For method I the concentration-distance (CT) curve at some time t is wholly fixed by the two parameters c = c1 (at the boundary r = a) and r = TO (corresponding to c = 0 ) , and these parameters are determined to fit the boundary conditions of diffusional flow. For method I1 the c-7 curve gets another degree of freedom by the assumption c = a0 ad u2r2. To get a fixed curve another boundary condition must be applied and this is really 1,he integrated form of the condition a t r = a. These two forms of the same boundary condition (cf. (10) and (12) in ref. 2) are not equivalent in the approximation procedure, as one of them is deduced from the other by applying the original (time-dependent) diffusion equation. However, in applying the integrated boundary condition the c -r curve is approximated with a straight line and thuci the accuracy of the calculations decreases, as the real er curve increases in curvature at small s-values. For a plane sheet method I really suggests a linear e x curve (21).l Thus we might suppose that the second degree curve used for calculation of D by method I1 provides a greater accuracy. For this case eq. 3 is still valid, if we substitute (Crank’s symbols)
+ +
r = z,A/27a = 1
(5)
but eq. 4 murk be exchanged with
!-
q1-2)
c1 =
a
CH
- + 2 ( 11 - : ) a
CH
(6)
To demonstratmethe accuracy of the approximation methods, we have made some calculations for an infinite bath (Z/a + a). From (3), (5) and (6) we get cJcB -+ 1 and Z/a.d(cl/cB) --t ( s / ~ 1/2) d(zo/a) and finally
+
$! (i+ a) =
x
(z (7)-
1)*
(7)
Calculations by method (11) from are compared with calculations by method (I) from (24)’ and “exact” calculations by Crank (Table II’) in Table 11. Evidently our assumption on the accuracy of the approximation is verified. For the cylinder case the increasing curvature of the CT curve with decreasing r is observed in the steady-state approximation (8). This influences the accuracy of method I1 and it already was proposed earlier2 to restrict calculations to ro/a > 0.30 (deeper penetration also corresponds to very small changes in external bath concentration, if the bath ratio is not too small). In applying the “integrated boundary condition” with the linear er curve in method 11, however, the error because of the curvature is compensated to an appreciable extent, as the contribution of the volume element at distance r to the total sorbed quantity is (c s) 2 ?r r dr, thus decreasing in proportion to r . In Table I1 we have made calculations in analogy to those for the plane sheet and the results give some idea about the decreasing accuracy as ro/a decreases. For the sphere the same sort of discussion is applicable. There is still larger increase of curvature with decreasing r , cj. (17),l and in spite of the compensation because of the rapidly decreasing volume elements 4+dr the error of method I1 increases
’
+
March, 19GO
NOTES
more rapidly with decreasing ro/a than for the cylinder as demonstrated in Table I1 (now A in eq. 3 is subst#ituted with V / 2 a ) . Also from the formal point of view method I is less troublesome than method 11. We conclude that the approximation method suggested2 is very useful for a plane sheet to get a great accuracy. It has formal advantages to the steady-state approximation method for a cylinder and also a great,er accuracy in the range indicated here. But it lhas no formal advantage to the steady-state aplproximation method for a sphere and a greater accuracy in a rather narrow range.
Subsequent examination of the data failed to reveal any significant effectof over- or under-cycling or of unequal sweep rates, or of the degree of resolution upon the magnitude or direction of the effect. It should be emphasized that optimal resolution wtts not attained, nor waa bad resolution tolerated. “Ringing” waa never observed for any of the image lines, or for the Cla satellites of (CH,),Si, but was attained a t the ClS lines of the other compounds in about half of the measurements. The widths of the image signals a t half-maximum were 0.70 to 0.85 c./sec. in nearly all cases, while the CI3 satellites (when not ringing) were, if anything, narrower than the corresponding C12 images, except for (CH& Si, for which they averaged ca. 0.12 c./sec. wider. The effect of sweep direction is fairly large, the C13 isotope effect appearing to be larger by 0.0022p.p.m. when the sweep is toward decreasing field, or smaller by the sanie amount when the opposite sweep is used exclusively. Although such error was avoided, it would not have concealed the isotope effect.
PROTON NlJCLEAR SPIN RESONANCE SPECTROSCOPY. XI. A CARBON-13 ISOTOPE EFFECT BY GEORGE VANDYKETIERS Contrtbulion Yo. 166 from the Central Research D e p l . , A4~nncsota Mtning .E. Mfg. Co , Sl. Paul 19, Mrnnesotu Received October 16, 1969
Recently n nuclear spin resonance (n.s.r.) “isotope effect” of CI3upon attached fluorine atoms has been discovered.’ The rather unexpectedly large shifts were always found to be in the direction corresponding to greater shielding by CL3than by C12. Though no shift was found for protons, the much smaller effect anticipated by analogy with the deuterium shifts2would not have been detected. As both the sign and the relative magnitude of such a C13 effect might prove theoretically interpretable, a morc elaborate experimental procedure has been used in the present work. Experimental The compounds studied were examined neat in the customary 5 mm. 0.d. aample tubes, from whirh air had been swept by means of a brisk stream of bubbles of prepurified nitrogen; however, air-saturated CHClj and ( CH3),Si were found to give the same results. The C13 isotopic isomers were present at their natural abnndances. The n.8.r. spectrometer and measurement techniques have been d e ~ c r i b e d . ~For the present study separate reference compounds were not employed, the exceedingly strong sharp signal from the normal (CI2) compound being used instead; “image” lines (akio called “side-bands”) are readily produced from it by audio-oscillator modulation of the magnetic field. When Cl3 ia present, the proton signal is split by it into a doublet, the. coupling constant bring ca. 100 to 250 c./sec. The positions of these two weak CL3satellite lines are measured, sepaxately, rclative to the strong C1* central peak by use of the “image” lines.3 The small but reproducible difference found in earh case results from the isotopic shift, aa otherwise the high- and low-field C13 romponents would be equally spared from the C12central line. Errors random in nature were counteracted by multiple repetition of measurements; all data have been used and weighted e ually. Measurements were made over a threeweek perio8. A t each session six sweeps were run on each of the two C13 lineal for each of the compounds studied. In addition to the routine alternation of sweep direction,a which virtually eliminaki errors due to differential saturation or to “ringing” of the signals, in most rases rare waa taken in the magnet cyrling to obtain a “flat” field and hcnre very symmetrical peaks for both dirertions of sweep; ciweep rates also were controlled to be equal (within 10’70) in both directions. (1) P. C . Lauterbur, private communication: I am indeed grateful for this advance information, without which it is unlikely that the preaent work would have been begun. (2) G . V. D. Tiers, J . Am. Chcm. Soc., 79, 5585 (1957). J . Chcm. Phys.. 49, 963 (1958). (3) G. V. D. Tiers, ’ P H I 8 JOURNAL, 60, 1151 (1958).
373
TABLE I THE EXCESSN.S.R. SHIELDING,A7, PRODUCED BY 0 1 , RELATIVE TO C1*, IN SEVERAL COMPOUNDS Compound
No. of meas.
A T , p.p.m.0 (Cia-Cl*)
J(C”H), c/sb
Shielding value, I C
(CH3)rSi 18 f0.0042 118.20 1O.OOO CHjI 18 .0012 151.17 7.843 CHZClz 12 ,0042 178.24 4.720 CHCla 18 ,0059 209.17 2.755 a Standard deviat,ion of the averaged value was ~k0.0012 p.p.m. in each case. Standard deviation of the averaged value was f0.09 r./sec. in each case. Measured in dilute solution in cc14, as described in ref. 3.
+ + +
Results and Discussion The results presented in Table I demonstrate a small but statistically significant excess shielding by CI3 in three molecules, namely, CHC13, CHBC12 and (CH3)$i. I n the case of CH31the shift is not large enough to be considered as established. The center of the doublet corresponding to protons attached directly to C13 is found a t shielding values higher by ca. 0.004 p.p.m. than the line due to the normal (CI2)compound. This effect is of the same sign but only about 1/44 as large as the corresponding effect upon fluorine.’ The same ratio, 1/40, has been observed for the deuterium isotope effect upon proton and fluorine shieldings.2 I t appears unrelated to the ratios found for the coupling constants in the same compounds, ca. 1/2 for J(C1a-H)/J(C13-F) and ca. 1/4 for J (D-H) /J (D-F) . The data in Table I seem to indicate a significant variability in the magnitude of the effect. The variation observed is not simply related either to the coupling constants, to the relative shieldings, or even to the number of substituents; however, the experimental uncertainty is such as not to warrant more detailed studies at this time. The conclusions reached above are of course entirely dependent upon the successful elimination of directed error in the measurements. Potential errors and the precautions taken have already been discussed in the Experimental section. There may well be further, unrecognized, sources of error; for example, treatment of the C13HC13 spectrum as an “AX” case rather than as an “AB” case‘ in fact must result in an apparent excess shielding by C13, even if there were no isotope effect. By (4) J. A . Pople. W. G. Schneider and H. J . Bernstein. “High-Resolution Nuclear Magnetic Resonance,” McCraw-Hill, Book C o . , Inc., N e w York, N . Y., 1969. pp. 118-123.
