NOTES
Dec., 1957 and has shown that the average time is twice the minimum. Since the l n e t i c relation between concentration and time is, in general, non-linear, we must average the concentration itself, not the time. The fraction of molecules having reaction times between 1 and t dt, Ft dt, is given by1
+
1669
quires a considerable calming length to develop, even for low Reynolds numbers, and it appears that the geometry of the system must be well known in order that one may even guess at the nature of the velocity profile.
where U is the conventional unit step function, and to is the minimum transit time. It should be noted that this function is independent of the size or shape of the reaction tube. Diffusion in the direction of flow is assumed negligible. For a firstorder reaction the concentration-time relation is (2)
ct = a e - k t t
which must be averaged with Ft
,
3
2 9.
Fig. 1.
= concn. a t time t co = initial concn. CI
61 = true rate constant
c = true average concn.
The usual procedure is t o calculate an apparent rate constant, k,, from the average time, t and the observed average concentration C (4)
= Coe-kat
What is ultimately desired is a relation between the true and apparent rate constants, which may be obtained from equations 1-4 1 + = -In 2 '$I/*
=
2.8
conventional half-time = In -, 2 Ei( -+IE kr
J: 7 ds
A similar equation for a second-order reaction with the concentration-time relation
kt '-1
- 2~~ + 2~~~In -
"1 (V) (l+,
- TO1n TO
=
CRYSTALLINE INTERMEDIATES I N THE HYDROLYTIC DEGRADATION OF CALCIUM POLYMETAPHOSPHATE BY EARLH. BROWN,JAMES R. LEHR,JAMES P. SMITH, WALTERE. BROWN AND ALVAW. FRAZIER
Diuiaion of Chemical Development, Tennessee Valley Authority Wilson Dam, Ala. Received June 84, 1967
is readily derived by the same procedure 1
An extensive search of the literature of kinetic flow systems was made in an effort t o determine the effect of velocity distribution. However, most of the work was done at low conversions where the correction is negligible. (In some cases effects of packing may be due to a change in velocity profile rather than a surface reaction.) I n summary, it appears that there is an appreciable correction to be made in flow systems when the velocity profile is parabolic, but the attainment of this distribution must be investigated for each system. This could be done by actual observation of the flow pattern with colored indicating streams.
(7)
e
coktia = 2
The magnitude of the correction may be seen in Fig. 1 where the per cent. error, ( k t / k a - l)lOO, is plotted against the ratio of the reaction time to the half-time, 8. The error is small for very short reaction times but rapidly becomes appreciable near the half-time. It is a frequent procedure in flow systems to use relatively more long reaction times at low temperatures and more short times at high temperatures. Thus an appreciable error may result in both the frequency factor and activation energy, both of which will be low. All of the foregoing arguments are based on the assumption of fully developed laminar flow and a parabolic velocity distribution. Laminar flow re-
Vitreous calcium polymetaphosphate, prepared by heating monocalcium phosphate, forms a viscous water-immiscible liquid when treated with water. Apparently a coacervate, the sirupy hydrated polymer, contains 30 to 60% [Ca(POa)z], and is analogous to hydrated magnesium polymetaphosphate.' A film of the coacervate dries reversibly in the open air to a glassy product containing about 18% water. Essentially the same product is formed upon addition of acetone or ethanol. Upon standing for several days in the absence of extra water, the viscous liquid becomes first a gel and then a mass of crystals-all one species (intermediate I)-which, if kept in the mother liquor, is converted to crystalline orthophosphate. With extra water, the gross degradation pattern is about the same except that another crystalline phase (intermediate TI) appears between intermediate I and orthophosphate. Also, intermediate I is con(1)
R. Pfanstiel and R. K.Iler, J . Am. Chrrn. rSoc., 78, 5510 (1956).
NOTES
1670
verted slowly to intermediate I1 upon standing in water. The optical properties and X-ray patterns of the intermediates differ from those of calcium phosphates heretofore reported. Solutions of the intermediates in 5% acetic acid yield pyrophosphate chromatograms. Intermediate I crystallizes as colorless spherulites with occasional projecting monocrystal zones suitable for some optical measurements. The refractive indexes vary among preparations, and are Q = 1.527-1.535, p 1.535-1.537, y = 1.552-1.557. The crystals are biaxial, probably (+). Angle 21' is large. Different preparations of intermediate 1 yield identical X-ray patterns notwithstanding the variations in refractive indexes. The spacings, d , and intensities, I, are shown in Table I. The chemical compositions of different preparations place the empirical formula within the limits Hl.2P207.2H20 to Ca1.5HP207.2H20.
Vol. 61
AN INHERENT DIFFICULTY I N SPECTROPHOTOMETRIC STUDIES OF ION-PAIRS BY STEPHEN R. COHEN Contribution from Metcalf Research Laboratory, Brown Univeraity, Providence, R . I . Recaived August 14, 1067
In the last few years a large number of studies have been made of the association of complex ions (usually cations) with oppositely charged i0ns.l Most of these studies have been carried out spectrophotometrically; upon mixing the complex cation and associating anion a new absorption band was produced in the visible or ultraviolet regions, or the ultraviolet absorption edge of the anion was displaced toward longer wave lengths. I n these cases it has been assumed that an electron was transferred from the anion of the pair to the complex cation, or a t least from the anion to a water molecule in the first hydration layer surrounding the cation (or vice versa). If true, then only the ionTABLE r pairs with the associating ions in close proximity X-RAY PATTERNS (Cu K a RADIATION;CAMERA DIAM., are detected. Even when, as with the association 14.32 CM.) of Ba++, Mg++, and K+, with Fe(CN)d-, the -Intermediate IIntermediate I1 change in absorption spectrum seems to be due to d, b. I a, A. I d, A. I polarization of the complex ion, only short range 10.7 VS 8.01 S 2.32 VW ion-pairs can be detected because of the decrease 6.74 VW 6.95 S 2.23 WM of the polarizing electric field with distance. I n 6.08 VW 5.48 vw 2.09 WM some cases, where the ions are associated by co4.66 W 5.20 2.00 w VW valent, or other more or less specific short range 3.58 M 4.47 vw 1.89 W "chemical" bonds, distant ion-pairs contribute 3.43 W 3.98 1.73 WM W little and may be ignored. I n others the bond is 3.14 VS 3.73 F 1.56 WM almost entirely due to electrostatic attraction be2.78 VW 3.42 1.54 W M tween the oppositely charged ions. With such 2.69 W MS 1.48 W 3.21 systems the Bjerrum-Fuoss theory predicts that if 2.30 W 3.10 MS 1.44 W they are multiply charged, two ions may be sep2.03 MS 1.40 W 3.04 M arated by a comparatively large distance and still 2.01 WM 2.95 1.38 W WM form an ion-pair. It follows that spectral measure1.83 VW 2.76 WM 1.36 W ments, and other methods of measuring association 174 vw 2.65 M 1 33 w which do not detect distant ion-pairs, will give 1.57 W W W 1.17 2.50 lower association constants for electrostatic ion1.53 WM 1.14 W pairs than conductivity and electromotive force 1.39 vw measurements which do. 1.34 VW The difference in equilibrium constants can be estimated following Bjerrum's derivation of the Intermediate I1 crystallizes as colorless euhedral triclinic prisms of parallelepipedon habit. The equations for ion-pairing.2 The association conusual forms are the (OlO), (001) and (201), modi- stant for an ion-pair is fied commonly by a small (hkl) form. Crystals K = (4~N/1000) 'r2dr e-mz2e3/DrkT (1) frequently are elongated along a and sometimes S a along b and c. The parallelepipedon profile where r is the distance between the centers of two angles are 120.5" (201 A OOl), 77.5" (on 201) and ions and D is the dielectric constant of the medium. 108" (201 A OlO), which corresponds to the y I n this treatment two oppositely charged ions beof X-ray. The crystals are biaxial (+) with 2V tween a, the distance of minimum approach, and (calcd.) = 68.6". The refractive indexes are a = q = -xlx2e2/2DkT, .the minimum value of the 1.540, 0 = 1.549, y = 1.568. The chemical com- integrand, are considered to form an ion-pair. position indicates an empirical formula of Ca2P207. In the usual derivation equation 1 is transformed to 2H20. K = -(4~hr/1000)(z,z~ea//okT)a&(b) (2) The powder X-ray pattern of intermediate I1 is shown in Table I. The lattice constants, as deter- where &(b) is defined by mine! by single-crystal X-ray diffraction, are a = 6.70 A., b = 7.38 A., c = 8.31 B., a = 85" 2', p = 102" 48' and y = 107" 23'. The unit cell contains 2(2Ca0.Pz05.2H20). The calculated density (2.(1) See S. R. Cohen and R. A. Plane, THISJOURNAL, 61, 1096 51 g./cc.) is in good agreement with the density (19571, for references to typical studies. (2.46 g./cc.) estimated from the indexes of refrac(2) N. Bjerrum, "Selected Papers," Einar Munkagaard, Copention by means of the Gladstone and Dale equation. hagen, 1949, p. 108.
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t
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i 1 1
t I
tI
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