COMMUNICATION TO THE EDITOR
Dee., 1958
1607
12.93. Hence the experimentally found molecular refraction confirms the isocyanate structure for this substance. Initial attempts to purify the crude product, from the reaction of (CH3)2S04and KOCN, by fractional distillation at atmospheric pressure, did not give satisfactory results. The main difficulty under these conditions is that extensive decomposition and polymerization occur.
The material can be purified, however, by distilling under a vacuum of 10-12 mm. a t room temperature. A capillary b.p. determination shows that a small quantity of some volatile material comes off at 29-30' and the main product boils a t 44-45' with decomposition. The CHINCO is difficult f o burn in a standard C and H combustion train and consistently gives low carbon values.
(2) The molecular refractions were calculated from atomic refractivities in: A. I. Vogel, "Textbook of Practical Organic Chemistry, 3d Ed., Longmans, Green & Co., New York, N. Y.,1956,p. 1036.
Calcd. for CHyNCO: C, 42.11; H, 5.26; N, 24.56. Found: C, 40.03, 40.23; H, 5.38, 5.24; N, 24.14, 24.32.
COMMUNICATION TO THE EDITOR THEORY OF T H E TRANSIENT STATE I N T H E ULTRACENTRIFUGE
Sir: Although Archibald has presented both exact' and approximate2treatments of the ultracentrifuge differential equation
ac D - - w2srC = 0,r ar
where p =
An =
r1, r2
(r - r l ) / H ;
T
=
(w2s?/H)t; a = w2s?H/2D
and =
2
[I
+
-
l ) l u n ( ~ eXp(-ap)C(p,O)dp )
(C)
(3)
recent investigator~~.~,5 have preferred to adapt The perturbation treatment has been carried to the centrifuge the more manageable solution through first order for un and second order for En, of Mason and Weaver6 obtained for the case of corrections to the latter vanishing in first order; gravity sedimentation from h2€,(2) .. .; U n = U,(O) + Xu,(" + . . .; e, = En(0)
+
A,
=
An(0)
The results are bC
D dX - - KC
= 0,x = XI,52
(G)
Un(W =
B , -I/
z[f,(k)(p)
B,
= 1
cos
m p
+ hA,(I) + . . .
+ (a/n.)gn'k'(p)
+ (a/n?r)2;f,@) = g,(O)
+
(4)
sin nrp]
= 1
I n order to determine the validity of this proce- fn(') = Bn[ap(l - p) - ( p - l / 2 ) ] - a/4; g,(l) = dure the centrifuge equation (C) has been solved Bn [ l / a ( P - 1/2)1 - a/4 by a perturbation treatment7 which in zero order ento) = B , ( n ~ ) ~ / 2 a en(1) ; = 0 yields the Mason-Weaver result. The findings en(2)/cn(0) = 3(n?r)-'[l (az/9)(1 15/n2?r2)] (5) are. presented in this preliminary report. If X is defined by the relation For the usual case of interest, C(p, 0) = Coand
+
+
- rI)/(rz + r1) = H/2? (1) B.A.(0)/Co then equations (C) for X = 0 (a2?= const.) have 4A,(')/An(") the same form as equations (G) and it is possible X = (rz
to obtain the transient concentration distribution in the ultracentrifuge as a power series expansion in X. I n the usual ultracentrifuge Y = 5 em. and H = 1 cm. so that X = 0.1 and the series converges rapidly. The form of the solution is (1) W. J. Archibald, Phys. Rev., 6 4 , 371 (1938); Ann. N . Y . Acad. Sci., 43, Art. 5, 211 (1942). (2) W. J . Archibald, J . A p p l . Phus., 18,362 (1947). (3) D.A. Yphantis, and D. F. Waugh, THISJOURNAL, 60, 623,630 (19563. (4) R. A. Pasternak, G . M . Nazarian, and J. R. Vinograd, Nature, 179, 92 (1957). (5) K. E. Van Holde, and R. L. Baldwin, THISJOURNAL, 62, 734
(1958).
M.Mason and W. Weaver, Phys. Rev., 23,412 (1924). (7) G. M. Nazarian, doctoral dissertation, Department of Chemis-
(6)
try, California Institute of Technology, 1957.
= =
+
4B,-%a[l ( - l ) n + l exp(-a)]/n2?r2 a[32/n2?r2Bn- 11 - 2[3 [tanh CY/^) ] (6)
The availability of the extremely accurate formula for en provides the basis for an experimental method of determining diffusion constants since for a fixed location in the cell, we have from (2) C ( t ) = C,,
+ KI exp(-ht) + KZexp(--k2t) + . . .
-
(7)
where kn = 2 a ( D / H 2 ) e n . For a of order of magnin2 so that kp = 4k1, k3 = 9kl. tude unity, En Also, if we choose the fixed location to be in the vicinity of p = 1/4 this makes uz = 0 whereas u1 is close to its maximum value giving iKzl
