COMMUNICATIONS TO THE EDITOR
Sept., 1963
of 2.3 instead of 3.15 as predicted by the0ry.l The problem of how to cope with the errors engendered by the square wave aissumption is the subject of this communication. The nonlinear differential equation describing the rate of change of concentrations of radicals C, being generated a t the rate / ( t ) and terminated bimolecularly a t a rate ktC2, is
1943
0.70
d 0.65 ,%
‘ /I
0.60
0.55
This equation can be rendered dimensionless by substitution of R’= C - = - radical concn. with pulsed radiation CB steady state radical concn. also R I = -R = - reaction rate with pulsed radiation R S steady state reaction rate
I’(t) =
--
Io
=
initiation rate steady state initiation
t ’ = -t _ T
T’
=
total time radical lifetime
T == period of radiation radical lifetime
7
where T = radical lifetime = (kJo)-1/2 Substituting the above and transforming eq. 1 into a difference equation, we obtain for the time interval [(n 1) - nl
+
or =
R’,
+ [I* - R’n’] At
(3) where At = T ’ / N ; n = 0, 1, 2 , ...N ; RTo- R’N; and N = total number of intervals per period. Equation 2 has been solved on an IBM 1620 using Euler’s method. TIVQcases were considered: (1) A square wave or variable period T’, a minimum value of 0, a maximum value of 1, a “light period’’ of T / 2 , and an average value of l/z over a period; ( 2 ) a sinusoid of the same period, maximum, minimum, and average value over a period of ‘/z I‘(t) = ‘/z - ‘/z cos (2nt‘)lT‘. Figure 1 dernonstra,tes the widely divergent results obtained for sine and square wave driving forces of equal total energy input. The magnitude of the differences in radical lifetimes varies with the average ratio of R/Rs over a period, R’; in the region of 2’.0.64 it is as much as an order of magnitude (the horizontal displacement of the two curves). It is, therefore, apparent that large errors are generated when square waves with round corners are used.se2 The authors are currently generalizing the program to handle kinetic schemes of the form dC - = I ( t ) - kC“ dt R‘n+1
(1) J. A. Ghormley, Radiation Res., 6, 247 (1956); see also A. 0. Allen, “Radiation Chemistry of Water,” D. Van Nostrand Go., Inc., Princeton, N. J., 1961. (2) E. J. Ilrtrt, a n d M. S. Matheson, Dzscussions Faraday Soc., 12, 169 (1962).
0.50 1
10
100
1000
T‘= T/r. Figure 1.
and other driving forces. We also have obtained a solution to eq. 1 by power series expansion and are developing methods for estimating the form of the distortion of a square wave form from the ratio of the high and low frequency values of R’. Acknowledgment.-The support of the Sational Science Foundation under contract G-19114 is gratefully acknowledged, as is the help of Dr. F. Lehman and the staff of the Sewark College of Engineering Computer Center. DEPARTMENT OF CHEMISTRY AND CHEMICAL EKGINEEILING STEVENS INSTITUTE OF TECHKOLOGY HOBOKEN, NEWJERSEY RECEIVED APRIL25, 1963
3.STEINBERG E. J. HENLEY
T H E ACTIVITY OF ASSOCIATION COLLOIDS; ABOVE T H E CRITICAL MICELLE CONCENTRATION Sir: The point of view that micelle formation in an association colloid is akin to a phase separation frequentliy has led to statementss-* that the activity of monomeric particles, as well as that of micelles in equilibrium with them, remains constant irrespective of the total concentration once the critical micelle concentratio:n (c.m.c.) is exceeded. An obvious argument against this assumption is that an activity independent of concentration must result in an enormously increaseld turbidity. This so-called critical opalescence is readily visible in systems in which the condition of constant activity really prevails, whereas solutions of association colloids always give rather low turbidities corresponding to molecular weights of micelles of the order of tens or hundreds of thousands. More direct information about the constancy (or increase) of monomer activity should be given by dialysis through membranes impervious to micelles. Once the constant activity was reached a t the c.m.c.., (1) (a) G. Stainsby and A. E. Alexander, Trans. Faraday Soc., 46, 527 (1950); (b) K. Shinoda, Bull. Chem. SOC.J a p a n , 26. 101 (1953). (2) E. Hutchinson, A. Inaba, and L. G. Bailey, Z . Physik. Chem. (Frankfurt), 6, 344 (1955). (3) E. D. Goddard and C . C. Benson, Can. J . Chem., 36,986 (1957). (4) E. Matijevic and B. A. Pethica, Trans. Faraday Soc., 64, 587 (1958). (5) E. Hutchinson, Z. physik. Chem. (Frankfurt),21, 38 (1959). (6) J. T. Davies and E. K. Rideal, “Interfacial Phenomena,” Academic Press, New York, N. Y . , 1961, p. 201. (7) K. Shinoda and E. Hutchinson, J . Phys. Chem., 66, 577 (1962). (8) F. M. Fowkes, ibid., 66, 1843 (1962).
1944
COMMUNICATIOSS TO THE EDITOR drbubbles
1
KT
Celophano t u b i n q
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