Figure 2. Proposed structure of HFBA-NDMA derivative
C3F7, arises by elimination of CO2 from the molecular ion, a process which is far less likely to occur in the ring isomer of this compound because of the greater separation of the two carbonyl groups. Loss of CO2 through a rearrangement from compounds where carbonyl groups are closely positioned such as imides is established (9). Cleavage of the ion at rnle 224, with a hydrogen rearrangement gives the +NH=C-C3F.i ion a t rnle 196. The fragment a t rnle 221 arises by loss of CO and a fluorine atom from the molecular ion, while the mle 205 fragment arises directly by loss of fluorine from the mle 224 fragment. The ion a t mle 149 occurs by fragmentation of the alkyl fluoro chain, giving loss of C2F5 from the molecular ion and mle 105 occurs by loss of C2F5 from the mle 224 (M-C02)+ fragment, this being a common form of cleavage of the CO-C3F7 moiety. The mass spectra of the azetidines and 2-azetidinones, to which this compound is closely related, have been well documented (9) and the ring splits by 1,2 and 3,4 or 2,3 and 1,4 bond cleavage (IO).In the NDMA derivative, 2-heptafluorobutyryldiazetidin-3-one, there are strong ions a t rnle 43 consisting of CH3CO and CH3N2 fragments. This is consistent with the behavior of azetidinones. Note that the N-N bond is not broken and that the CH3N2 fragment can also arise from the fragment at mle 224. If the structure contained an unsaturated three-membered ring, the fragment
CH&O would be absent and further evidence against this structure is provided by the NMR data described below. A proton Fourier transform magnetic resonance spectrum was obtained of the reaction mixture of HFBANDMA, using pyridine-& and CDC13 in place of the protonated materials for the reaction. Peaks ascribable to the reaction products were observed for chemical shifts on the region of 6 = 3.2 to 3.6.This indicates the absence of unsaturation, and is consistent with the proposed structure. Interference from the other components of the reaction made further interpretation difficult and, in view of the carcinogenic nature of the nitrosamines, a larger scale preparation and purification was not carried out.
ACKNOWLEDGMENT The authors thank C. P. Richards of Laboratory of the Government Chemist for the NMR data and D. E. Games of University College, Cardiff, for the field desorption spectra.
LITERATURE CITED J. B. Brooks, C. C. Alley, and R. Jones, Anal. Chem., 44, 1881 (1972). T. A. Gough, K. Sugden, and K. S. Webb, Anal. Chem., 47, 509 (1975). H. D. Beckey. lnt. J. Mass Spectrom. /on Phys., 2, 500 (1969). F. W. Rollgen and H. D. Beckey, Int. J. Mass Spectrom. /on Phys., 12, 465 (1973) (5) R. M. Silverstein and G. C. Bassier "Spectrometric Identification of Organic Compounds", 2nd ed. John Wiley and Sons, New York, 1972, pp 27-31. (6) R . B. Greenwald and E. C. Taylor, J. Am. Chem. SOC., 90, 5272 (1968). (7) C. A. Bache and D. J. Liske, Anal. Chem., 39, 786 (1967). (8)W. R. McLean. D. L. Stanton. and G. E. Penketh, Analyst (London),98, 432 (1973). (9) H. Btidzikiewicz. C. Djerassi, and D. H. Williams, "Mass Spectrometry of Organic Compounds", Holden-Day, San Francisco, Calif., 1967, p 364. (10) M. B. Jackson, T. M. Spotswood, and J. H. Bowie, Org. Mass Spectrom., 1, 857 (1968). (1) (2) (3) (4)
RECEIVEDfor review September 15, 1975. Accepted November 21, 1975. The Government Chemist is thanked for permission to publish this paper.
Semi-Empirical Analytical Approximation to the Normal Probability Integral T. S. Buys Department of Chemistry, University of South Africa, P.O. Box 392, Pretoria 000 1, South Africa
K. de Clerk" Department of Physical, Theoretical, and Organic Chemistry, University of Pretoria, Hillcrest, Pretoria 0002, South Africa
A function of the form f ( x ) = ( 1 f ( y * ~ / 4 2 ) ) - ' /with ~ y as a fitting function, is shown to be a good analytical approximation to erf x which is simply related to the normal probability integral.
Peak shapes generated by a wide variety of instrumentation can often be satisfactorily approximated by a Gaussian distribution function ( I )
where j is the mean and u2 the variance of the distribution in x . Quantification is usually related to the area under the signal curve so that the normal probability integral
is frequently encountered. The error function, erf(x), is simply related to P ( x ) by P ( x )=
l/2
I + (A ",)I 1
erf
-X-
(3)
Both P ( x ) and erf(x) are normally regarded as numerical functions which can be generated by means of suitable series expansions. In view of its importance in theoretical formulations, it would be of some interest to enquire whether a simple functional approximation could be constructed. Both high and low x approximations are available (e.g. ref. 2), viz., ANALYTICAL CHEMISTRY, VOL. 48, NO. 3, MARCH 1976
585
L
x-
2
3
A
Flgure 2. Dependenceof y on X
0 0 I----
0
_ ~ ~ _ _ _ ~3 ~ ~ _ _ _ _ 2
x
Figure 1. Illustration of some approximations of erf X e-x2
1- -
erf(x)
(large x )
(4)
X V 5
If, instead of the block input, the same mass (area) is introduced initially in the form of a Gaussian distribution function
where
and erf(x)
2x
(small x )
-
v5
(5)
but their applicabilities are limited as can be appreciated from Figure 1. An Analytical Approximation for erf(x).The present approach to an analytical formulation for erf(x) stems from the recognition of the fact that this function can also be regarded as being generated by solving the ordinary diffusion equation
aC _
a2C
-
D
at
an analytical solution for Equation 6 can be obtained. The expression for the corresponding peak maximum a t t is now (11)
Since Equations 8 and 11differ only in initial conditions which are subjected to identical time-dependent processes as described by the differential equation, it is to be expected that limt-,
~
y:)
( +-
C m g = ci, 1
Cmg/Cmp= 1
(12)
If, in addition, it is assumed that Ci, = Ci,, i.e.,
with a block input function (at t = 0)
limt-o C,,/C,,
Ci(0,x) = Ci, = mi/wi, -w;/2 Ix Iw J 2
(7)
mi is the input mass (or area) and wi the total width. The maximum of the peak a t any subsequent time t is then given by
= 1
(13)
one would not expect C,,/C,, to differ much from unity for finite values oft; Le., it is expected that C,,/C,, would be a weak function of t which will be easy to describe empirically in terms of a fitting function. Quantitatively, such
Table I. Comparison of Various Approximations to erf (x)
X
erf(x)
Eq. 1 6 , B = 0.3300977, C = 2.142997
0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3.6 3.8 4.0
0 0.2227026 0.4283924 0.6038561 0.7421010 0.8427008 0.9103140 0.9522851 0.9763484 0.9890905 0.9953223 0.9981372 0.9993115 0.9997640 0.9999250 0.9999779 0.9999940 0.9999985 0.9999996 0,9999999 1.o
0 0.2223467 0.4273998 0.6031104 0.7422639 0.8433926 0.9107621 0.9521316 0.975747 0.9883732 0.9947256 0.9977385 0.9990853 0.9996518 0.9998755 0.9999583 0.9999869 0.9999962 0,9999990 0.9999997 0.9999999
Sum of squares of deviations
Deviation from
0 -0.3559 -0.9926 -0.74 57 0.1629 0.6918 0.4481 -0.1535 -0.6014 -0.7173 -0.5967 -0.3987 -0.2262 -0.11 22 -0.04 9 5 -0.01 9 6 -0.00 7 1 -0.0023 -0.0006 -0.0002 -0.0001
erf(x)
x
10’
3.855325 X
Eq. 1 6 , B = 0.3331385,C= 2
Deviation from erf(x) x 10’
0 0.2229469 0.4298393 0.6067411 0.7451159 0.8441317 0.9094845 0.9497724 0.9732482 0.9862917 0.9932371 0.9967892 0.9985346 0.9993579 0.9997302 0.9998915 0.9999582 0.9999847 0.9999946 0.9999982 0.9999994
0 0.2443 1.4469 2.8850 3.0149 1.4309 -0.8295 -2.5127 -3.1 0 0 2 -2.7988 -2.0852 -1.3480 -0.7769 -0.406 1 -0.1 9 4 8 -0.0864 -0.035 8 -0.0138 -0.0050 -0.0017 -0.0006 5.304034 X
10-5
586
ANALYTICAL CHEMISTRY, VOL. 48,
NO. 3,
MARCH 1976
an empirical fitting parameter, y, can be generated by setting C,, = C,, for all values o f t . From the equality a t t = 0, it follows that (see Equations 7 and 10) ui
= 7Wi/(27r)1'2
(14)
where
form will find application in situations where an analytical expression for erf(x) is required. Some of the approximations are illustrated in Figure 1. The function [l (K/ 4 ~ ~ ) ] mis- a~first / ~ approximation of Equation 16 with y = 1. The more exact approximations cannot be discerned graphically and are compared numerically in Table I.
+
LIST O F SYMBOLS
y = mip/mig
B = fitting parameter in y = exp ( - B X C ) C = concentration a t time t (Equation 6) Ci = value of C at t = 0
while equating Equations 8 and 11yields
where
x = Wi/4(Dt)"2
(17)
As a first approximation, y can be set equal to unity, Le., equal masses (mi, = mi,) are assumed. This corresponds t o a previous application ( 3 ) .The actual dependence of y on X can be numerically determined and is shown in Figure 2. A suitable fitting function for this curve is exp(-EXC) where E and C are constants. A simultaneous nonlinear least squares fit on B and C in Equation 12, using a Marquardt-type algorithm (for a brief outline, see, e.g., ref. ( 4 ) ) ,yielded E = 0.3300977, C = 2.142997. For analytical purposes, C = 2 is a convenient choice for which the corresponding best value for B was found as E = 0.3331385. A comparison of the goodness of the fit is given in Table I and Figure 1. DISCUSSION I t is apparent from these results that increasing accuracy is paid for in terms of increasing functional complexity and a suitable compromise will be dictated by the demands of every specific situation. Generally speaking, the form erf(x) = (1
+ n-e-1/3x2/4x2)-1/2
appears to be surprisingly accurate and it is hoped that this
Ci, = Ci for a Gaussian input Ci, = Ci for a block input C,, = maximum of C for a Gaussian input C,, = maximum of C for a block input D = diffusion coefficient mi = input mass (or area of input distribution) mi, = mi for Gaussian input mip = mi for block input P ( x ) = normal probability integral t = time L U ~= width of block input X = convenient parameter (Equation 17) x = axial coordinate f = mean of distribution Greek
symbols
y = convenient fitting function (rni,/mi,) @(XI = Gaussian distribution function u = standard deviation
LITERATURE C I T E D (1) G. A. Korn and T. M. Korn, "Mathematical Handbook for Scientists and Engineers", McGraw-Hill. New York, 1961, p 564. (2) H. B. Dwight, "Tables of Integrals and Other Mathematical Data", Macmillan, New York, 1966, p 136. (3) K. De Clerk, T. S.Buys, and V. Pretorius. Separ. Sci., 6,759 (1971). (4) N. Draper and H. Smith, "Applied Regression Analysis", Wiley, New York, 1967.
RECEIVEDfor review March 24, 1975. Accepted October 6, 1975.
General Purpose Minicomputer Data Acquisition and Control System Based on CAMAC G. W. Bushnell, T. K. Davies, and A. D. Kirk' Department of Chemistry, University of Victoria, Victoria, B.C., Canada V8W 2 Y 2
S. K. P. Wong Academic Systems Services, University of Victoria, Victoria, B.C., Canada V8 W 2Y2
A data acqulsition and control system of an extremely powerful and flexible nature based on a single CAMAC crate/ controller combined with a PDP-11 computer has been assembled. As the software support the BASIC interpreter has been modified to incorporate a set of CAMAC commands. The result is a total system which can be qulckly configured to a particular experimental situation, and then can be quickly programmed to run the experiment and collect data, using a high level, simple-to-learn, interactive language. Applications to time averaglng, kinetics by NMR, and automation of a four-circle x-ray diffractometer are described.
In the past few years, there have appeared a plethora of papers describing specific applications of computers to analytical and other data acquisition and control situations. Frequently, the systems have a dedicated minicomputer, which may consequently be under-utilized. Usually the interfacing is specific to the particular application and therefore of no general purpose use, and generally considerable time and/or cost is involved in software development. Furthermore, the cost per application of this approach is very high, although time-sharing and the advent of microcomputers provide alternative solutions to this problem, but ANALYTICAL CHEMISTRY, VOL. 48, NO. 3, MARCH 1976
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