Determining Extremums
letters Safer Dye for Dry-Column Chromatography
To the Editor: I should like to make a comment on the article on "dry column" chromatoeranhv. .. . .. J. CHEM. EDUC.. 50. 367 (19731.One ot the recommended dyes to use in this experiment is I,-dimethslaminoazohi~nzene.I should like to uoint out that' this is one of the compounds listed in the 0ccupational Safety and Health Act (OSHA) 1910.93C as a known Dotent carcinoeen and as such should not he used. I think this is particularly true in an undergraduate laboratow where the students think that anv chemicals thev use arirelatively safe. For those who may not have seen the carcinogen listing from OSHA 1910.93C Compound
C. A. Registry Number
2-Aeetylaminofluorene 4-Aminodiphenyl Benzidine (and its salts) 3,3'-Dichlarohenzidine (and its salts) 4-Dimethylsminoazohenzene
a-Naphthylamine 8-Naphthylamine 4-Nitrobiphenyl N-Nitrosodimethylamine 8-Propiolaetone Bis-Chloromethylether Methyl ehloromethyl ether 4,4'-Methylene (his)-2-chloroaniline Ethyleneimine Thomas Maier 76 Mt. Morency Drive Rochester, New York 14612
To the Editor: T. Maiers has called our attention to the fact that one of the dves used in the illustrative exneriment described in our article on "dry-column" chrdmatography, p-dimethylaminoazohenzene, is listed in the Occupational Safety and Health Act (although from the information that we have been able to gather, i t is not a "known notent carcinogen"). ~ l t h o u ~all h compounds used in the laboratories must be considered hazardous and handled as such, we do agree that this particular dye should probably not he used in routine experiments in an undergraduate laboratory and should be replaced by another dye of similar rr value. We have found a useful and inexpensive replacement to be 1,4-di-p-toluidinoanthraquinone,available from Allied Chemical Corporation, Special Chemicals Division, 40 Rector Street, New York, N.Y. 10006, as "D&C Green 6" (at $23.50 a pound). Bernard Loev Smith Kline & French Laboratories Philadelphia, Pennsylvania 19101 496 I
Journal ot Chemical Education
To the Editor: In a recent note: "Iterative Method for Solving Equilibrium Problems by Free-Energy Minimization," [J. CHEM. ED., 50, 299 (1973)l Scott Davidson presented an elegant scheme to numerically determine an extremum in a function of one variable and further indicated the scheme could be used to find the solution of some equations of the form ffx) = 0. Some extensions of this line of thought are worth noting. Actually solutions to most wellbehaved functions of the form ffx) = b can be solved by this method. It is accomplished by finding an extremum in g2W, where gfx) = ffx) - b gfx) will cross zero at x such that ffx) = b and thus dg(x)/dx = 2g(x)g'(x) = 0 if g'fx) is finite a t this value of x. As a further consideration it can he noted that a set of experimentally determined numbers a t hand is frequently desired to he fit to some model, which fit in general involves obtaining estimates for parameters associated with the model. If the prohlem is linear, the least squares technique is appropriate. Most often the problem is nonlinear and an iterative scheme is employed. In essence the least squares formalism seeks an extremum in the man of the wkighted sum of squared deviations of calculated values from experimental values (WSSD) as a function of the model parameters. If there are n independent parameters, the n-tuple a t which the extremum occurs in n + 1 dimensional space is the set of best estimators for the parameters. In any problem it is expected that the map of the WSSD should minimize a t the best estimates of the model parameters. For the nonlinear case in general neither uniqueness of an extremum, nor convergence to an extremum is guaranteed. A generalization of the method suggested by Davidson can be used in solving this problem as well. Near the extremum a general quadratic function can he used to approximate the surface of the WSSD. The function suggested is XtAX + BX + C = S where t denotes the transpose, C is a number, B is a 1 by M array, A is an M by M symmetric square array and M is the number of parameters in the model. N = ( ( M + 1 ) ( M 2))/2 numbers in A, B and C characterize the quadratic function of the model parameters of the M-tuple X. N determinations of the WSSD near the initial guesses of the model parameters will give the numbers for the initial arrays Ao, Bo and Co. The first estimate of the best fit parameters can then he evaluated as XI' = -'h Bdo-'. The process can he continued until no further significant changes in the estimator for the M-tunle of the extremum are calculated. This numerical approach involves more computation than the usual technique in which the problem is linearized hv using a truncated Taylor expansion of the model funitions. For a discussion of this technique see "Statistics in Physical Sciences," by W. C. Hamilton, Ronald Press, New York, 1964. The larger the number of model parameters the more sienificant is the difference in the number of computations required for this technique over the Taylor expansion technique. The technique nroposed here is simpler to program, however, since
