Statistical Method in Isotope Analysis Sir: Genty recently published an article on the application of statistical method to isotope analysis ( I ) . It contains an elaborate and valuable discussion on the number of isotopic variants of molecules, the probability of obtaining each variant, and gives expressions for determining isotope ratios. Since reading the article, I have been puzzled by the discrepancy between some of his equations and those used in our laboratory (2, 3 ) . Recently, I have come to the conclusion that several of Genty's equations contain an extra term. The major equations of the article are correct, so Genty's methods of verification did not uncover the extra term. The basic flaw occurs in an unnumbered equation following his equation ( 4 ) : Or,
PI = F , [ A " ' ]
. . .[A"']
a
(Y
'. . .[A'"] '
(1)
His Equation 5 elaborates ( I ): P i -
Or,
. . [A"']Orp
(3) N is the multinomial coefficient. Since most of Genty's equations use ratios of (1) or (21, the extra term is eliminated; the equations concerned with isotope ratios are correct. Fourteen equations, each introduced by Pi, should be modified. The number within the parentheses in the following list refers to the number of the equation in Genty's article: Pi, pi (51, p2(8), P3(14), P4(16), p5(17), p6, p7, p8, pg, Pg, PI 0, and Plo. Except for P1 and the numerical examples, simple deletion of the p n term is sufficient to correct the expression. LITERATURE CITED (1)C.Genty, Anal. Chem., 45, 505 (1973). (2)E. McLaughlin and R. W. Rozett, J. Organomet., 52, 261 (1973). (3)R. W. Rozett, Anal. Chem., 46, 2085 (1974). (4) W. Feller, "Probability Theory and Its Applications," 3rd ed., Wiley and Sons, New York, N.Y., 1970,Vol. 1, p 167.
Richard W. Rozettl
?Z ! l x a , ! . ..ai!. . .CYP! p"
[A"']
.
P , = N,[A'i']Ori. .[A"']='.
a.
Or
. . ,[A"'] '. . .[A'"'] '
(2)
The symbols are Genty's. The term l / p n should not be present. If it is removed, the equation correctly defines the binomial distribution when two isotopes are present, and the multinomial distribution in the general case ( 4 ) . Equation 1 should read:
Chemistry Department Fordham University Bronx, New York 10458
RECEIVEDfor review August 14, 1974. Accepted October 31, 1974.
'
Present temporary address, Research Division, Center for Experimental Design & Data Analysis, Environmental Data Service, 3300 Whitehaven St., N.W., Washington, D.C. 20235.
AIDS FOR ANALYTICAL CHEMISTS Reusable Glass-Bound pH Indicators G. Bruce Harper' Department of Chemistry, Brock University, St. Catharines, Ontario, Can Ontario, Canada
For many years, enzymes have been insolubilized on various carriers. The binding of enzymes to highly porous glass was pioneered by H. Weetall of Corning Glass. The binding modes used are included in a brochure "Corning Biomaterial Supports" available from Pierce Chemical Co., Rockford Ill. 61105. In the process of activating high-surface area glass in the binding of enzymes, this author produced a pale orange sintered glass funnel (Kimax 350-ml-80F). Attempts to wash out the color were unsuccessful. The funnel turned reversibly deep red when washed with acids, a discovery which led to the development of further glass-bound pH indicators. Conventionally, indicators have been used by dissolving the indicator chemical entity in the liquid to be tested, or by coating paper with the chemical entity and then contacting the coated carrier with the liquid to be tested. In Present address, Department of Chemistry, University of Toronto, Toronto, M5S ZAl,Canada. 348
*
nd Department of Chemistry, University of Toronto, Toronto,
whatever form indicators are presently used, a quantity of the indicator is required for each test. The amount of indicators consumed in an industry or in an active laboratory can be both substantial and costly. Furthermore, bound indicators are superior in the determination of pH's of weakly or nonbuffered solutions. Typical pH papers in nonbuffered solutions need long immersions for color change and dyes bleed into solutions. Also, a chromatographic effect causes free dyes to creep on paper, forming zones of uneven color. The use of bound indicators removes such difficulties. Glass-bound pH indicators have several advantages over soluble indicators: 1) They can be used indefinitely with quantitative recovery of indicator. They exhibit continued and apparently constant activity, as indicated by the intensity of color changes, over a period of at least one year a t room temperature, upon exposure to most organic and aqueous assay conditions. 2) They are unsusceptible to mi-
ANALYTICAL CHEMISTRY, VOL. 47, NO. 2, FEBRUARY 1975
