these types of solvent impurities would interfere in syntheses or reaction kinetics in much the same way as water.

(7) A. I. Vogel, “Textbook of Practical Organic Chemistry”, Longmans,

(8) J. P. Kennedy and S. Rengachara, Adv. Polym. Sci., 14, 1 (1974). (9) J. Tranchant, Bull. SOC. Chim. fr., 2216 (1968).

1961.

LITERATURE CITED (10) H. G. Streim, E. A. Boyce, and J. R. Smith, Anal. Chem., 33, 85 (1961). (1 1) H. S. Knight and F. T. Weiss, Anal. Chem., 34, 749 (1962). (12) J. W. Forbes, Anal. Chem., 34, 1125(1962). (13) L. F. Fieser, and M. Fieser, “Reagents for Organic Synthesis”, J. Wiley,

(1) A. S. Brown, J. Chem. Phys., 19, 1226 (1951). (2) J. H. Bower, J. Res. Nat. Bur. Stand., 12, 241 (1934). (3) F. Trusell and H. Diehl, Anal. Chem., 35, 674 (1963). (4) R. L. Meeker, F. Critchfield, and E. T. Bishop, Anal. Chem., 34, 1510

New York, 1967.

(1962). (5) B. D. Pearson and J. E. Ollerenshaw, Chem. lnd. (London), 370 (1966). (6) T. H. Bates, J. V. F. Best, T. F. Williams, Nature (London), 188, 469 for review June 299 lg7& Accepted August 2 3 9

(1960). 1976.

119621 \

(5) B. D. Pearson and J. E. Ollerenshaw, Chem. lnd. (London), 370 (1966). (6) T. H. Bates, J. V. F. Best, T. F. Williams, Nature (London), 188, 469 for review June 299 lg7& Accepted August 2 3 9

(1960). 1976.

Nonlinear Calibration Curves

Lowell M. Schwartz

Department of Chemistry, University of Massachusetts, Boston, Mass. 02 125

As is well known, the procedure by which an instrument or procedure is “calibrated” or “standardized” involves mea- suring the responses yi of the instrument or procedure at a number of known settings or concentrations xi and then plotting the results to form a calibration or standard curve. Subsequently in analysis, measurements of responses Yi lead to determinations of corresponding unknown values X i by reading from the curve. In the special case of a linear cali- bration curve, the unknown values can alternatively be cal- culated from the equation

Yi - y X i = ? + - b

where Z and y are the averages of xi and yi, respectively, and b is the slope. In addition, the analyst wishes to know the level of confidence of an X, determination. In the linear case the random variable Xi is a nonlinear function of the random variables Yi, 7, and b and, consequently, even if these latter three variables are normally distributed, Xi is in general not normally distributed (1). However, if the scatter of the y z about the calibration line is sufficiently small, X i will be ap- proximately normally distributed and under such conditions the variance estimate of X, is calculated as (1)

where u2 is a uniform population variance of both Yi and yi and Yi is the mean of N replicate measurements a t the same unknown X, . Also confidence limits associated with X i can be calculated from an exact formula ( 2 ) which is valid even if the scatter of yi about the calibration line is not small.

When the calibration curve is nonlinear, the estimation of uncertainties may be more difficult. An approximate treat- ment would be valid if the scatter of a given Yi subtends only a small segment of the nonlinear curve. By linearizing the curve locally about the (X i , Yi) point of interest, an approxi- mate variance or standard error in Xi could be calculated from Equation 2 modified to account for the localization. Another approach can be used when the curvature in the subtended segment is sufficiently large that linearization is not appro- priate. This method uses several well-established numerical and statistical techniques: Given a number of response vs. setting observations, yi vs. x i , a polynominal representation can be determined. Assuming that the calibration procedure has accounted properly for any systematic errors, the re- sponses are still subject to random or indeterminate uncer-

tainty. Consequently, it is prudent to make measurements a t a number of distinct x, far in excess of the appropriate number of terms in the polynomial required to follow the intrinsic curvature of the response curve. Hence the polynomial need not pass through all experimental points, subject as they are to spurious random fluctuation. Finding the polynomial is then an overdetermined mathematical problem which is solved by a curvilinear least-squares technique, and the use of orthogonal polynomials (3) in this connection is particularly well-suited as has been noted ( 4 ) .

The selection of the appropriate polynomial degree requires statistical criteria, If each y , random variable is known to scatter normally about its mean with a common population variance u2, the calibrating polynomial should be selected with the minimum degree which reduces the variance of residuals to a value near u2. Clearly a polynomial of lesser degree will not properly follow the response curvature and the variance of residuals will be inflated by the resulting systematic error. A polynomial of greater degree, on the other hand, will follow the spurious random fluctuation of the data. If, as is frequently the case, there is no a priori knowledge of the population variance u2, this value and the appropriate polynomial degree can be estimated by monitoring the effect of successively adding terms using F tests (5). In other cases, the response measurements may be of unequal reliability and each may scatter about its mean with an individual population variance U T . Hence, a weighted least-squares procedure is required using weighting factors w, for each point which are inversely proportional to of. The polynomial degree is selected to reduce the variance of residuals to the average of UT.

Having selected the appropriate polynomial degree m, the calibration curve in terms of orthogonal polynomials is of the form

(3)

where Pj ( x ) are polynomials of degree j , each of which involves parameters dependent on all the x i (and wi # 1 if appropri- ate). The procedure for calculating the coefficients a, also provides estimates of the variances var(aj) (6). If the con- ventional power series representation

of the calibrating polynomial is required, formulas are avail- able (6) for decoding the a, into the cj and the var(u,) into the variances var(cj) and covariances cov(c,, c k ) .

The determinations of the unknowns X i corresponding to the analytical measurements Yi involve solution of either

ANALYTICAL CHEMISTRY, VOL. 48, NO. 14, DECEMBER 1976 2287

Figure 1. Sketch illustrating the effect of calibration line curvature on the skewness of the X, distribution function

Equation 3 or 4 with Y, substituted for y on the left. Although the procedure is simple enough when the curve is plotted graphically, the solution becomes a formidable algebraic problem as m increase and one of m possible roots must be chosen. These solutions are generally obtained numerically using the Newton-Raphson method (6) to find x = X , yielding the appropriate zero value of the function G ( x ) = Y, - y ( x ) . The procedure is iterative requiring an initial good guess at the value X , but this requirement is not a problem in practice as the analyst generally knows fairly well an approximate X,.

The next step is to find an uncertainty in X,. As stated earlier, even in the case of a linear calibration curve, normally distributed y , and Y, do not in general yield normally dis- tributed X,. But if the calibration line has significant curva- ture, the distribution function of a given X , will be expected to show greater deviation from normality. This effect is il- lustrated by the sketch in Figure l. Y, l and Yaz are two mea- surements of the same unknown X, equally spaced about the mean Y, of the Y, distribution function. The corresponding two unknown determinations are X,, and X a z . The X, dis- tribution function is skewed as the mean x, does not coincide with the peak which is a t the X value corresponding to y,, i.e. a t X ( 7 , ) . What is the best value of the unknown X, and its uncertainty is open to question. Only the normal distribution function can be completely characterized by the two statistical parameters “mean” and “standard deviation” and these universally understood to represent “best” and “uncertainty”, respectively. As the distribution function diverges from nor- mality, these particular statistical parameters lose their un- ambiguous meanings and other alternatives may become more appropriate to express the random variability of the deter- mination.

Nevertheless, an approach to unknown uncertainty esti- mates through nonlinear response curves involves random error propagation by Monte Carlo simulation (7). The un- certainty in the determination of X, is due to two factors: (1) to uncertainty in the corresponding Y, as expressed, say, as s y, its estimated standard deviation and (2) to the uncertainty in the calibrating curve itself as reflected in the variances or standard deviations of the coefficients a, or c]. The combined effects of these factors on X , could be found experimentally

by replicate determinations of the calibration curve and for each, replicate Y, measurements. Alternatively, this project can be simulated using the random number generating capabilities of a digital computer. The calculation of X , from Y, via the Newton-Raphson procedure can be repeated any number of times with Y, scattered normally about its mean with standard deviation sy, and with each polynomial coef- ficient scattered about its mean with appropriate variance. The resulting table of X, then contains all the statistical in- formation on the distribution function of that random vari- able, and the analyst can easily calculate whatever statistical parameters he wishes in order to transmit the unknown value and associated uncertainty.

A digital computer program written in FORTRAN IV code which performs the calculation discussed in this paper is of- fered to interested readers and a listing together with details of operation and sample calculations will be sent on request to the author. The program has been submitted to Quantum Chemistry Program Exchange, Chemistry Department, In- diana University, Bloomington, Ind. 47401 from whom a punched card or magnetic tape listing may be purchased. Briefly the program features are as follows: Input required 1. yL vs. x, data 2. Responses Y, Optional input 1. Weighting factors w, corresponding to ( x , , y , ) 2. An a priori estimate of the population variance of y , 3. Approximate X, corresponding to Y, 4. Standard deviations sy , corresponding to Y, Default procedures 1. If w, are not supplied, all w, = 1. 2. If a priori u2 is not supplied, i t is estimated by F testing. 3. If approximate X , are not supplied, the program constructs

a table of y ( x , ) and searches this table for X, (Y , ) by linear interpolation or extrapolation. If y is not a monotonically changing function of x, this procedure may not find the proper root and so approximate X, must be supplied as input.

4. If standard deviations sy , are not supplied, these are as- sumed to be -.

Calculation procedure 1. Orthogonal polynomials are fit to the n y , vs. x, (and w,)

data points up to a maximum degree 15 or (n - 1) which- ever is less.

2. If c2 is prescribed, the polynomial is truncated at the degree m which reduces the variance of residuals of this value. If u2 is not supplied, the polynomial is truncated at degree m, such that higher degree terms fail the F test for significance at the 95% confidence level.

3. Orthogonal and power series coefficients and their vari- ances are recalculated for the appropriate mth degree calibration curve.

4. Values of unknowns X , corresponding to responses Y, are calculated from the calibration curve using the Newton- Raphson procedure.

5 . Procedure 4 is repeated 100 times but with the Y, scattered random normally with standard deviations sy, and with the polynomial coefficients also scattered random normally.

Results printed out 1. Residual variances, F ratios, and tabulated Fo 05 used in

F testing in procedure 2 2. Orthogonal and power series coefficients and variances for

the mth degree calibrating polynomial 3. A table of corresponding Y,, X , ( Y,) , the calculated mean

5?, from the Monte Carlo simulations, and the calculated standard deviation of the Monte Carlo X , about x,.

A comparison of X, (Y,) and x, reflects the curvature of the calibrating curve a t this point as illustrated in Figure 1. The

2288 ANALYTICAL CHEMISTRY, VOL. 48, NO. 14, DECEMBER 1976

standard deviation of the Monte Carlo X, may be a crude measure of the uncertainty associated with X,. It approaches the well-understood meaning as the distribution of X , ap- proaches normality. A reviewer suggests tha t a simple test should be devised to help an analyst decide if the deviation from normality in X , is sufficient to warrant using the more sophisticated Monte Carlo method over the local linearization method. One possibility is to draw the calibration curve and as in Figure 1 to lay off ordinates YL2 and YlZ at, say, 2 s y d n - tervals each from the mean yL. Reading from the curve X( Y,), X ( Y L 1 ) and X( Y , J , the analyst has a measure of skewness of the X distribution as the difference between X(yL) and 112 [ X ( Y , , ) + X(Y , , ) ] as is shown in Figure 1. This simple test ignores the effect of the scatter of y, on the X distribution and hence should be used with caution when this scatter is large.

LITERATURE CITED C. A. Bennett and N. L. Franklin, "Statistical Analysis in Chemistry and the Chemical Industry", John Wiley and Sons, New York, 1954, sec. 6.24. i. M. Koithoff, E. B. Sandeil, E. J. Meehan. and S. Bruckenstein, "Quantitative Chemical Analysis", Fourth ed., Macmillan, New York, 1969, Chapter 16. G. E. Forsythe, J. SOC. lnd. Appl. Math., 5 (2), 74 (1957). G. Henderson and A. Gajjar. J. Chern. Educ., 48,693 (1971). 0. L. Davis and P. L. Goldsmith, "Statistical Methods in Research and Pro- duction", Fourth revised ed., Oliver and Boyd, Edinburgh, 1972, Chapter 0. A. Raiston, "A First Course in Numerical Analysis", McGraw-Hill Book Co., New York, 1965. L. M. Schwartz, Anal. Chern., 47, 963 (1975).

RECEIVED for review May 13,1976. Accepted September 7 , 1976.

Vapor Phase Silylation of Laboratory Glassware

David C. Fenitnore,* Chester M. Davis, James H. Whitford, and Charles A. Harrington

Texas Research lnstitute of Mental Sciences, Texas Medical Center, Houston, Texas 77030

Irreversible adsorption of sample material on the surfaces of laboratory glassware is a problem encountered frequently in determinations of microgram and submicrogram amounts of polar compounds. The severity of the problem can be such as to constitute the major contributing factor limiting the sensitivity of certain assay procedures. For this reason many laboratories, particularly those involved in assays of samples of biological origin, inactivate the surfaces of glassware used in sample preparation by treatment with various silylating reagents. These procedures, which are often used for inacti- vation of gas chromatographic column materials ( I , 2), usually involve treatment of the glass surface with hexamethyldi- silazane, dimethyldichlorosilane, or mixtures of these or similar silylating reagents dissolved in solvents such as toluene or pyridine. The glassware is then rinsed in additional solvent followed by further rinsing with methanol to hydrolyze any unreacted dimethyldichlorosilane if that compound is used in the reagent mixture.

Such treatment produces a very satisfactory nonadsorptive surface, but where large amounts of glassware are processed, the excessive volumes of solvent employed create problems of disposal or reclamation. We therefore developed a glassware silylation procedure similar to that used by some investigators in preparing glass capillary columns for gas chromatography (3 ) . The treatment utilizes small amounts of hexamethyl- disilazane in the vapor phase a t elevated temperature and completely eliminates the need for solvents in the process.

HMDS I Figure 1. Tube with fritted glass element for introduction of hexa- methyldisilazane vapor to vacuum oven

Chloromethylsilanes are not required as catalysts under these conditions which also eliminates problems arising from for- mation of hydrogen chloride. Large amounts of glassware can be prepared with minimal manipulation, low cost, and absence of environmental hazards.

EXPERIMENTAL Apparatus. Vapor phase silylation was performed in a vacuum

oven (Blue M Model POM-16VC-2) evacuated by a mechanical vac- uum pump (Welch Duo-Seal Model 1402) with the exhaust vented to the laboratory fume hood system. The inlet port to the oven was modified with standard pipe and tube fittings to permit the attach- ment of a small glass reservoir with an internal fritted glass gas dis- persion element for vaporization of the silylating reagent (Figure 1). Atmosphere admitted to the inlet of the tube passed through a drying tube containing a desiccant.

Reagent. Hexamethyldisilazane (HMDS) was obtained from Pierce Chemical Company, Rockford, Ill., and was used as re- ceived.

Procedure. Glassware was washed thoroughly using a commercial

Figure 2. Recovery of AQ-tetrahydrocannabinol-3H from silylated (0) and untreated (0) glassware. Vertical lines indicate standard deviation of six determinations measured as disintegrations per minute (DPM)

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