of sensitivity in either the MS or GProC. The specific activity of the radioactivity will influence the optimal split ratio to be employed. However, the sensitivity of the GProC is dependent on residence time in the detector. Therefore, the slower flow rates (30-50 cm3/min)are preferred. If flow rates slower than 30 cm3/min are desired or are necessary for separation, the flow through the GProC must be increased to 30 cm3/minby adding make-up carrier to maintain optimum counting efficiency. In addition, sensitivity is also very dependent on the carrier gas-quench gas ratio. The quench gas should be approximately 10%of the carrier gas flow. The addition of quench gas also reduces the lag time between the appearance of the ion current in the MS and the signal at the GProC. The lag time can be accurately
established using radioactive standards with similar retention times to the unknown compounds. The transfer line to the GProC is heated to at least 20 “C greater than the GC oven temperature to prevent condensation losses.
DISCUSSION The techniques just described offer several advantages, the greatest of which is the ability t o determine rapidly the component associated with the radioactivity in the total ion current of a complex mixture without laborious fraction trapping and liquid scintillation counting. A second advantage results from the use of the effluent splitter: it is the ability to run much higher flow rates and larger diameter columns in the GC than are usually possible with GC-MS. The third advantage is that when 1-or 2-pl solvent injections are made with a 1 : l O
(1part to the MS) effluent splitter, the solvent can be allowed t o enter the MS without dramatic increases in pressure. Therefore, if the radioactivity coelutes with the solvent, a mass spectrum can still be obtained. The detection limit of the GProC is dependent upon the flux of radioactivity in the detector. Therefore, the sharper the eluting GC peak, the greater the sensitivity. The detection limit of the GProC (2.5 signal-to-noise ratio) is 660 dpm/min. For convenience, the GProC with its own recorder can be mounted on a cart and wheeled up directly adjacent to the GC-MS unit. This allows rapid conversion from one instrument to another, and limits the length of the transfer lines. This technique has been successfully applied to the identification of metabolites of alkyl halides, halogenated aromatic hydrocarbons, pesticides, and experimental drugs. An illustrative example is the identification of phenylglyoxylic acid, mandelic acid, hippuric acid, and benzoic acid as the major urinary metabolites of styrene in the rat. As seen in Figure 2 the total ion chromatogram from the GC-MS contained several components; however, the peaks containing the radioactive metabolites were readily identified from the GProC scan.
RECEIVEDfor review June 17,1976. Accepted September 16, 1976.
Radiotracer Method for the Determination of Solvent Water Content David Roger Burfield Department of Chemistry, University of Malaya, Kuala Lumpur 22-1 7, Malaysia
The practice of modern chemistry increasingly demands the use of rigorously dried solvents. This requirement is commonly exemplified in such diverse fields as kinetic studies involving moisture sensitive components, and organic syntheses. Although quantitative information concerning solvent drying is sparse, qualitative comparisons abound, e.g., commenting on the relative drying efficiencies of sodium and calcium hydride for benzene, one author observes ( I ) “One day drying over calcium hydride was found to be as effective as 6 months refluxing over sodium”. Methods reported in the literature for determining the relative or absolute efficiencies of drying agents include gravimetric (2, 3), infrared (4, 5 ) , and functional test (1, 6) methods. None of these methods is completely satisfactory, particularly for the determination of the absolute efficiencies of drying agents a t very low moisture contents. The elegant gravimetric method employed by Trussell and Diehl(3) and earlier workers ( 2 )for gas drying is limited in that the results cannot be directly extrapolated to drying in the liquid phase, since drying efficiencies and the relative order of desiccants are markedly dependent on solvent type. The infrared method ( 4 , 5 ) ,which appears to be an extremely useful rapid technique for determining the moisture content of solvents, has been employed ( 5 )to examine the rapidity and efficiency of a few drying agents for benzene, diethyl ether, and ethyl acetate. Since the detection limit of this method appears to be about 10 pglg of water, the method is not applicable to “super-dry” solvents. The term “super-dry” has been used earlier in a qualitative sense, e.g., by Vogel (7) but in this context it is taken to mean solvents containing less than 1 pg/g of water. The functional test method is widely used in qualitative comparison of solvent drying for use in rate or other physical
measurements. The method is dependent on the assumption that removal of water leads to an optimization of the physical quantity. Examples of this method are the purification of benzene for dielectric constant measurements ( I ) , and drying of reaction components for the study of the cationic polymerization of @-pinene(6).I t should be emphasized that the results of this method are not only qualitative but frequently also ambiguous. Thus the drying agent may serve to remove or introduce impurities other than water, and more importantly the physical quantity may not be optimized by complete exclusion of water, viz., the rate of cationic polymerization which with certain catalysts goes through a maximum as the water content is reduced (8). Besides the above methods used for comparative studies, numerous procedures for the determination of the water content of solvents have been reported and reviewed in the literature (9).The most important of these methods is without doubt the Karl Fischer titration which may be used down to a few pg/g of water, but it is susceptible to contamination errors a t low water concentrations. Other notable methods include gasometric techniques, involving the liberation of acetylene from calcium carbide and its subsequent determination by GLC ( I 0 , l I )or IR (12).These procedures appear to be highly sensitive, but tedious to carry out. Although invaluable for the determination of solvent water content, none of the above methods are suited for a survey of drying efficiency for super-dry solvents. This paper describes a new approach to the problem. Drying efficiency is determined by addition of a specified amount of tritium-labeled water to a rigorously dried solvent and subsequent determination of the decrease in activity of the solvent after treatment with various drying agents. The method
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Table I. Determination of Water Content in Dioxane by Near-Infrared and Tritium Labeling Methods
Drying time, ha
Specific activity of dioxane, counts per min/g x
Water content, mg/g Tritium Nearlabeling IR
48.6 2.00b 2.00b 36.6 1.51 1.43 27.2 1.12 1.03 2 16.3 0.67 0.63 4 3.81 0.16 0.40 6 0.972 0.04 0.04 24 Static drying at room temperature using 5% w/w 4-A molecular sieves. Represents amount of water initially added to dry solvent. 0
1
Table 11. Comparison of the Efficiencies of Various Drying Agents for Dioxanea-
Drying agent
Amount used, % w/w
Residual water content after 24 h (pg/g) (tritium labeling)
4-.A molecular sieve 4.9 40 Calcium hydride 5.1 30 Sodium 6.2 20 400 Phosphorus 4.6 pentoxide O1 These figures apply to static drying at room temperature. Initial water content = 2.0 mdp. is rapid and is applicable to the accurate determination of extremely low water concentrations. EXPERIMENTAL Reagents. Reagent grade dioxane was purified and dried by a method described in detail elsewhere (13).4-Methyl-1-pentenewas: purified and dried by fractional distillation from calcium hydride. Molecular sieves (4A) were activated by heating overnight at 250 O C . Sodium pellets were washed with petroleum ether and cut under dry nitrogen to give small 2-mm cubes. Calcium hydride was finely powdered under dry nitrogen before use. Reagent grade phosphorous pentoxide was used without treatment. Tritium labeled water was purchased from the Radiochemical Centre, England, at an initial activity of 5 Ci/ml and was diluted with appropriate quantities of inactive water. Techniques. The water content was determined by the near-IR method described earlier ( 4 ) with a Unicam SP700 Spectrophotometer. Radioactive samples were assayed in a scintillation solution containing 0.4 g POPOP, and 4.0 g PPO per liter of toluene with a Beckman Model LS-100 liquid scintillation spectrometer. Dilution analysis was employed with high activity samples. Solvents were initially rigorously dried and a known amount of labeled water was introduced. In the case of 4-methyl-l-pentene,prolonged stirring was necessary to dissolve even 200 pg/g of water; initially water droplets were observable.
RESULTS AND DISCUSSION Initial experiments were conducted to test the validity of the method. Thus the drying of dioxane by 4-A molecular sieves was followed by both the near-IR and tritium labeling methods. (The near-IR method was chosen because i t is simple to operate and is reliable ( 4 ) ) .The two methods gave closely comparable results (Table I). The near-IR method was performed in triplicate and found to give readings reproducible to about f 2 0 pglg. Subject to sampling procedures and sufficiently high activities, the radioactive method is capable of a precision of f l % over the whole range of water content and thus is significantly superior to the near-IR method at low concentrations. For tritiated water of very high specific ac2286
e
Table 111. Preparation of “Super-Dry” 4-Methyl-lpentenen Specific activity of Water hydrocarbon content, (counts per min/g) wdg 9.35 x 106 232 Initial sample After stirring 6.08 x 103 0.150 over CaH2 After distillation 1.66 x 103 0.041 from CaH2 0 Dried with 3.5% w/w calcium hydride.
tivity ( 5 Ci/g), assuming a counting efficiency of bo%, the detection limit of the tracer method is about 10 pg of water, whereas, for the near-IR the detection limit in this study was found to be about 10 pg. Preliminary results for a series of different drying agents are summarized in Table 11. Dioxane is a notably difficult solvent to rigorously dry; and sodium, calcium hydride, and molecular sieves at moderate loadings and under static drying conditions do not reduce the water content t o “super-dry” levels. Phosphorous pentoxide, long considered the ultimate drying standard ( 3 ) ,is seen to be even less effective. Drying of hydrocarbons poses less of a problem and the efficiency of calcium hydride for 4-methyl-1-pentene is shown in Table 111. Merely stirring the hydrocarbon over finely powdered calcium hydride overnight was found to reduce the moisture content to 0.15 lg/g. Further drying of the sample by refluxing for 4 h and subsequent fractional distillation further decreased the water content to 0.041 pg/g. This example illustrates the strength of the tritium tracer method in its ability to accuately detect very low water concentrations. Besides the rapidity of sampling and assay, and the ability to detect very low water contents, the method has the additional advantage that it is not susceptible to contamination after sampling. Samples may thus be accumulated and activities determined batchwise. Although the method has significant advantages over other techniques, there are a number of factors which limit its general applicability. First, the method cannot be used with solvents containing readily exchangeable hydrogens since it is based on the total activity of the sample which is assumed to be present as labeled water. Tritium exchange with solvents, such as alcohols (Equation 11, HOT
+ ROH + H20 + ROT
would clearly invalidate the basis of the method, since the labeled alcohol would be indistinguishable from the presence of labeled water. These exchange reactions will also interfere in the case of solvents containing small amounts of hydroxylic or similar impurities. Second, for desiccants such as sodium metal or calcium hydride which undergo chemical reactions with water, a kinetic isotope effect should be observable, because of the difference in reactivity between the oxygenprotium and oxygen-tritium bonds. Tritium, the heavier isotope, has a smaller rate constant for bond rupture and consequently the actual water content of the sample will be smaller by a factor of kH/kT than the experimental value. I t should be possible to determine this factor by comparison of the results from labeling experiments with other direct water determination methods. Both of these factors give rise to an inflated value for the solvent water content, so the method can safely give the maximum water content of the solvent under test. Furthermore, the exchange with hydroxylic or similar impurities is not completely disadvantageous since frequently
ANALYTICAL CHEMISTRY, VOL. 48, NO. 14, DECEMBER 1976
these types of solvent impurities would interfere in syntheses or reaction kinetics in much the same way as water.
LITERATURE CITED (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 119621 (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 (1960).
(7)A. I. Vogel, “Textbook of Practical Organic Chemistry”, Longmans, 1961. (8)J. P. Kennedy and S. Rengachara, Adv. Polym. Sci., 14, 1 (1974). (9)J. Tranchant, Bull. SOC. Chim. fr., 2216 (1968). (10) H. G. Streim, E. A. Boyce, and J. R. Smith, Anal. Chem., 33, 85 (1961). (11) 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, New York, 1967.
\
for review June
299
lg7& Accepted August
239
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 measuring 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 calibration curve, the unknown values can alternatively be calculated from the equation
Xi=?+-
Yi
-y
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 approximately 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 at 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 treatment 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 approximate 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 appropriate. 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 responses are still subject to random or indeterminate uncer-
tainty. Consequently, it is prudent to make measurements at 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 UT. 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 # 1if appropriate). The procedure for calculating the coefficients a, also provides estimates of the variances var(aj) (6). If the conventional power series representation
of the calibrating polynomial is required, formulas are available (6) for decoding the a, into the c j 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
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